1. Introduction and statement of main results

In 2015, M. Calanchi and B. Ruf have established a weighted Trudinger–Moser inequality in the unit ball $\mathcal {B}$ of $\mathbb {R}^N,\ N \geq 2.$ Such type of inequality is not new and many inequalities of Trudinger–Moser type defined in weighted Sobolev spaces have been proved; we can for example cite [Reference Abreu and Fernandez1–Reference Albuquerque3, Reference Aouaoui and Albuquerque5, Reference Calanchi13–Reference Calanchi and Ruf16, Reference de Oliveira and do Ò21, Reference do Ò and de Souza22, Reference Furtado, Medeiros and Severo26, Reference Lam29, Reference Nguyen and Takahashi33]. The majority of those works considered the restriction to radial functions, and in [Reference Lam29] although the weight is not necessarily radial but its growth permits to pass to the radial case through some radial rearrangement. This interest to reduce the inequality to the radial functions is mainly motivated by their ability to increase the maximal growth of the integrability. The weight that M. Calanchi and B. Ruf considered is of logarithmic type and turned out to be of great interest. More precisely, they introduced the subspace $W_{0,rad}^{1,N}(\mathcal {B}, \sigma _{\beta })$ defined as the radial functions of the completion of $C_0^{\infty }(\mathcal {B})$ with respect to the norm

\[ \left\| u\right\|_{\sigma_{\beta}}^N=\int_{\mathcal{B}}\sigma_{\beta}(x)\left|\nabla u\right|^N\,{\rm d}x, \]

where $\sigma _{\beta }(x)=(\log \frac {1}{\left | x\right |})^{\beta (N-1)}$ or $\sigma _{\beta }(x)=(\log \frac {e}{\left | x\right |})^{\beta (N-1)}$, $0 < \beta \leq 1,\ x \in \mathcal {B}.$ In [Reference Calanchi and Ruf16, theorem 1], M. Calanchi and B. Ruf proved the following result: for $0 < \beta < 1,$ we have

• •

  (1.1)\begin{equation} \int_{\mathcal{B}}\,{\rm e}^{\left|u\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x <{+} \infty,\ \forall\ u \in W_{0,rad}^{1,N}(\mathcal{B}, \sigma_{\beta}). \end{equation}

• •

  (1.2)\begin{equation} \sup\left\{\int_{\mathcal{B}}\,{\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x,\ u \in W^{1,N}_{0,rad}(\mathcal{B}, \sigma_{\beta}),\ \left\|u\right\|_{ \sigma_{\beta}} \leq 1\right\} <{+} \infty\Leftrightarrow\alpha \leq \alpha_{N,\beta}, \end{equation}
  where $\alpha _{N, \beta } =N[\omega _{N-1}^{\frac {1}{N-1}}(1-\beta )]^{\frac {1}{1-\beta }}$ and $\omega _{N-1}$ is the area of the unit sphere in $\mathbb {R}^N.$

The case $N = 2$ has been considered in a previous work (see [Reference Calanchi and Ruf15]). Note that when $\beta = 0,$ (1.2) recovers the classical Trudinger–Moser inequality (see [Reference Moser31, Reference Trudinger38]). Next, M. Calanchi and B. Ruf considered the case when $\beta =1.$ In this case, the specific behaviour of the weight function has an impact on the corresponding embeddings. In fact, the maximal growth ${\rm e}^{\left |s\right |^{N'}}$ proved in the classical Trudinger–Moser inequality significantly increased such that a doubly exponential growth is now permitted. More precisely, they proved the following result given in [Reference Calanchi and Ruf16, theorem 4]:

• •

  (1.3)\begin{equation} \int_{\mathcal{B}}\,{\rm e}^{{\rm e}^{\left|u\right|^{N'}}}\,{\rm d}x<{+}\infty,\ \forall u\in W_{0,rad}^{1,N}(\mathcal{B}, \sigma_1),\ \text{where}\ \sigma_1(x) = \left(\log \frac{e}{\left|x\right|}\right)^{N-1}. \end{equation}

• •

  (1.4)\begin{equation} \sup\left\{\int_{\mathcal{B}}\,{\rm e}^{a\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} | u|^{N'}}}\,{\rm d}x,\ u \in W_{0,rad}^{1,N}(\mathcal{B}, \sigma_1),\ \left\|u\right\|_{\sigma_1} \leq 1 \right\}<{+}\infty \Longleftrightarrow a\leq N. \end{equation}

The proof of (1.2) in the critical case $\alpha = \alpha _{N,\beta }$ is mainly based on a suitable change of variable combined to some integral inequality due to M.A. Leckband. In [Reference Nguyen34], V.H. Nguyen provided a simpler proof of (1.2) in which he proved that the function

\[ \beta \longmapsto MT(N,\alpha,\beta) = \sup\left\{\int_{\mathcal{B}}\,{\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x,\ u \in W^{1,N}_{0,rad}(\mathcal{B}, \sigma_{\beta}),\ \left\|u\right\|_{ \sigma_{\beta}} \leq 1\right\} \]

is decreasing on $[0, 1).$ Moreover, V.H. Nguyen proved the existence of maximizer for this inequality when $\beta$ is sufficiently small. The question of the attainability of the inequality (1.2) has been also considered by P. Roy in [Reference Roy35] for the case $N = 2,$ and in [Reference Roy36] for higher dimensions. Taking advantage of these new Trudinger–Moser inequalities defined on the unit ball $\mathcal {B}$ in $\mathbb {R}^N,$ some authors studied an elliptic problem having a doubly exponential growth at infinity. It mainly consists in the following equation

\[ \left\{\begin{array}{@{}ll} -\text{div}(\sigma_1(x)|\nabla u|^{N-2}\nabla u)=f(x,u), & \text{in}\ \mathcal{B},\\ u>0, & \text{in}\ \mathcal{B},\\ u=0, & \text{on}\ \partial\mathcal{B}, \end{array}\right. \]

where the nonlinear term $f(x, u)$ is a continuous function, radial in $x\in \mathcal {B}$ and has a critical doubly exponential growth at infinity, which means that there exists a positive constant $\alpha _0$ such that

\[ \displaystyle\lim_{s \to +\infty}\frac{f(x,s)}{{\rm e}^{N\,{\rm e}^{\alpha |s|^{N^{\prime}}}}}=\left\{\begin{array}{@{}ll} 0, & \text{for all}\ \alpha>\alpha_0,\\ + \infty, & \text{for all}\ \alpha\leq\alpha_0. \end{array}\right. \]

M. Calanchi, B. Ruf and F. Sani proved in [Reference Calanchi, Ruf and Sani17] the existence of a nontrivial radial solution for the case $N = 2.$ This result has been recently generalized by C. Zhang in [Reference Zhang40] for higher dimensions. When we try to extend (1.1)–(1.4) to the whole Euclidean space $\mathbb {R}^N,\ N \geq 2,$ we face many obstacles which mainly consist of the embedding and denseness properties of the functional space that we construct by extending the weight outside of the unit ball $\mathcal {B}.$ For the first attempts, we worked with the weight $\sigma _{\beta }(x) = (\log \frac {e}{\left |x\right |})^{\beta (N-1)},\ \left |x\right | < 1.$ In [Reference Aouaoui and Jlel6], we considered a radial weight $w_{\beta }$ defined by

(1.5)\begin{equation} w_{\beta}(x)=\left\{\begin{array}{@{}ll}\displaystyle \left(\log\left(\dfrac{e}{|x|}\right)\right)^{\beta(N-1)}\ & \text{if}\ |x|<1,\\ \displaystyle \chi(\left|x\right|) & \text{if}\ |x|\geq1, \end{array} \right.\end{equation}

where, $0 <\beta \leq 1$ and $\chi : [1, + \infty [ \rightarrow ]0, + \infty [$ is a continuous function such that $\chi (1) = 1,\ \displaystyle {\inf _{t \geq 1} \chi (t) > 0.}$ Denoted by $Y_{\beta }$ the weighted Sobolev space

\[ Y_{\beta}=\left\{ u\in W_{rad}^{1,N}(\mathbb{R}^N)\ ;\ \displaystyle\int_{\mathbb{R}^N}w_{\beta}(x)|\nabla u|^N\,{\rm d}x<{+}\infty\right\} \]

and we equip it with the standard Sobolev norm

\[ \left\|u\right\|_{Y_{\beta}}^N=\int_{\mathbb{R}^N}|\nabla u|^Nw_{\beta}(x)\,{\rm d}x+\int_{\mathbb{R}^N}\left| u\right|^N\,{\rm d}x. \]

We obtained the following extensions of (1.1) and (1.2) to the whole space $\mathbb {R}^N:$ Let $N\geq 2$ and $w_{\beta }$ be defined by (1.5). Then, for all $\alpha >0$ and $u \in Y_{\beta },$ we have

(1.6)\begin{equation} \int_{\mathbb{R}^N} \left({\rm e}^{\alpha\left| u\right|^{\frac{N^{\prime}}{1-\beta}}}-S_{N-2}\left(\alpha \left| u\right|^{\frac{N'}{1-\beta}}\right)\right)\,{\rm d}x<{+}\infty, \end{equation}

where

\[ S_{N-2}(t)=\sum_{k=0}^{N-2}\frac{t^k}{k!},\ t \geq 0. \]

Moreover, if $\alpha < \alpha _{N,\beta }$, then

(1.7)\begin{equation} \displaystyle\sup_{u\in Y_{\beta},\ \left\| u\right\|_{Y_{\beta}}\leq1}\int_{\mathbb{R}^N} \left({\rm e}^{\alpha\left| u\right|^{\frac{N'}{1-\beta}}}-S_{N-2}\left(\alpha, \left| u\right|^{\frac{N'}{1-\beta}}\right)\right)\,{\rm d}x<{+}\infty, \end{equation}

and if $\alpha > \alpha _{N,\beta }$, then the supremum in (1.7) becomes infinite. For the value $\alpha = \alpha _{N,\beta },$ the supremum in (1.7) is not necessarily finite. However, the sharpness of the Trudinger–Moser inequality could be recovered by considering a different functional space. More precisely, for $0<\beta <1,$ we define $Y_{\beta }'$ as the space of all the radial functions of the completion of $C_0^{\infty }(\mathbb {R}^N)$ with respect to the norm

\begin{align*} \left\| u\right\|_{Y_{\beta}'}& =\left|\nabla u\right|_{L^N(\mathbb{R}^N,w_{\beta})}+\left| u\right|_{L^{d'_{\beta}}(\mathbb{R}^N)}=\left(\int_{\mathbb{R}^N}w_{\beta}(x)\left| \nabla u\right|^N\,{\rm d}x\right)^{\frac{1}{N}}\nonumber\\& \quad +\left(\int_{\mathbb{R}^N} \left| u\right|^{d'_{\beta}}\,{\rm d}x\right)^{\frac{1}{d'_{\beta}}}, \end{align*}

where $d'_{\beta }=\frac {N'(1-\beta )}{N'-1+\beta }.$ For that space, we obtained the following sharp Trudinger–Moser inequality which can be considered as another extension of (1.2): Let $0<\beta \leq \frac {1}{N'+1}$ and $w_{\beta }$ be defined by (1.5). Then,

\[ \sup_{u\in Y_{\beta}^{\prime},\ \left\| u\right\|_{Y_{\beta}^{\prime}}\leq1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left| u\right|^{\frac{N'}{1-\beta}}}-S_{N-2}\left(\alpha \left| u\right|^{\frac{N'}{1-\beta}}\right)\right){\rm d}x<{+}\infty\Leftrightarrow \alpha \leq \alpha_{N,\beta}. \]

The value $\beta =1$ is a kind of second order limiting case. In [Reference Aouaoui and Jlel6], we established the following extension of (1.3) and (1.4):

• • For all $\alpha >0$ and $u\in Y_1,$ there holds

  (1.8)\begin{equation} \int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\left| u\right|^{N'}}-1\right)}-S_{N-2}\left( \alpha\left({\rm e}^{\left| u\right|^{N'}}-1\right)\right)\right){\rm d}x<{+}\infty. \end{equation}

• • If $a\leq N\,{\rm e}^{-(\inf \limits _{s\geq 1}\chi (s))^{-\frac {1}{N(N-1)}}}$, then

  (1.9)\begin{equation} \sup_{u\in Y_1,\ \left\| u\right\|_{Y_1}\leq1}\int_{\mathbb{R}^n}\left({\rm e}^{a\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}-S_{N-2}\left(a\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)\right)\right){\rm d}x<{+}\infty. \end{equation}

• • If $\displaystyle a>N\exp \left (\frac {1}{N-1}\int _0^{+\infty }\log ^N(1+t)\,{\rm e}^{-Nt}\,{\rm d}t\right )$, then

  (1.10)\begin{equation} \sup_{u\in Y_1,\ \left\| u\right\|_{Y_1}\leq 1}\int_{\mathbb{R}^N}\left({\rm e}^{a\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}-S_{N-2}\left(a\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)\right)\right){\rm d}x={+}\infty. \end{equation}

Note that the previous results come as generalizations of earlier works dealing with the case $N = 2.$ See [Reference Aouaoui4, Reference Aouaoui and Jlel7]. A further interesting extensions of (1.1)–(1.4) to the whole Euclidean space $\mathbb {R}^N$ has been provided in [Reference Aouaoui and Jlel8, Reference Aouaoui and Jlel10].

In this paper, we consider two types of weights. First, for the case when $0 < \beta < 1,$ we consider the weight defined by

(1.11)\begin{equation} w_{\beta}(x)=\left\{\begin{array}{@{}ll}\displaystyle (- \log (\left|x\right|))^{\beta(N-1)} & \text{if}\ |x|<1,\\ \displaystyle \chi(\left|x\right|) & \text{if}\ |x|\geq1, \end{array} \right. \end{equation}

where $\chi : [1, + \infty [ \rightarrow ]0, + \infty [$ is a continuous function such that $\chi (1) = 0.$ Moreover, the function $\chi$ is chosen such that $w_{\beta }$ satisfies (1.12), that is, $w_{\beta }$ belongs to the Muckenhoupt's class $A_N$ (we also say that $w_{\beta }$ is an $A_N-$weight), that is

(1.12)\begin{equation} \sup \left(\frac{1}{\left|B\right|} \int_{B} w_{\beta}(x)\,{\rm d}x\right)\left( \frac{1}{\left|B\right|} \int_{B} (w_{\beta}(x))^{\frac{1}{1-N}}\,{\rm d}x\right)^{N-1} <{+} \infty, \end{equation}

where the supremum is taken over all balls $B$ in $\mathbb {R}^N.$ The importance of this property of the weight $w_{\beta }$ lies in the fact that it implies that $C_{0}^{\infty }( \mathbb {R}^N)$ is dense in the space $E_{\beta }$ (see, for instance, [Reference Cavalheiro18, Reference Kilpeläinen28, Reference Nakai, Tomita and Yabuta32] and references therein). An interesting example of such a function $\chi$ is given by: $\chi (t) = \log ^{\gamma } (t),\ \gamma > 0$ (see [Reference Haroske27]). In particular, one can consider the weight

\[ w_{\beta}(x) = \left|\log \left|x\right|\right|^{\beta(N-1)},\ x \in \mathbb{R}^N\backslash{\left\{0\right\}}. \]

That last weight can be seen as a natural extension of $(- \log \left |x\right |)^{\beta (N-1)}$ defined on $\mathcal {B} = \left \{x \in \mathbb {R}^N,\ \left |x\right | < 1\right \}$ and considered in [Reference Calanchi and Ruf16]. Second, for the case $\beta = 1,$ we consider the weight

(1.13)\begin{equation} w_{1}(x)=\left\{\begin{array}{@{}ll}\displaystyle (1- \log (\left|x\right|))^{N-1} & \text{if}\ |x|<1,\\ \displaystyle \chi(\left|x\right|) & \text{if}\ |x|\geq1, \end{array} \right. \end{equation}

where $\chi : [1, + \infty [ \rightarrow ]0, + \infty [$ is a continuous function such that $\chi (1) = 1$ and $w_1$ belongs to the Muckenhoupt's class $A_N.$ Here, are some examples of such a function $\chi.$

• • $\chi$ can be any continuous and positive function such that $\chi (1) = 1$ and

  \[ 0 < \inf_{t \geq 1} \chi(t) \leq \sup_{t \geq 1} \chi(t) <{+}\infty. \]

• • $\chi (t) = t^{\alpha },\ 0 < \alpha < N(N-1).$

• • $\chi (t) = 1 + \log ^{\gamma } t,\ \gamma > 0.$

For details about these examples, we refer to [Reference Aouaoui and Jlel8]. Let $1 < q < N.$ For $0 < \beta \leq 1 ,$ denote by $E_{q,\beta }$ the weighted Sobolev space

\[ E_{q,\beta} = \left\{u \in D^{1,q}_r( \mathbb{R}^N),\ \int_{\mathbb{R}^N} w_{\beta}(x) \left| \nabla u\right|^N\,{\rm d}x <{+} \infty \right\}, \]

where $D^{1,q}_r( \mathbb {R}^N) = \left \{u \in L^{q^*}( \mathbb {R}^N),\ u\ \text {radial},\ \displaystyle {\int _{\mathbb {R}^N} \left | \nabla u\right |^q\,{\rm d}x < + \infty }\right \}$ and $q^* = \frac {Nq}{N-q}.$ We first equip the functional space $E_{q,\beta }$ with the norm

\[ \left\|u\right\|_{E_{q,\beta}} = \left(\int_{\mathbb{R}^N}\left|\nabla u\right|^Nw_{\beta}(x)\,{\rm d}x + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}} \right)^{\frac{1}{N}},\ u \in E_{q ,\beta}. \]

The first result in the present work concerns the case $0 < \beta < 1$ and the norm $\left \|\ \cdot \ \right \|_{E_{q,\beta }}.$ It is given by the following Trudinger–Moser inequality.

Theorem 1.1 Let $0 < \beta < 1$ and $w_{\beta }$ be defined by (1.11). Let $j_{\beta } = \inf \left \{j \geq 1,\ j \geq \frac {(1- \beta )q^*}{N'}\right \}.$

For all $\alpha >0$ and $u \in E_{q,\beta },$ we have

(1.14)\begin{equation} \int_{\mathbb{R}^N} \left({\rm e}^{\alpha\left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x<{+}\infty, \end{equation}

where $\displaystyle {S_{j_{\beta }-1}(t) = \sum _{j=0}^{j_{\beta }-1} \frac {t^j}{j!},\ t \in [0, + \infty [.}$ Moreover, if $\alpha < \alpha _{N,\beta }$, then

(1.15)\begin{equation} \displaystyle\sup_{u\in E_{q,\beta},\ \left\| u\right\|_{E_{q,\beta}}\leq1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x<{+}\infty, \end{equation}

and if $\alpha > \alpha _{N,\beta }$, then

(1.16)\begin{equation} \displaystyle\sup_{u\in E_{q,\beta},\ \left\| u\right\|_{E_{q,\beta}}\leq1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x={+}\infty. \end{equation}

The second result in this paper concerns the case $\beta = 1.$ More precisely, we prove the following theorem:

Theorem 1.2 Let $w_1$ be defined by (1.13). Let $j_1 = \inf \left \{j \geq 1,\ j \geq \frac {q^*}{N'}\right \}.$ Set

(1.17)\begin{equation} C_{q,N} = \omega_{N-1}^{-\frac{1}{q}} \left( \frac{q-1}{N-q}\right)^{\frac{q-1}{q}}. \end{equation}

For all $\alpha >0$ and $u\in E_{q,1},$ we have

(1.18)\begin{equation} \int_{\mathbb{R}^N} \left({\rm e}^{\alpha\left({\rm e}^{\left| u\right|^{N'}}-1\right)}-S_{j_1-1} \left(\alpha \left({\rm e}^{\left|u\right|^{N'}}-1\right)\right)\right){\rm d}x<{+}\infty, \end{equation}

where $\displaystyle { S_{j_{1}-1}(t) = \sum _{j=0}^{j_{1}-1} \frac {t^j}{j!},\ t \in [0, + \infty [.}$ Moreover, if $\alpha \leq N\,{\rm e}^{-\omega _{N-1}^{\frac {1}{N-1}} C_{q,N}^{N'}},$ then

(1.19)\begin{equation} \sup_{u\in E_{q,1},\ \left\| u\right\|_{E_{q,1}}\leq1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}-S_{j_1-1} \left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}-1\right)\right)\right){\rm d}x<{+}\infty, \end{equation}

and if

\[ \alpha > N \exp\left(\frac{\omega_{N-1}^{\frac{N}{q}-1}}{N-1} \left(\int_0^{+ \infty} \frac{{\rm e}^{(q-N)t}}{(1+t)^q}\,{\rm d}t\right)^{\frac{N}{q}}\right), \]

then

(1.20)\begin{equation} \sup_{u\in E_{q,1},\ \left\| u\right\|_{E_{q,1}}\leq 1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}-S_{j_1-1} \left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}-1\right)\right)\right){\rm d}x={+}\infty. \end{equation}

Note that the Trudinger–Moser inequalities proved in theorems 1.1 and 1.2 are not necessarily sharp. However, as we will see, this sharpness can be recovered when we consider another norm on the space $E_{q,\beta }$ equivalent to $\left \|\ \cdot \ \right \|_{E_{q,\beta }}$ and given by:

\begin{align*} & \left\|u\right\|_{q,\beta} = \left| \nabla u\right|_{L_{w_{\beta}}^N( \mathbb{R}^N)} + \left| \nabla u\right|_{L^q( \mathbb{R}^N)} = \left(\int_{\mathbb{R}^N}\left|\nabla u\right|^Nw_{\beta}(x)\,{\rm d}x\right)^{\frac{1}{N}}\\ & \quad + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}},\ \text{for}\ 0 < \beta < 1, \end{align*}

and

\[ \left\|u\right\|_{q,1} = \left(\int_{\mathbb{R}^N}\left|\nabla u\right|^Nw_{1}(x)\,{\rm d}x\right)^{\frac{1}{N}} + \left( \int_{\left|x\right| \geq 1} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}},\ \text{for}\ \beta =1. \]

The equivalence of this norm and $\left \|\ \cdot \ \right \|_{E_{q,\beta }}$ is proved below (see remark 1.8). Using the new norm $\left \|\ \cdot \ \right \|_{q,\beta },$ we can establish the following sharp Trudinger–Moser inequalities.

Theorem 1.3 Let $0 < \beta < 1$ and $w_{\beta }$ be defined by (1.11). We have,

(1.21)\begin{equation} \displaystyle\sup_{u\in E_{q,\beta},\ \left\| u\right\|_{q,\beta}\leq1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x<{+}\infty \Leftrightarrow \alpha \leq \alpha_{N,\beta}. \end{equation}

Theorem 1.4 Let $w_1$ be defined by (1.13). We have,

(1.22)\begin{align} & \sup_{u\in E_{q,1},\ \left\| u\right\|_{q,1}\leq 1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}-S_{j_1-1} \left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}-1\right)\right)\right){\rm d}x\nonumber\\ & \quad <{+}\infty \Leftrightarrow \alpha \leq N. \end{align}

Comparing to previously cited works, there are many novelty aspects in the present work that we have to highlight. First, we are considering the case when the weight $w_{\beta },\ 0 < \beta < 1,$ vanishes at $x \in \mathbb {R}^N$ such that $\left |x\right | = 1.$ In fact, as it was mentioned above, we did not consider such a case and we preferred take $w_{\beta } (x) = (1- \log \left |x\right |)^{\beta (N-1)},\ 0 < \left |x\right | < 1$ in such a way that $\inf \limits _{x \in \mathbb {R}^N} w_{\beta }(x) > 0.$ The second aspect of novelty consists on taking only the gradient of the function to define the norms $\left \|\ \cdot \right \|_{E_{q,\beta }}$ and $\left \|\ \cdot \ \right \|_{q,\beta }.$ The combination of the norms of the gradient in two different Lebesgue spaces which are $L^N_{w_{\beta }}( \mathbb {R}^N)$ and $L^q(\mathbb {R}^N)$ has a real impact on the obtained inequalities. At this stage, we have to mention the work [Reference Carvalho, Figueiredo, Furtado and Medeiros19] in which the authors proved that

(1.23)\begin{equation} \sup_{u \in E^{N,q}, \left\|u\right\|_{E^{N,q}} \leq 1} \int_{\mathbb{R}^N} \Phi_{\alpha,j_0}(u)\,{\rm d}x <{+} \infty,\ \forall\ 0 < \alpha < \alpha_N = N \omega_{N-1}^{\frac{1}{N-1}}, \end{equation}

and

(1.24)\begin{equation} \sup_{u \in E^{N,q}, \left\|u\right\|_{E^{N,q}} \leq 1} \int_{\mathbb{R}^N} \Phi_{\alpha,j_0}(u)\,{\rm d}x={+} \infty,\ \forall\ \alpha > \alpha_N, \end{equation}

where $1 < q < N,\ E^{N,q}$ is defined as the completion of $C_0^{\infty }( \mathbb {R}^N)$ with respect to the norm

\[ \left\|u\right\|_{E^{N,q}} = \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^N\,{\rm d}x + \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\right)^{\frac{1}{N}}, \]

and

\[ \Phi_{\alpha,j_0}(u) = {\rm e}^{\alpha \left|u\right|^{N'}} - \sum_{j=0}^{j_0-1} \frac{\alpha^j}{j!} \left|u\right|^{jN'},\ j_0 = \inf\left\{j \in \mathbb{N},\ j \geq \frac{q^*}{N'}\right\}. \]

So, we can clearly note that this result can be recovered when we take $\beta = 0$ in theorem 1.1. In other words, our present work can be partially seen as a generalization of [Reference Carvalho, Figueiredo, Furtado and Medeiros19] (when we choose $\chi \equiv 1$). But in contrast with [Reference Carvalho, Figueiredo, Furtado and Medeiros19], we are able here to establish the sharpness of the inequality by introducing the new norm $\left \|\ \cdot \ \right \|_{q,\beta }.$ Obviously, this result of sharpness also holds for (1.23) provided that we pass from the norm $\left \|\ \cdot \ \right \|_{E^{N,q}}$ to the new one given by

\[ \left\|u\right\|= \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^N\,{\rm d}x\right)^{\frac{1}{N}} + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}},\ u \in E^{N,q}. \]

This leads us to the next point of novelty in the present work. It mainly consists on the sharpness of the inequalities (1.21) and (1.22). Actually, we have to highlight that such a sharp inequalities have been obtained in [Reference Aouaoui and Jlel9] for the case when $0< \beta < 1$ and its singular generalization proved in [Reference Aouaoui and Jlel11] for the case when $0 < \beta < 1$ or even when $\beta = 1$ (i.e., for the doubly exponential growth case). In our present work and due to the existence of the term $\left | \nabla u\right |_{L^q( \mathbb {R}^N)},$ we are able to guarantee the same sharpness property of the inequalities for the both cases $0 < \beta < 1$ and also $\beta = 1.$ Finally, we establish an improvement of (1.15), (1.19), (1.21) and (1.22) for weakly convergent sequences in $E_{q,\beta },\ 0 < \beta \leq 1$ with constants larger than those found in (1.15), (1.19), (1.21), and (1.22). These results are completely new. Moreover, inequality (1.25) proved below is also an improvement of the inequality proved by J.L. Carvalho, G.M. Figueiredo, M.F. Furtado, and E. Medeiros in [Reference Carvalho, Figueiredo, Furtado and Medeiros19]. The proof of these new results is mainly based on some version of the Concentration-Compactness principle due to P.L. Lions in [Reference Lions30].

The next result concerns the norm $\left \|\ \cdot \ \right \|_{q,\beta },\ 0 < \beta \leq 1,$ and it consists in some improvements of the inequalities (1.21) and (1.22). At first glance and in a natural way, the reader is expecting to find that these improvements can be obtained by a simple change of the norm $\left \|\ \cdot \ \right \|_{E_{q,\beta }},\ 0 < \beta \leq 1$ which appears in the expression of $P_{N,\beta }(u)$ in theorem 1.5 by $\left \|\ \cdot \ \right \|_{q,\beta }.$ But, due to the difference of the ‘geometric structure’ of the two norms, the situation is less easier than it seems.

Remark 1.7 Obviously, all the results obtained for the weight $w_{\beta }$ given by (1.11) hold true when we take

\[ w_{\beta}(x)=\left\{\begin{array}{@{}ll}\displaystyle (1- \log (\left|x\right|))^{\beta(N-1)} & \text{if}\ |x|<1,\\ \displaystyle \chi(\left|x\right|) & \text{if}\ |x|\geq1, \end{array} \right. \]

where $\chi : [1, + \infty [ \rightarrow ]0, + \infty [$ is a continuous function such that $\chi (1) = 1$ and $w_{\beta } \in A_N.$

In the last part of this work, we apply the Trudinger–Moser inequalities established in theorem 1.2 to study some elliptic quasilinear equation defined in $\mathbb {R}^N$ and containing a nonlinearities having a doubly exponential growth at infinity. More precisely, we prove the existence of at least one nontrivial solution to the equation

\[ - \text{div}\left(w_1(x) \left| \nabla u\right|^{N-2} \nabla u\right) - \Delta_q u = f(u),\ \text{in}\ \mathbb{R}^N,\ N \geq 2, \]

where $f : \mathbb {R} \rightarrow \mathbb {R}$ is a continuous function enjoying a doubly exponential growth at infinity governed by the inequality (1.22). In the mathematical literature, the first equation involving an operator with non-standard growth of the type $(p, N)-$Laplacian with $0 < p < N$ appeared in [Reference Yang and Perera39] where the problem was studied in a bounded domain and where the nonlinear term has an exponential growth governed by the classical Trudinger–Moser inequality. In [Reference Yang and Perera39], the authors obtained an existence result via a suitable minimax argument. This work was followed by [Reference Figueiredo and Nunes24] where the Nehari manifold approach has been used to obtain an existence result always by assuming the boundedness of the domain. For equations in the entire space, we can quote the following recent works [Reference Carvalho, Figueiredo, Furtado and Medeiros19, Reference Chen, Fiscella, Pucci and Tang20, Reference Fiscella and Pucci25] which deal with exponential growth governed by classical non-weighted Trudinger–Moser inequality.

Remark 1.8 We can easily show that, for $0 < \beta \leq 1,$ the norms $\left \|\ \cdot \ \right \|_{q,\beta }$ and $\left \|\ \cdot \ \right \|_{E_{q,\beta }}$ are equivalent. The case when $0 < \beta < 1$ is rather evident, we only prove the equivalence of the norms when $\beta = 1.$ For that aim, let $u \in E_{q,1}.$ We have

\begin{align*} \int_{\left|x\right| < 1} \left| \nabla u\right|^q\,{\rm d}x & = \int_{\left|x\right| < 1} \left| \nabla u\right|^q w_1^{\frac{q}{N}} w_1^{- \frac{q}{N}}\,{\rm d}x \leq \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_1(x)\,{\rm d}x\right)^{\frac{q}{N}}\\ & \quad \left(\int_{\left|x\right| < 1} w_1^{- \frac{q}{N-q}}\,{\rm d}x\right)^{\frac{N-q}{N}}. \end{align*}

Since $\displaystyle { \int _0^1 (1- \log r)^{-q(N-1)/N-q} r^{N-1}\,{\rm d}r < + \infty,}$ then $\displaystyle {\int _{\left |x\right | < 1} w_1^{-q/N-q}\,{\rm d}x < + \infty.}$ Consequently, there exists a positive constant $M_0$ such that

\[ \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} \leq M_0 \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_1\,{\rm d}x\right)^{\frac{1}{N}}. \]

Thus,

(1.33)\begin{equation} \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} + \left( \int_{\left|x\right| \geq 1} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} \leq (1+M_0) \left\|u\right\|_{q,1}. \end{equation}

Now, using the following elementary inequality,

\[ (a+b)^{\alpha} \leq a^{\alpha} + b^{\alpha},\ \forall\ a,b \geq 0,\ \forall\ 0 < \alpha < 1, \]

we infer from (1.33) that

\begin{align*} \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} & = \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^q\,{\rm d}x + \int_{\left|x\right| \geq 1} \left|\nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} \\ & \leq \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} + \left( \int_{\left|x\right| \geq 1} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} \\ & \leq (1+M_0) \left\|u\right\|_{q,1}. \end{align*}

Hence,

(1.34)\begin{equation} \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^N w_1\,{\rm d}x\right)^{\frac{1}{N}} + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} \leq (2+M_0) \left\|u\right\|_{q,1}. \end{equation}

Now, having in mind that

\begin{align*} \left\|u\right\|_{E_{q,1}} & = \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^N w_1\,{\rm d}x + \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^q dx\right)^{\frac{N}{q}}\right)^{\frac{1}{N}} \leq \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^N w_1\,{\rm d}x\right)^{\frac{1}{N}}\\ & \quad + \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}}, \end{align*}

by (1.34) we obtain

(1.35)\begin{equation} \left\|u\right\|_{E_{q,1}} \leq (2+M_0) \left\|u\right\|_{q,1}. \end{equation}

On the other hand, taking into account that the function $x \longmapsto x^{\frac {1}{N}}$ is concave on $[0, + \infty [,$ we get

\[ a^{\frac{1}{N}} + b^{\frac{1}{N}} \leq 2^{1- \frac{1}{N}}(a+b)^{\frac{1}{N}},\ \forall\ a,b \geq 0. \]

It follows that,

\[ \left\|u\right\|_{E_{q,1}} \geq 2^{ \frac{1}{N} -1} \left(\left( \int_{\mathbb{R}^N} \left| \nabla u\right|^N w_1\,{\rm d}x\right)^{\frac{1}{N}} + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}}\right) \geq 2^{ \frac{1}{N} -1} \left\|u\right\|_{q,1}. \]

Combining that last inequality with (1.35), we deduce that

\[ 2^{ \frac{1}{N} -1} \left\|u\right\|_{q,1} \leq \left\|u\right\|_{E_{q,1}} \leq (2+M_0) \left\|u\right\|_{q,1}. \]

Remark 1.9 A pertinent question is why when $\beta = 1,$ we change the form of the norm $\left \|\ \cdot \ \right \|_{q,1}$ by taking only the integral over the set $\left \{x \in \mathbb {R}^N,\ \left |x\right | \geq 1 \right \}.$ In fact, one can naturally expect that this last norm takes the form

\[ \left\|u\right\|_{q,1}^{(1)} = \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^N w_1\,{\rm d}x\right)^{\frac{1}{N}} + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}}. \]

Taking that last norm, we can easily adapt the proof of theorem 1.4 to prove that, if $\alpha \leq N,$ then

\[ \sup_{u\in E_{q,1},\ \left\| u\right\|_{q,1}^{(1)}\leq 1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}\!-\!1\right)}\!-\!S_{j_1-1} \left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}\!-\!1\right)\right)\right){\rm d}x\!<\!{+}\infty. \]

The problem lies in the construction of a sequence (if there exists) $(u_k)_k \subset E_{q,1}$ such that $\left \|u_k\right \|_{q,1}^{(1)} \leq 1$ and

\[ \int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u_k\right|^{N'}}-1\right)}-S_{j_1-1} \left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u_k\right|^{N'}}-1\right)\right)\right){\rm d}x \to + \infty,\ k \to + \infty. \]

We do not know the existence of such a sequence.

2. Proof of theorem 1.1

We start by proving (1.14). For that aim, fix $\alpha > 0$ and $u \in E_{q,\beta }.$ We have

(2.1)\begin{align} & \int_{\mathbb{R}^N} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x\nonumber\\ & \quad = \int_{\left|x\right| \geq 1} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x\nonumber\\ & \qquad + \int_{\left|x\right| < 1} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x. \end{align}

On the one hand, we have

(2.2)\begin{equation} \int_{\left|x\right| \geq 1} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x = \sum_{j=j_{\beta}}^{+ \infty} \frac{\alpha^j}{j!} \int_{\left|x\right| \geq 1} \left|u\right|^{\frac{jN'}{1- \beta}} {\rm d}x. \end{equation}

Since $u$ belongs to $D^{1,q}_r(\mathbb {R}^N),$ then by the radial lemma (see [Reference Su, Wang and Willem37, lemma 1]), we know that

(2.3)\begin{equation} \left|u(x)\right| \leq C_{q,N} \left|x\right|^{- \frac{N-q}{q}} \left| \nabla u\right|_{L^q( \mathbb{R}^N)},\ \forall\ x \neq 0, \end{equation}

where $C_{q,N}$ is given by (1.17). For $j \geq j_{\beta },$ we have $\frac {jN'}{1- \beta } \geq q^*.$ By (2.3), it yields

(2.4)\begin{equation} \left|u(x)\right|^{\frac{jN'}{1- \beta} - q^*} \leq {C_{q,N}}^{\frac{jN'}{1- \beta}-q^*}\left| \nabla u\right|_{L^{q}( \mathbb{R}^N)}^{\frac{jN'}{1- \beta} -q^*},\ \forall\ x \in \mathbb{R}^N,\ \left|x\right| \geq 1. \end{equation}

By (2.4), we infer

(2.5)\begin{align} \sum_{j=j_{\beta}}^{+ \infty} \frac{\alpha^j}{j!} \int_{\left|x\right| \geq 1} \left|u\right|^{\frac{jN'}{1- \beta}} \,{\rm d}x & \leq \sum_{j= j_{\beta}}^{+ \infty} \frac{\alpha^j}{j!} {C_{q,N}}^{\frac{jN'}{1- \beta}-q^*} \left|\nabla u\right|_{L^{q}( \mathbb{R}^N)}^{\frac{jN'}{1- \beta} -q^*} \int_{\left|x\right| \geq 1} \left|u\right|^{q^*}\,{\rm d}x\nonumber\\ & \leq \sum_{j= j_{\beta}}^{+ \infty} \frac{\alpha^j}{j!} {C_{q,N}}^{\frac{jN'}{1- \beta}-q^*} \left|\nabla u\right|_{L^{q}( \mathbb{R}^N)}^{\frac{jN'}{1- \beta}-q^*} \left|u\right|_{L^{q^*}( \mathbb{R}^N)}^{q^*} \nonumber\\ & \leq C'\,{\rm e}^{\alpha {C_{q,N}}^{\frac{N'}{1-\beta}} \left|\nabla u\right|_{L^{q}( \mathbb{R}^N)}^{\frac{N'}{1- \beta}}}, \end{align}

where we used the continuous embedding $D^{1,q}_r( \mathbb {R}^N) \hookrightarrow L^{q^*}( \mathbb {R}^N).$ Putting (2.5) in (2.2), we obtain

(2.6)\begin{equation} \int_{\left|x\right| \geq 1} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \leq C'\,{\rm e}^{\alpha {C_{q,N}}^{\frac{N'}{1-\beta}} \left\|u\right\|_{E_{q, \beta}}^{\frac{N'}{1- \beta}}}. \end{equation}

Now, in order to estimate the second integral in (2.1), set

(2.7)\begin{equation} v(x) = \left\{\begin{array}{@{}cc} u(x) - u(e_1), & 0 \leq \left|x\right| < 1, \\ 0, & \left|x\right| \geq 1, \end{array} \right. \end{equation}

where $e_1= (1,0,\cdots,0)$ is the first vector in the canonical basis of $\mathbb {R}^N.$ Clearly, $v \in W^{1,N}_{0,rad}(\mathcal {B},\sigma _{\beta }),$ with $\sigma _{\beta }(x) = (- \log \left |x\right |)^{\beta (N-1)},\ x \in \mathcal {B}.$ An elementary calculus gives the following inequality: for all $\epsilon >0$, we have

(2.8)\begin{equation} (a+b)^{\frac{N'}{1-\beta}}\leq(1+\epsilon)a^{\frac{N'}{1-\beta}}+\frac{1+\epsilon}{\left((1+\epsilon)^{\frac{1-\beta}{N'-1+\beta}}-1\right)^{\frac{N'-1+\beta}{1-\beta}}}b^{\frac{N'}{1-\beta}},\ \forall\ a,b \geq 0. \end{equation}

Fix $0 < \epsilon < 1.$ By (2.8), we get

(2.9)\begin{align} & \int_{\left|x\right| < 1} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \nonumber\\ & \leq \int_{\left|x\right| < 1}\,{\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x \nonumber\\ & \leq \int_{\left|x\right| < 1}\,{\rm e}^{\alpha (\left|v\right| + \left|u(e_1)\right|)^{\frac{N'}{1- \beta}}}\,{\rm d}x \nonumber\\ & \leq \exp\left( \frac{\alpha(1+ \epsilon)}{\left((1+\epsilon)^{\frac{1-\beta}{N'-1+\beta}}-1\right)^{\frac{N'-1+\beta}{1-\beta}}} \left|u(e_1)\right|^{\frac{N'}{1- \beta}}\right) \int_{\left|x\right| < 1}\,{\rm e}^{\alpha (1+ \epsilon) \left|v\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x. \end{align}

By (1.1), we know that

\[ \int_{\left|x\right| < 1}\,{\rm e}^{\alpha (1+ \epsilon) \left|v\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x <{+} \infty, \]

and by consequence (2.9) leads to

(2.10)\begin{equation} \int_{\left|x\right| < 1} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x <{+}\infty. \end{equation}

Combining (2.10) and (2.6), we deduce that (1.14) holds.

Now, we prove (1.15). By (2.6), it yields

(2.11)\begin{equation} \sup_{u \in E_{q,\beta},\ \left\| u\right\|_{E_{q,\beta}}\leq 1}\int_{\left|x\right| \geq 1}\left(e^{\alpha_{N,\beta}\left| u\right|^{\frac{N'}{1-\beta}}}\!-\!S_{j_{\beta}-1}\left(\alpha_{N, \beta} \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x \!\leq\! C'\,{\rm e}^{\alpha_{N,\beta} {C_{q,N}}^{\frac{N'}{1-\beta}}}. \end{equation}

Next, let $\alpha < \alpha _{N,\beta }.$ Clearly, there exists $\epsilon > 0$ such that $\alpha (1+ \epsilon ) < \alpha _{N,\beta }.$ Let $u \in E_{q, \beta }$ be such that $\left \|u\right \|_{E_{q,\beta }} \leq 1.$ Having in mind that $v$ defined by (2.7) belongs to $W^{1,N}_{0,rad}(\mathcal {B}, \sigma _{\beta })$ and

\[ \int_{\left|x\right| < 1} \left| \nabla v\right|^N \sigma_{\beta}(x)\,{\rm d}x = \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_{\beta}(x)\,{\rm d}x \leq 1, \]

then, by the virtue of (1.2), we infer that there exists a positive constant $C_{\beta } > 0$ such that

(2.12)\begin{equation} \int_{\left|x\right| < 1}\,{\rm e}^{\alpha(1+ \epsilon) \left|v\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x \leq \int_{\left|x\right| < 1}\,{\rm e}^{\alpha_{N,\beta} \left|v\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x \leq C_{\beta}. \end{equation}

Moreover, by (2.3), we know that $\left |u(e_1)\right | \leq C_{q,N}.$ Hence, by (2.9) and (2.12) we obtain

(2.13)\begin{equation} \int_{\left|x\right| < 1} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \!\leq\! \exp\left( \frac{\alpha_{N,\beta} C_{q,N}^{\frac{N'}{1- \beta}}}{\left((1\!+\!\epsilon)^{\frac{1\!-\!\beta}{N'-1+\beta}}\!-\!1\right)^{\frac{N'-1+\beta}{1-\beta}}} \right) C_{\beta}. \end{equation}

Combining (2.13) and (2.11), we deduce that (1.15) holds.

For $u \in E_{q,\beta },$ set $\psi (t) = \omega _{N-1}^{\frac {1}{N}} u(x)$ with $\left |x\right | = {\rm e}^{-t},\ t \in \mathbb {R}.$ A direct computation gives:

\begin{align*} & \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_{\beta}(x) \,{\rm d}x = \int_0^{+ \infty} t^{\beta (N-1)} \left| \psi'(t)\right|^N\,{\rm d}t,\\ & \int_{\left|x\right| \geq 1} w_{\beta}(x) \left| \nabla u\right|^N\,{\rm d}x = \int_{- \infty}^0 \chi({\rm e}^{{-}t}) \left| \psi'(t)\right|^{N}\,{\rm d}t,\\ & \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x = \omega_{N-1}^{1- \frac{q}{N}} \int_{-\infty}^{+ \infty} \left| \psi'(t)\right|^q\,{\rm e}^{(q-N)t}\,{\rm d}t, \end{align*}

and

\begin{align*} & \int_{\mathbb{R}^N} \left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \\ & = \omega_{N-1} \int_{- \infty}^{+ \infty} \left({\rm e}^{\alpha \omega_{N-1}^{-\frac{1}{(N-1)(1- \beta)}} \left| \psi(t)\right|^{\frac{N'}{1- \beta}}}\right.\nonumber\\ & \left.\quad -S_{j_{\beta}-1} \left(\alpha \omega_{N-1}^{-\frac{1}{(N-1)(1- \beta)}} \left| \psi(t)\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm e}^{{-}Nt}\,{\rm d}t. \end{align*}

Let $\gamma > 0$ to be fixed later. Consider the sequence of test functions:

\[ \psi_k(t) = \left\{\begin{array}{cc} k^{( \beta -1)(\gamma + \dfrac{1}{N}) + \gamma} t, & 0 \leq t \leq k^{- \gamma}, \\ k^{\dfrac{\beta -1}{N}} t^{1- \beta}, & k^{- \gamma} \leq t \leq k, \\ k^{\dfrac{1- \beta}{N'}}, & t \geq k, \\ 0, & t \leq 0. \end{array} \right. \]

For $k \geq 1,$ define $u_k \in E_{q,\beta }$ by $\psi _k(t) = \omega _{N-1}^{\frac {1}{N}} u_k(x),\ \left |x\right | = {\rm e}^{-t},\ t \in \mathbb {R}.$ We have

(2.14)\begin{align} \int_{\mathbb{R}^N} \left| \nabla u_k\right|^N w_{\beta}(x)\,{\rm d}x & = \int_0^k t^{\beta(N-1)} \left| \psi'_k(t)\right|^N\,{\rm d}t\nonumber\\ & = \int_{k^{- \gamma}}^k \frac{(1- \beta)^N t^{- \beta}}{k^{1- \beta}}\,{\rm d}t + \int_0^{k^{- \gamma}} t^{\beta(N-1)} k^{N((\beta -1)( \gamma + \frac{1}{N}) + \gamma)}\,{\rm d}t\nonumber\\ & = (1- \beta)^{N-1} + k^{(\beta-1)(\gamma+1)} \left(\frac{1}{1+ \beta(N-1)} - (1- \beta)^{N-1}\right). \end{align}

(2.15)\begin{align} \int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x & = \omega_{N-1}^{1- \frac{q}{N}} \frac{k^{q((\beta-1)(\gamma + \frac{1}{N}) + \gamma)}}{N-q} \left(1- {\rm e}^{(q-N)k^{-\gamma}}\right)\nonumber\\ & \quad + \omega_{N-1}^{1- \frac{q}{N}}\frac{(1- \beta)^k}{k^{\frac{(1- \beta)q}{N}}} \int_{k^{-\gamma}}^k \frac{{\rm e}^{(q-N)t}}{t^{q \beta}}\,{\rm d}t. \end{align}

Choosing $\gamma$ small enough such that $( \beta -1)(\gamma + \frac {1}{N}) + \gamma < 0.$ By this choice, we get

\[ \omega_{N-1}^{1- \frac{q}{N}} \frac{k^{q((\beta-1)(\gamma + \frac{1}{N}) + \gamma)}}{N-q} \left(1- {\rm e}^{(q-N)k^{-\gamma}}\right) \to 0,\ k \to + \infty. \]

On the other hand,

\[ \frac{(1- \beta)^k}{k^{\frac{(1- \beta)q}{N}}} \int_{k^{-\gamma}}^k \frac{{\rm e}^{(q-N)t}}{t^{q \beta}}\,{\rm d}t \leq \frac{(1- \beta)^k}{k^{\frac{(1- \beta)q}{N}}} \frac{k^{\gamma q \beta}}{N-q} \left({\rm e}^{(q-N)k^{- \gamma}} - {\rm e}^{(q-N)k}\right) \to 0,\ k \to + \infty. \]

Thus, by (2.15), we infer

\[ \int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x \to 0,\ k \to + \infty. \]

Now, taking into account that $(\beta -1)( \gamma +1) < 0,$ it follows from (2.14) that

\[ \int_{\mathbb{R}^N} \left| \nabla u_k\right|^N w_{\beta}(x)\,{\rm d}x \to ( 1- \beta)^{N-1},\ k \to + \infty. \]

Hence,

\[ \left\|u_k\right\|_{E_{q ,\beta}} = \left(\left| \nabla u_k\right|_{L^N_{w_{\beta}}(\mathbb{R}^N)}^N + \left| \nabla u_k\right|_{L^q( \mathbb{R}^N)}^{N}\right)^{\frac{1}{N}} \to (1- \beta)^{\frac{1}{N'}},\ k \to + \infty. \]

Set $\widetilde {u_k} = \frac {u_k}{\left \|u_k\right \|_{E_{q,\beta }}}.$ For $\alpha > 0,$ we have

(2.16)\begin{align} & \sup_{u \in E_{q,\beta},\ \left\|u\right\|_{E_{q,\beta}} \leq 1} \int_{\mathbb{R}^N}\left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} - S_{j_{\beta}-1}\left( \alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x\nonumber\\ & \geq \int_{\mathbb{R}^N} \left({\rm e}^{\alpha \left|\widetilde{u_k}\right|^{\frac{N'}{1- \beta}}} - S_{j_{\beta}-1}\left( \alpha \left|\widetilde{u_k}\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \nonumber\\ & = \omega_{N-1} \int_{-\infty}^{+ \infty} \left({\rm e}^{\alpha \omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} \left| \frac{\psi_k(t)}{\left\|u_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}}\right.\nonumber\\ & \left. - S_{j_{\beta}-1}\left(\alpha \omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} \left| \frac{\psi_k(t)}{\left\|u_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}\right)\right){\rm e}^{{-}Nt}\,{\rm d}t \nonumber\\ & \geq \omega_{N-1} \int_{k}^{+ \infty} \left({\rm e}^{\alpha \omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} \left| \frac{\psi_k(t)}{\left\|u_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}}\right.\nonumber\\ & \left. - S_{j_{\beta}-1}\left(\alpha \omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} \left| \frac{\psi_k(t)}{\left\|u_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}\right)\right){\rm e}^{{-}Nt}\,{\rm d}t \nonumber\\ & = \omega_{N-1} \frac{{\rm e}^{{-}Nk}}{N} \left({\rm e}^{\alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} k}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}}} -S_{j_{\beta}-1}\left(\alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} k}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}}\right)\right). \end{align}

Clearly, for $0 \leq j \leq j_{\beta }-1,$ we have

\[ {\rm e}^{{-}Nk} k^j \frac{1}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}} \to 0,\ k \to + \infty. \]

Thus,

(2.17)\begin{equation} S_{j_{\beta}-1}\left(\alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} k}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}}\right) \to 0,\ k\to + \infty. \end{equation}

Moreover,

\[ {\rm e}^{{-}Nk}\,{\rm e}^{\alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} k}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}}} = {\rm e}^{k\left({-}N + \alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}}}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}}\right)}. \]

Having in mind that $\left \|u_k\right \|_{E_{q,\beta }} \to (1- \beta )^{1/N'},$ then

\[{-}N + \alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}}}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}} \to -N + \alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}}}{(1- \beta)^{\frac{1}{1- \beta}}}. \]

If $\alpha > \alpha _{N,\beta },$ then

\[{-}N + \alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}}}{(1- \beta)^{\frac{1}{1- \beta}}} > 0. \]

Consequently,

(2.18)\begin{equation} {\rm e}^{{-}Nk}\,{\rm e}^{\alpha \frac{\omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} k}{\left\|u_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}}} \to + \infty,\ k \to + \infty. \end{equation}

Combining (2.18) and (2.17), we deduce from (2.16) that

\[ \sup_{u \in E_{q,\beta},\ \left\|u\right\|_{E_{q,\beta}} \leq 1} \int_{\mathbb{R}^N}\left({\rm e}^{\alpha \left|u\right|^{\frac{N'}{1- \beta}}} - S_{j_{\beta}-1}\left( \alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x ={+} \infty,\ \forall\ \alpha > \alpha_{N,\beta}. \]

We conclude that (1.16) holds.

3. Proof of theorem 1.2

Let $\alpha > 0$ and $u \in E_{q,1}.$ We have

(3.1)\begin{equation} \int_{\left|x\right| \geq 1} \left({\rm e}^{\alpha ({\rm e}^{\left|u\right|^{N'}}-1)} - S_{j_1-1} \left(\alpha ({\rm e}^{\left|u\right|^{N'}}-1)\right)\right) {\rm d}x = \sum_{j=j_1}^{+ \infty} \frac{\alpha^j}{j!} \int_{\left|x\right| \geq 1} \left({\rm e}^{\left|u\right|^{N'}}-1\right)^j\,{\rm d}x. \end{equation}

Using the monotony of the function defined on $[0, + \infty [$ by $s \longmapsto \frac {e^s -1}{s},$ from (2.3) it yields

\[ {\rm e}^{\left|u(x)\right|^{N'}} -1 \leq \frac{{\rm e}^{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}} -1}{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}} \left|u(x)\right|^{N'},\ \forall\ x \in \mathbb{R}^N,\ \left|x\right| \geq 1. \]

Observe that $j \geq j_1 \Leftrightarrow jN' \geq q^{*}.$ Thus, for $j \geq j_1,$ we have

\begin{align*} \left({\rm e}^{\left|u(x)\right|^{N'}} -1\right)^j & \leq \left(\frac{{\rm e}^{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}} -1}{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}}\right)^j \left|u(x)\right|^{jN'} \\ & \leq \left(\frac{{\rm e}^{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}} -1}{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}}\right)^j \left|u(x)\right|^{jN'-q^*} \left|u(x)\right|^{q^*} \\ & \leq \left(\frac{{\rm e}^{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}} -1}{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}}\right)^j \left(C_{q,N} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}\right)^{jN' -q^*}\left|u(x)\right|^{q^*},\\ & \quad \forall\ x \in \mathbb{R}^N,\ \left|x\right| \geq 1. \end{align*}

Putting that last inequality in (3.1), we infer

(3.2)\begin{align} & \int_{\left|x\right| \geq 1} \left({\rm e}^{\alpha ({\rm e}^{\left|u\right|^{N'}}-1)} - S_{j_1-1} \left(\alpha ({\rm e}^{\left|u\right|^{N'}}-1)\right)\right) {\rm d}x \nonumber\\ & \leq \sum_{j=j_1}^{+ \infty}\frac{\alpha^j}{j!}\left(\frac{{\rm e}^{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}} -1}{C_{q,N}^{N'} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N'}}\right)^j \left(C_{q,N} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}\right)^{jN' -q^*} \int_{\left|x\right| \geq 1} \left|u\right|^{q^*}\,{\rm d}x \nonumber\\ & \leq C'' \sum_{j=j_1}^{+ \infty} \frac{\alpha^j}{j!} \left({\rm e}^{C_{q,N}^{N'} \left| \nabla u\right|_{L^q(\mathbb{R}^N)}^{N'}} -1\right)^j \nonumber\\ & \leq C'' \sum_{j=0}^{+ \infty} \frac{\alpha^j}{j!} \left({\rm e}^{C_{q,N}^{N'} \left\|u\right\|_{E_{q,1}}^{N'}} -1\right)^j \nonumber\\ & = C'' \exp\left(\alpha \left({\rm e}^{C_{q,N}^{N'} \left\|u\right\|_{E_{q,1}}^{N'}} -1\right) \right). \end{align}

Next, by (2.8) one can easily deduce the following inequality:

(3.3)\begin{equation} (a+b)^{N'} \leq (1+ \epsilon) a^{N'} + \frac{1+ \epsilon}{\left((1+ \epsilon)^{\frac{1}{N'-1}} -1\right)^{N'-1}} b^{N'},\ \forall\ a,b \geq 0,\ \forall\ \epsilon > 0. \end{equation}

By (3.3), we have

\[ \left|u(x)\right|^{N'} \leq (1+ \epsilon) \left|v(x)\right|^{N'} + \frac{1+ \epsilon}{\left((1+ \epsilon)^{\frac{1}{N'-1}} -1\right)^{N'-1}} \left|u(e_1)\right|^{N'},\ \forall\ x \in \mathbb{R}^N,\ \left|x\right| < 1, \]

where $v$ is given by (2.7). Thus,

(3.4)\begin{align} & \int_{\left|x\right| < 1}\,{\rm e}^{\alpha\,{\rm e}^{\left|u\right|^{N'}}}\,{\rm d}x\nonumber\\ & \leq \int_{\left|x\right| < 1} \exp\left( \alpha \exp\left(\frac{1+ \epsilon}{\left((1+ \epsilon)^{\frac{1}{N'-1}} -1\right)^{N'-1}} \left|u(e_1)\right|^{N'}\right) {\rm e}^{(1+ \epsilon)\left|v\right|^{N'}}\right){\rm d}x\nonumber\\ & = \int_{\left|x\right| < 1} \exp\left( \alpha \exp\left(\frac{\left|u(e_1)\right|^{N'}}{\left(1-(1+ \epsilon)^{1-N}\right)^{\frac{1}{N-1}}} \right) {\rm e}^{(1+ \epsilon)\left|v\right|^{N'}}\right){\rm d}x. \end{align}

Clearly, $v \in W^{1,N}_{0,rad}(\mathcal {B},\sigma _{1}).$ By (1.3) and (3.4), it follows

(3.5)\begin{equation} \int_{\left|x\right| < 1}\,{\rm e}^{\alpha {\rm e}^{\left|u\right|^{N'}}}\,{\rm d}x <{+} \infty. \end{equation}

Combining (3.5) and (3.2), we easily see that (1.18) holds.

The next step in the proof of theorem 1.2 consists of proving (1.19). First, observe that by (3.2), we have

(3.6)\begin{align} & \sup_{u \in E_{q,1},\ \left\|u\right\|_{E_{q,1}} \leq 1} \int_{\left|x\right| \geq 1} \left({\rm e}^{\alpha\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}- S_{j_1-1} \left(\alpha ({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}-1)\right) \right){\rm d}x \nonumber\\ & \leq C'' \exp\left(\alpha \left({\rm e}^{C_{q,N}^{N'} \omega_{N-1}^{\frac{1}{N-1}}} -1\right) \right). \end{align}

Let $u \in E_{q,1}$ be such that $\left \|u\right \|_{E_{q,1}} \leq 1.$ Note that if $\displaystyle { \int _{\left |x\right | < 1} \left | \nabla u\right |^N w_{1}(x)\,{\rm d}x = 1,}$ then $\displaystyle { \int _{\mathbb {R}^N} \left | \nabla u\right |^q\,{\rm d}x = 0 }$ which implies that $u = 0.$ If $\displaystyle { \int _{\left |x\right | < 1} \left | \nabla u\right |^N w_{1}(x)\,{\rm d}x = 0,}$ then $v = 0$ (where $v$ is given by (2.7)) and by consequence $u(x) = u(e_1),\ \forall \ x \in \mathbb {R}^N,\ 0 < \left |x\right | < 1.$ Hence, by (2.3), we get

\[ \int_{\left|x\right|<1}\,{\rm e}^{\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u(x)\right|^{N'}}}\,{\rm d}x \!=\! \int_{\left|x\right|<1}\,{\rm e}^{\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u(e_1)\right|^{N'}}}\,{\rm d}x \!\leq\! \frac{\omega_{N-1}}{N}\,{\rm e}^{\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}C_{q,N}^{N'}}},\ \forall\ \alpha \!>\! 0. \]

Thus, without loss of generality, we can assume that

\[ 0 < \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_{1}(x)\,{\rm d}x < 1. \]

Choose $\epsilon > 0$ such that

\[ \frac{1}{(1+ \epsilon)^{N-1}} = \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_1(x)\,{\rm d}x. \]

Using again (2.3), it yields

\begin{align*} \left|u(e_1)\right|^{N} & \leq C_{q,N}^{N} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{N} \\ & = C_{q,N}^{N} \left(1- \int_{\mathbb{R}^N} \left| \nabla u\right|^N w_1(x)\,{\rm d}x\right) \\ & \leq C_{q,N}^{N} \left(1- \int_{\left|x\right| <1} \left| \nabla u\right|^N w_1(x)\,{\rm d}x\right) \\ & = C_{q,N}^{N} \left(1- \frac{1}{(1+ \epsilon)^{N-1}}\right). \end{align*}

Thus,

(3.7)\begin{equation} \left|u(e_1)\right|^{N'} \leq C_{q,N}^{N'} \left(1- \frac{1}{(1+ \epsilon)^{N-1}}\right)^{\frac{1}{N-1}}. \end{equation}

In view of (3.7) and (3.4), we infer

(3.8)\begin{align} & \int_{\left|x\right| < 1}\,{\rm e}^{\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}}\,{\rm d}x \nonumber\\ & \leq \int_{\left|x\right| < 1} \exp\left( \alpha \exp\left(\frac{\omega_{N-1}^{\frac{1}{N-1}}\left|u(e_1)\right|^{N'}}{\left(1-(1+ \epsilon)^{1-N}\right)^{\frac{1}{N-1}}} \right) {\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}(1+ \epsilon)\left|v\right|^{N'}}\right){\rm d}x \nonumber\\ & \leq \int_{\left|x\right| < 1} \exp\left( \alpha \exp\left(\frac{\omega_{N-1}^{\frac{1}{N-1}} C_{q,N}^{N'} \left(1- (1+ \epsilon)^{1-N}\right)^{\frac{1}{N-1}}}{\left(1-(1+ \epsilon)^{1-N}\right)^{\frac{1}{N-1}}} \right) {\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}(1+ \epsilon)\left|v\right|^{N'}}\right){\rm d}x. \nonumber\\ & = \int_{\left|x\right| < 1} \exp\left( \alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} C_{q,N}^{N'}}\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}(1+ \epsilon)\left|v\right|^{N'}}\right){\rm d}x \nonumber\\ & = \int_{\left|x\right|<1} \exp \left(\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} C_{q,N}^{N'}}\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} \left|\widetilde{v}\right|^{N'}}\right) {\rm d}x, \end{align}

where $\widetilde {v} = (1 + \epsilon )^{\frac {1}{N'}} v.$ Assume that $\alpha \,{\rm e}^{\omega _{N-1}^{\frac {1}{N-1}} C_{q,N}^{N'}} \leq N.$ Taking into account that $\widetilde {v} \in W_{0,rad}^{1,N}(\mathcal {B}, \sigma _1)$ and

\[ \int_{\left|x\right| < 1} \left| \nabla \widetilde{v}\right|^N w_1(x)\,{\rm d}x =1, \]

then we deduce from (1.4) that

(3.9)\begin{equation} \sup_{u \in E_{q,1},\ \left\|u\right\|_{E_{q,1}} \leq 1} \int_{\left|x\right| < 1}\,{\rm e}^{\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}}\,{\rm d}x <{+}\infty. \end{equation}

Plainly, (1.19) immediately follows from (3.9) and (3.6).

The end of the proof of theorem 1.2 consists of showing (1.20). For that aim, we make a change of variable similar to the case $0 < \beta < 1.$ More precisely, for $u \in E_{q,1},$ set $\psi (t) = \omega _{N-1}^{\frac {1}{N}} u(x)$ with $\left |x\right | = {\rm e}^{-t},\ t \in \mathbb {R}.$ We have

\begin{align*} & \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_{1}(x)\,{\rm d}x = \int_0^{+ \infty} ( 1 + t) ^{N-1} \left| \psi'(t)\right|^N\,{\rm d}t,\\ & \int_{\left|x\right| \geq 1} \left| \nabla u\right|^N w_{1}(x)\,{\rm d}x = \int_{- \infty}^0 \chi({\rm e}^{{-}t}) \left| \psi'(t)\right|^{N}\,{\rm d}t,\\ & \int_{\mathbb{R}^N} \left| \nabla u\right|^{q}\,{\rm d}x = \omega_{N-1}^{1-\frac{q}{N}} \int_{- \infty}^{+ \infty} \left|\psi'(t)\right|^{q}\,{\rm e}^{(q-N)t}\,{\rm d}t, \end{align*}

and

\begin{align*} & \int_{\mathbb{R}^N} \left({\rm e}^{\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} \left|u\right|^{N'}} -1\right)} -S_{j_1-1}\left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} \left|u\right|^{N'}} -1\right)\right)\right) {\rm d}x \\ & = \omega_{N-1} \int_{- \infty}^{+ \infty} \left({\rm e}^{ \alpha \left({\rm e}^{\left| \psi(t)\right|^{N'}} -1\right)} -S_{j_1-1}\left(\alpha \left({\rm e}^{\left| \psi(t)\right|^{N'}} -1\right)\right)\right) {\rm e}^{{-}Nt}\,{\rm d}t. \end{align*}

For $k \geq 0,$ consider the family of test functions:

(3.10)\begin{equation} \psi_k(t) = \left\{\begin{array}{@{}cc} \dfrac{ \log(1+t)}{ (\log(k+1))^{\dfrac{1}{N}}}, & 0 \leq t \leq k, \\ ( \log(k+1))^{\dfrac{1}{N'}}, & t \geq k, \\ 0, & t \leq 0, \end{array} \right. \end{equation}

and define $u_k \in E_{q,1}$ by $\psi _k(t) = \omega _{N-1}^{\frac {1}{N}} u_k(x).$ We have

\[ \int_{\mathbb{R}^N} \left| \nabla u_k\right|^N w_1(x)\,{\rm d}x = 1, \]

and

\[ \int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x = \frac{\omega_{N-1}^{1- \frac{q}{N}}}{\left(\log (k+1)\right)^{\frac{q}{N}}}\int_0^k \frac{{\rm e}^{(q-N)t}}{(1+t)^q}\,{\rm d}t \to 0,\ k \to + \infty. \]

Observe that,

(3.11)\begin{align} \lim_{k \to + \infty} \log(k\!+\!1) \left(1\!- \frac{1}{\left\|u_k\right\|_{E_{q,1}}^{N'}}\right) & = \lim_{k \to + \infty} \log (k\!+\!1) \left(1\!- \frac{1}{\left(1\!+\! \left| \nabla u_k\right|_{L^q( \mathbb{R}^N)}^N\right)^{\frac{1}{N-1}}}\right) \nonumber\\ & = \lim_{k \to + \infty} \log(k+1) \left(\frac{1}{N-1} \left| \nabla u_k\right|_{L^q( \mathbb{R}^N)}^{N}\right). \end{align}

We have

\[ \lim_{k \to + \infty} \log (k+1) \left(\frac{1}{N-1} \left| \nabla u_k\right|_{L^q( \mathbb{R}^N)}^{N}\right) = \frac{\omega_{N-1}^{\frac{N}{q} -1}}{N-1} \left(\int_0^{+ \infty} \frac{{\rm e}^{(q-N)t}}{(1+t)^q}\,{\rm d}t\right)^{\frac{N}{q}}, \]

which, by (3.11), leads to

(3.12)\begin{equation} \lim_{k \to + \infty} \log(k+1) \left(1- \frac{1}{\left\|u_k\right\|_{E_{q,1}}^{N'}}\right) = \frac{\omega_{N-1}^{\frac{N}{q} -1}}{N-1} \left(\int_0^{+ \infty} \frac{{\rm e}^{(q-N)t}}{(1+t)^q}\,{\rm d}t\right)^{\frac{N}{q}}. \end{equation}

Set $\widetilde {u_k}=\frac {u_k}{\left \|u_k\right \|_{E_{q,1}}}.$ For $\alpha > 0,$ we have

(3.13)\begin{align} & \sup_{u\in E_{q,1},\ \left\| u\right\|_{E_{q,1}}\leq1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{n-1}^{\frac{1}{n-1}}\left| u\right|^{N'}}-1\right)}-S_{j_1-1} \left(\alpha \left({\rm e}^{\omega_{n-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}-1\right)\right)\right){\rm d}x \nonumber\\ & \geq \int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{n-1}^{\frac{1}{N-1}}\left| \widetilde{u_k}\right|^{N'}}-1\right)}-S_{j_1-1} \left(\alpha \left({\rm e}^{\omega_{n-1}^{\frac{1}{n-1}}\left|\widetilde{u_k}\right|^{N'}}-1\right)\right)\right){\rm d}x \nonumber\\ & = \omega_{N-1} \int_{- \infty}^{+ \infty} \left({\rm e}^{ \alpha \left({\rm e}^{\left| \frac{ \psi_k(t)}{\left\|u_k\right\|_{E_{q,1}}}\right|^{N'}} -1\right)} -S_{j_1-1}\left(\alpha \left({\rm e}^{\left| \frac{\psi_k(t)}{\left\|u_k\right\|_{E_{q,1}}}\right|^{N'}} -1\right)\right)\right) {\rm e}^{{-}Nt}\,{\rm d}t \nonumber\\ & \geq \omega_{N-1} \int_{k}^{+ \infty} \left({\rm e}^{ \alpha \left({\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}}} -1\right)} -S_{j_1-1}\left(\alpha \left({\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}}} -1\right)\right)\right) {\rm e}^{{-}Nt}\,{\rm d}t\nonumber\\ & = \omega_{N-1} \frac{{\rm e}^{{-}Nk}}{N}\left({\rm e}^{ \alpha \left({\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}}} -1\right)} -S_{j_1-1}\left(\alpha \left({\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}}} -1\right)\right)\right). \end{align}

Taking into account that

\begin{align*} -Nk + \alpha \left({\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}}} -1\right) & ={-}N(k+1) + \alpha\,{\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}}} - \alpha + N \\ & = (k+1) \left({-}N + \alpha\,{\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}} - \log(k+1)}\right) - \alpha +N, \end{align*}

and using (3.12) we obtain that

\[{-}Nk + \alpha \left({\rm e}^{\frac{\log (k+1)}{\left\|u_k\right\|_{E_{q,1}}^{N'}}} -1\right) \to + \infty,\ \forall\ \alpha > N \exp\left(\frac{\omega_{N-1}^{\frac{N}{q}-1}}{N-1} \left(\int_0^{+ \infty} \frac{{\rm e}^{(q-N)t}}{(1+t)^q}\,{\rm d}t\right)^{\frac{N}{q}}\right). \]

Finally, by (3.13), we conclude that, if

\[ \alpha > N \exp\left(\frac{\omega_{N-1}^{\frac{N}{q}-1}}{N-1} \left(\int_0^{+ \infty} \frac{{\rm e}^{(q-N)t}}{(1+t)^q}\,{\rm d}t\right)^{\frac{N}{q}}\right), \]

then

\[ \sup_{u\in E_{q,1},\ \left\| u\right\|_{E_{q,1}}\leq1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-\!1\right)}\!-\!S_{j_1-\!1} \left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}\!-\!1\right)\right)\right){\rm d}x \!=\!{+} \infty. \]

This ends the proof of theorem 1.2.

4. Proof of theorem 1.3

We claim that

(4.1)\begin{equation} \sup_{u \in E_{q,\beta},\ \left\| u\right\|_{q,\beta}\leq 1}\int_{\mathbb{R}^N}\left({\rm e}^{\alpha_{N,\beta}\left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha_{N, \beta} \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x <{+} \infty. \end{equation}

First, observe that arguing exactly as in the proof of (2.11), one can easily show that

(4.2)\begin{equation} \sup_{u \in E_{q,\beta},\ \left\| u\right\|_{q,\beta}\leq 1}\int_{\left|x\right| \geq 1}\left(e^{\alpha_{N,\beta}\left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha_{N, \beta} \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x <{+} \infty. \end{equation}

It remains to prove that

(4.3)\begin{equation} \sup_{u \in E_{q,\beta},\ \left\| u\right\|_{q,\beta}\leq 1}\int_{\left|x\right| < 1}\left({\rm e}^{\alpha_{N,\beta}\left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha_{N, \beta} \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x <{+} \infty. \end{equation}

For that aim, let $u \in E_{q,\beta }$ be such that $\left \|u\right \|_{q,\beta } \leq 1.$ Choose $\epsilon > 0$ such that

\[ (1+ \epsilon)^{(1- \beta)(N-1)} = \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_{\beta}(x)\,{\rm d}x\right)^{{-}1}. \]

Using inequality (2.8), it follows

(4.4)\begin{align} & \int_{\left|x\right| < 1}\,{\rm e}^{\alpha_{N, \beta} \left|u\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x \leq \exp\left(\frac{\alpha_{N, \beta}(1+\epsilon)\left|u(e_1)\right|^{\frac{N'}{1- \beta}}}{\left((1+\varepsilon)^{\frac{1-\beta}{N'-1+\beta}}-1\right)^{\frac{N'-1+\beta}{1-\beta}}}\right) \nonumber\\ & \quad \int_{\left|x\right| < 1}\,{\rm e}^{\alpha_{N, \beta} (1+ \varepsilon) \left|v\right|^{\frac{N'}{1-\beta}}}\,{\rm d}x. \end{align}

We have,

\[ \int_{\left|x\right| < 1} \left| \nabla ((1+ \epsilon)^{\frac{1- \beta}{N'}} v)\right|^N w_{\beta}(x)\,{\rm d}x = (1+ \epsilon)^{(1- \beta)(N-1)} \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_{\beta}(x)\,{\rm d}x = 1. \]

By the virtue of (1.2), we know that there exists a constant $C_{\beta } > 0$ such that

\[ \int_{\left|x\right| < 1}\,{\rm e}^{\alpha_{N, \beta} ( 1 + \epsilon) \left|v\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x \leq C_{\beta}. \]

On the other hand, by (2.3), we have

\begin{align*} \left|u(e_1)\right|^{\frac{N'}{1- \beta}} & \leq C_{q,N}^{\frac{N'}{1- \beta}} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^{\frac{N'}{1-\beta}} \\ & \leq C_{q,N}^{\frac{N'}{1- \beta}} \left(1- \left| \nabla u\right|_{L_{w_{\beta}}^N( \mathbb{R}^N)}\right)^{\frac{N'}{1- \beta}} \\ & \leq C_{q,N}^{\frac{N'}{1- \beta}} \left(1- \left( \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_{\beta}(x)\,{\rm d}x\right)^{\frac{1}{N}}\right)^{\frac{N'}{1- \beta}} \\ & \leq C_{q,N}^{\frac{N'}{1- \beta}} \left(1- (1+ \epsilon)^{\frac{(1- \beta)(1-N)}{N}}\right)^{\frac{N'}{1- \beta}} \\ & \leq C_{q,N}^{\frac{N'}{1- \beta}} (1+ \epsilon)^{{-}1} \left((1+ \epsilon)^{\frac{1- \beta}{N'}} -1\right)^{\frac{N'}{1- \beta}}. \end{align*}

Putting that last inequality in (4.4), we deduce that

(4.5)\begin{equation} \int_{\left|x\right| < 1}\,{\rm e}^{\alpha_{N, \beta} \left|u\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x\leq C_{\beta} \exp\left(\frac{\alpha_{N,\beta} C_{q,N}^{\frac{N'}{1- \beta}} \left((1+ \epsilon)^{\frac{1- \beta}{N'}} -1\right)^{\frac{N'}{1- \beta}}}{\left((1+\epsilon)^{\frac{1-\beta}{N'-1+\beta}}-1\right)^{\frac{N'-1+\beta}{1-\beta}}}\right). \end{equation}

Since $\frac {N'}{1- \beta } \geq \frac {N' -1+ \beta }{1- \beta },$ then the function defined on $]1, + \infty [$ by

\[ x \longmapsto \frac{\left(x^{\frac{1- \beta}{N'}} -1\right)^{\frac{N'}{1- \beta}}}{\left(x^{\frac{1- \beta}{N'-1+ \beta}} -1\right)^{\frac{N'-1+ \beta}{1- \beta}}} \]

is bounded. In view of (4.5), we can easily conclude that (4.3) follows. Combining (4.2) and (4.3), we deduce that (4.1) holds. Finally, if $\alpha > \alpha _{N,\beta },$ we proceed exactly as in the proof of theorem (1.1) keeping the same Moser sequence to prove that

\[ \sup_{u \in E_{q,\beta},\ \left\| u\right\|_{q,\beta}\leq 1}\int_{\left|x\right| < 1}\left({\rm e}^{\alpha \left| u\right|^{\frac{N'}{1-\beta}}}-S_{j_{\beta}-1}\left(\alpha \left|u\right|^{\frac{N'}{1- \beta}}\right)\right){\rm d}x ={+} \infty. \]

5. Proof of theorem 1.4

We claim that

(5.1)\begin{equation} \sup_{u \in E_{q,1},\ \left\|u\right\|_{q,1} \leq 1} \int_{\mathbb{R}^N} \left({\rm e}^{N\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}\!-\!S_{j_1-1} \left(N ({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}\!-\!1)\right)\right)dx \!<\!{+} \infty. \end{equation}

First, proceeding as in the proof of (3.6), one can easily see that

(5.2)\begin{equation} \sup_{u \in E_{q,1},\ \left\|u\right\|_{q,1} \leq 1} \int_{\left|x\right| \geq 1} \left({\rm e}^{N\left(\!{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left| u\right|^{N'}}-1\right)}\!-\!S_{j_1\!-1} \left(\!N ({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}-1)\!\right)\!\right){\rm d}x \!<\!{+} \infty. \end{equation}

Now, let $u \in E_{q,1}$ be such that $\left \|u\right \|_{q,1} \leq 1.$ Without loss of generality, we can assume that

\[ 0 < \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_1(x) \,{\rm d}x < 1. \]

Using the convexity of the function defined on $[0, + \infty [$ by $x \longmapsto {\rm e}^{x^{N'}},$ we can easily get the following inequality

\[ {\rm e}^{(a+b)^{N'}} \leq \frac{\epsilon}{1+ \epsilon}\,{\rm e}^{\left(\frac{1+ \epsilon}{\epsilon}\right)^{N'} a^{N'}} + \frac{1}{1+ \epsilon}\,{\rm e}^{(1+ \epsilon)^{N'} b^{N'}},\ \forall\ a,b \geq 0,\ \forall\ \epsilon > 0. \]

For $v$ defined as in (2.7), it yields

\[ {\rm e}^{ \left|u(x)\right|^{N'}} \leq \frac{\epsilon}{1+ \epsilon}\,{\rm e}^{\left(\frac{1+ \epsilon}{\epsilon}\right)^{N'} \left|u(e_1)\right|^{N'}} + \frac{1}{1+ \epsilon}\,{\rm e}^{(1+ \epsilon)^{N'} \left|v(x)\right|^{N'}},\ \forall\ x \in \mathbb{R}^N,\ \left|x\right| < 1. \]

Hence,

(5.3)\begin{align} & \int_{\left|x\right| < 1}\,{\rm e}^{\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}\left|u\right|^{N'}}}\,{\rm d}x \leq {\rm e}^{\alpha \frac{\epsilon}{1+ \epsilon}\,{\rm e}^{\left(\frac{1+ \epsilon}{\epsilon}\right)^{N'} \omega_{N-1}^{\frac{1}{N-1}}\left|u(e_1)\right|^{N'}}} \nonumber\\ & \quad \int_{\left|x\right| <1}\,{\rm e}^{\frac{\alpha}{1+ \epsilon}\,{\rm e}^{(1+ \epsilon)^{N'} \omega_{N-1}^{\frac{1}{N-1}}\left|v(x)\right|^{N'}}}\,{\rm d}x. \end{align}

Choose $\epsilon > 0$ such that

\[ (1+ \epsilon)^N \int_{\left|x\right| < 1} \left| \nabla u\right|^N w_1(x)\,{\rm d}x =1. \]

Clearly, $\widetilde {v} = (1+ \epsilon ) v \in W_{0,rad}^{1,N}(\mathcal {B}, \sigma _1).$ We have

\begin{align*} & \int_{\left|x\right| <1}\,{\rm e}^{\frac{\alpha}{1+ \epsilon}\,{\rm e}^{(1+ \epsilon)^{N'} \omega_{N-1}^{\frac{1}{N-1}}\left|v(x)\right|^{N'}}}\,{\rm d}x = \int_{\left|x\right| <1}\,{\rm e}^{\frac{\alpha}{1+ \epsilon}\,{\rm e}^{ \omega_{N-1}^{\frac{1}{N-1}}\left|\widetilde{v}(x)\right|^{N'}}}\,{\rm d}x\\ & \quad \leq \int_{\left|x\right| <1}\,{\rm e}^{\alpha\,{\rm e}^{ \omega_{N-1}^{\frac{1}{N-1}}\left|\widetilde{v}(x)\right|^{N'}}}\,{\rm d}x. \end{align*}

For $\alpha \leq N,$ by (1.4) we get

(5.4)\begin{align} & \int_{\left|x\right| <1}\,{\rm e}^{\frac{\alpha}{1+ \epsilon}\,{\rm e}^{ \omega_{N-1}^{\frac{1}{N-1}}\left|\widetilde{v}(x)\right|^{N'}}}\,{\rm d}x \nonumber\\ & \leq \int_{\left|x\right| <1}\,{\rm e}^{N\,{\rm e}^{ \omega_{N-1}^{\frac{1}{N-1}}\left|\widetilde{v}(x)\right|^{N'}}}\,{\rm d}x \nonumber\\ & \leq \sup\left\{\int_{\left|x\right| <1}\,{\rm e}^{N\,{\rm e}^{ \omega_{N-1}^{\frac{1}{N-1}}\left|z(x)\right|^{N'}}}\,{\rm d}x,\ z \in W_{0,rad}^{1,N}( \mathcal{B}, \sigma_1),\ \left\|z\right\|_{\sigma_1} \leq 1 \right\} <{+} \infty. \end{align}

On the other hand, by looking at the proof of (2.3) in [Reference Su, Wang and Willem37], we can easily see that we have a more precise inequality, that is

\[ \left|u(x)\right| \leq C_{q,N} \left|x\right|^{-\frac{N-q}{q}} \left(\int_{\left|z\right| \geq \left|x\right|} \left| \nabla u (z)\right|^q\,{\rm d}z\right)^{\frac{1}{q}},\ \forall\ x \neq 0. \]

It follows,

\begin{align*} \left|u(e_1)\right| & \leq C_{q,N} \left(\int_{\left|x\right| \geq 1} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{1}{q}} \\ & \leq C_{q,N} \left(1- \left| \nabla u\right|_{L^N_{w_1}(\mathbb{R}^N)}\right) \\ & \leq C_{q,N}\left(1- \left(\int_{\left|x\right| < 1} \left| \nabla u\right|^N w_1(x)\,{\rm d}x\right)^{\frac{1}{N}}\right) \\ & = C_{q,N} \left(1- \frac{1}{1+ \epsilon}\right). \end{align*}

Hence,

(5.5)\begin{equation} {\rm e}^{\alpha \frac{\epsilon}{1+ \epsilon}\,{\rm e}^{\left(\frac{1+ \epsilon}{\epsilon}\right)^{N'} \omega_{N-1}^{\frac{1}{N-1}}\left|u(e_1)\right|^{N'}}} \leq {\rm e}^{\alpha\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} C_{q,N}^{N'}}}. \end{equation}

Combining (5.5) and (5.4), we deduce from (5.3) that

(5.6)\begin{equation} \sup_{u \in E_{q,1},\ \left\|u\right\|_{q,1} \leq 1} \int_{\left|x\right| < 1}\,{\rm e}^{N {\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} \left|u\right|^{N'}}}\,{\rm d}x <{+} \infty. \end{equation}

In view of (5.6) and (5.2), we can conclude that (5.1) holds. The end of the proof gives a clear idea about the real reason of taking only the integral of $\left | \nabla u\right |^q$ over the set $\left \{x \in \mathbb {R}^N,\ \left |x\right | \geq 1 \right \}$ in the definition of the norm $\left \|u\right \|_{q,1}.$ In fact, we take the family of test functions given by (3.10) and we define as usual $u_k \in E_{q,1}$ by $u_k(x) = \omega _{N-1}^{-\frac {1}{N}} \psi _k(t),\ \left |x\right | = e^{-t},\ t \in \mathbb {R}.$ Observing that $u_k(x) = 0,\ \forall \ x \in \mathbb {R}^N,\ \left |x\right | \geq 1,$ we immediately get

\[ \left\|u_k\right\|_{q,1} = \left(\int_{\mathbb{R}^N} \left| \nabla u_k\right|^N w_1\,{\rm d}x\right)^{\frac{1}{N}} =\left(\int_{\left|x\right| < 1} \left| \nabla u_k\right|^N w_1\,{\rm d}x\right)^{\frac{1}{N}} =1,\ \forall\ k \geq 0. \]

Consequently, we are returning to the case of a sequence lying in $W_{0,rad}^{1,N}( \mathcal {B}, \sigma _1).$ Therefore, the conclusion follows.

6. Proof of theorem 1.5

1. 1. Case $0 < \beta < 1:$ Sub-case $0 < \left \|u\right \|_{E_{q,\beta }} < 1:$

   Assume by contradiction that for some $0 < p_1 < (\frac {1}{1 - \left \|u\right \|_{E_{q,\beta }}^N})^{\frac {1}{(1- \beta )(N-1)}}$ we have

   (6.1)\begin{equation} \sup_n \int_{\mathbb{R}^N} \left({\rm e}^{p_1 \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(p_1 \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x ={+} \infty. \end{equation}
   For $L \in ]0, + \infty [$ and $v \in E_{q,\beta },$ set
   (6.2)\begin{equation} G_L(v) = \left\{\begin{array}{cc} L, & \text{if}\ v > L, \\ - L, \text{if}\ v <{-}L, \\ v, & \text{if}\ \left|v\right| \leq L,\end{array} \right.\ \text{and}\ T_L(v) = v - G_L(v). \end{equation}
   Plainly, there exists $\epsilon > 0$ such that
   \[ \left(p_1(1+ \epsilon)\right)^{(1- \beta)(N-1)} < \frac{1}{1 - \left\|u\right\|_{E_{q,\beta}}^N}. \]
   Since $\left \|G_L(u)\right \|_{E_{q,\beta }} \to \left \|u\right \|_{E_{q,\beta }}$ as $L \to + \infty,$ then one can choose $L$ large enough such that
   (6.3)\begin{equation} \left(p_1(1+ \epsilon)\right)^{(1- \beta)(N-1)} < \frac{1}{1 - \left\|G_L(u)\right\|_{E_{q,\beta}}^N}. \end{equation}
   We claim that
   (6.4)\begin{align} & \limsup_{n \to + \infty} \left(\int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^N w_{\beta}(x)\,{\rm d}x+ \left(\int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\right)\nonumber\\ & \quad < \left( \frac{1}{p_1(1+ \epsilon)}\right)^{(1- \beta)(N-1)}. \end{align}
   Suppose that this does not hold. Then, there exists a subsequence of $(u_n)_n$ that we still denote by $(u_n)_n$ such that
   (6.5)\begin{align} & \int_{\mathbb{R}^N} \left| \nabla T_L(u_n) \right|^N w_{\beta}(x)\,{\rm d}x + \left(\int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\nonumber\\ & \quad \geq \left( \frac{1}{p_1(1+ \epsilon)}\right)^{(1- \beta)(N-1)},\ \forall\ n \geq 0. \end{align}
   Using (6.5), it yields
   (6.6)\begin{align} 1 & = \int_{\mathbb{R}^N} \left| \nabla u_n\right|^N w_{\beta}(x)\,{\rm d}x + \left( \int_{\mathbb{R}^N} \left| \nabla u_n\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\nonumber\\ & = \int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^N w_{\beta}(x)\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla G_L(u_n)\right|^N w_{\beta}(x)\,{\rm d}x \nonumber\\ & \quad + \left(\int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^q\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla G_L(u_n)\right|^q\,{\rm d}x\right)^{\frac{N}{q}} \nonumber\\ & \geq \int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^N w_{\beta}(x)\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla G_L(u_n)\right|^N w_{\beta}(x)\,{\rm d}x \nonumber\\ & \quad + \left(\int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^q\,{\rm d}x\right)^{\frac{N}{q}} + \left(\int_{\mathbb{R}^N} \left| \nabla G_L(u_n)\right|^q\,{\rm d}x\right)^{\frac{N}{q}} \nonumber\\ & \geq \left( \frac{1}{p_1(1+ \epsilon)}\right)^{(1- \beta)(N-1)} + \left\|G_L(u_n)\right\|_{E_{q,\beta}}^N. \end{align}
   Clearly $G_L(u_n) \rightharpoonup G_L(u)$ weakly in $E_{q,\beta }.$ Consequently, passing to the lower limit as $n$ tends to $+ \infty$ in (6.6), we obtain
   \[ 1 \geq \left\|G_L(u)\right\|_{E_{q,\beta}}^N + \left( \frac{1}{p_1(1+ \epsilon)}\right)^{(1- \beta)(N-1)}. \]
   Thus,
   \[ \left( p_1(1+ \epsilon)\right)^{(1- \beta)(N-1)} \geq \frac{1}{1- \left\|G_L(u)\right\|_{E_{q,\beta}}^N}, \]
   which is in contradiction with (6.3). Therefore, our claim (6.4) is true. Set
   \[ \Omega_{n,L} = \left\{x \in \mathbb{R}^N,\ \left|u_n(x)\right| \geq L\right\}. \]
   By (6.4), up to a subsequence,
   \[ \left\|\left(p_1(1+ \epsilon)\right)^{\frac{(1- \beta)}{N'}} T_L(u_n)\right\|_{ E_{q,\beta}} < 1,\ \forall\ n \geq 0. \]
   We have
   (6.7)\begin{align} & \int_{\Omega_{n,L}} \left({\rm e}^{\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right)\,{\rm d}x\nonumber\\ & = \int_{\Omega_{n,L} \cap \mathcal{B}} \left({\rm e}^{\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right)\,{\rm d}x \nonumber\\ & \quad + \int_{\Omega_{n,L} \cap \mathcal{B}^c} \left({\rm e}^{\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right)\,{\rm d}x. \end{align}
   On the one hand, by (2.8), we get
   (6.8)\begin{align} & \int_{\Omega_{n,L} \cap \mathcal{B}} \left({\rm e}^{\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \nonumber\\ & \leq \int_{\Omega_{n,L} \cap \mathcal{B}}\,{\rm e}^{\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}}\,{\rm d}x \nonumber\\ & \leq \int_{\Omega_{n,L} \cap \mathcal{B}}\,{\rm e}^{p_1 \alpha_{N,\beta} (1+ \epsilon) \left|u_n -L\right|^{\frac{N'}{1- \beta}}}\,{\rm e}^{p_1 \alpha_{N,\beta} A(\epsilon) L^{\frac{N'}{1- \beta}}}\,{\rm d}x \nonumber\\ & \leq {\rm e}^{p_1 \alpha_{N,\beta} A(\epsilon) L^{\frac{N'}{1- \beta}}} \int_{\Omega_{n,L} \cap \mathcal{B}}\,{\rm e}^{p_1 \alpha_{N,\beta} (1+ \epsilon) \left|T_L(u_n)\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x \nonumber\\ & \leq {\rm e}^{p_1 \alpha_{N,\beta} A(\epsilon) L^{\frac{N'}{1- \beta}}} \int_{\mathcal{B}}\,{\rm e}^{p_1 \alpha_{N,\beta} (1+ \epsilon) \left|T_L(u_n)\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x, \end{align}
   where $A(\epsilon ) = \frac {1+\epsilon }{((1+\epsilon )^{\frac {1-\beta }{N'-1+\beta }}-1)^{\frac {N'-1+\beta }{1-\beta }}}.$ Having in mind that $\left |u_n(x)\right | = \left |u_n(e_1)\right | \leq C_{q,N},\ \forall \ x \in \mathbb {R}^N,\ \left |x\right | = 1,$ then
   \[ T_L(u_n(x)) = 0,\ \forall\ x \in \mathbb{R}^N,\ \left|x\right| = 1,\ \forall\ L > C_{q,N}. \]
   Consequently, $T_L(u_n) \in W_{0,rad}^{1,N}( \mathcal {B}, \sigma _{\beta }),\ \forall \ L > C_{q,N}.$ Since
   \[ \int_{\mathcal{B}} \left| \nabla (p_1(1+ \epsilon))^{\frac{1- \beta}{N'}}T_L(u_n)\right|^N w_{\beta}(x)\,{\rm d}x < 1, \]
   then by (1.2), we infer
   \[ \sup_{n} \int_{\mathcal{B}}\,{\rm e}^{p_1 \alpha_{N,\beta} (1+ \epsilon) \left|T_L(u_n)\right|^{\frac{N'}{1- \beta}}}\,{\rm d}x <{+} \infty,\ \forall\ L > C_{q,N}. \]
   Putting that result in (6.8), we obtain
   (6.9)\begin{align} & \sup_n \int_{\Omega_{n,L} \cap \mathcal{B}} \left({\rm e}^{\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \nonumber\\ & \quad <{+} \infty,\ \forall\ L > C_{q,N}. \end{align}
   On the other hand, in view of (2.3), we know that
   \[ \left|u_n(x)\right| \leq C_{q,N} \left| \nabla u_n\right|_{L^q(\mathbb{R}^N)} \leq C_{q,N},\ \forall\ x \in \mathbb{R}^N,\ \left|x\right| \geq 1. \]
   Hence,
   \[ \Omega_{n,L}\cap \mathcal{B}^c = \emptyset,\ \forall\ L > C_{q,N}. \]
   We deduce from (6.9) and (6.7) that
   (6.10)\begin{equation} \sup_n \int_{\Omega_{n,L}} \left({\rm e}^{\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}} \!-\!S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \!<\!{+}\infty,\ \forall\ L \!>\! C_{q,N}. \end{equation}
   Next, observe that
   \begin{align*} & \int_{\Omega_{n,L}^c} \left({\rm e}^{p_1 \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \\ & = \int_{\left|u_n\right| < L} \sum_{j=j_{\beta}}^{+ \infty} \frac{(p_1 \alpha_{N,\beta})^j}{j!} \left|u_n\right|^{\frac{jN'}{1- \beta}}\,{\rm d}x \\ & = \int_{\left|u_n\right| < L} \sum_{j=j_{\beta}}^{+ \infty} \frac{(p_1 \alpha_{N,\beta})^j}{j!} \left|\frac{u_n}{L}\right|^{\frac{jN'}{1- \beta}} L^{\frac{jN'}{1- \beta}}\,{\rm d}x \\ & \leq c_4 \sum_{j=j_{\beta}}^{+ \infty} \frac{(p_1 \alpha_{N, \beta} L^{\frac{N'}{1- \beta}})^j}{j!} \int_{\left|u_n\right| < L}\left|\frac{u_n}{L}\right|^{j_{\beta}\frac{N'}{1- \beta}}\,{\rm d}x \\ & \leq c_4 \sum_{j=j_{\beta}}^{+ \infty} \frac{(p_1 \alpha_{N, \beta} L^{\frac{N'}{1- \beta}})^j}{j!} \int_{\left|u_n\right| < L} \left|\frac{u_n}{L}\right|^{q^{*}}\,{\rm d}x \\ & \leq c_4 \frac{{\rm e}^{p_1 \alpha_{N, \beta} L^{\frac{N'}{1- \beta}}}}{L^{q^*}} \int_{\mathbb{R}^N} \left|u_n\right|^{q^*}\,{\rm d}x \\ & \leq c_5 \frac{{\rm e}^{p_1 \alpha_{N, \beta} L^{\frac{N'}{1- \beta}}}}{L^{q^*}}, \end{align*}
   where we used the fact that $j_{\beta } \geq \frac {(1- \beta )q^*}{N'}$ together with the boundedness of the sequence $(u_n)_n$ in $L^{q^*}( \mathbb {R}^N).$ Therefore,
   (6.11)\begin{equation} \sup_{n} \int_{\Omega_{n,L}^c} \left(e^{p_1 \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x <{+} \infty. \end{equation}
   Combining (6.11) and (6.10), we conclude that
   \[ \sup_{n} \int_{\mathbb{R}^N} \left({\rm e}^{p_1 \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p_1 \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x <{+} \infty, \]
   which, in view of (6.1), leads to the expected contradiction. Case $\left \|u\right \|_{E_{q,\beta }} = 1:$ Since $u_n \rightharpoonup u$ weakly in $E_{q,\beta }$ which is uniformly convex, then $u_n \to u$ strongly in $E_{q,\beta }$ (see [Reference Brezis12, proposition 3.32]). We can easily adapt the arguments used in the proof of [Reference do Ó, Medeiros and Severo23, proposition1] to deduce that there exists $v \in E_{q,\beta }$ such that, up to a subsequence, $\left |u_n(x)\right | \leq v(x)$ a.e. $x \in \mathbb {R}^N,\ \forall \ n.$ Let $0 < p < + \infty.$ We have
   \begin{align*} & \sup_n \int_{\mathbb{R}^N} \left({\rm e}^{p \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(p \alpha_{N, \beta} \left|u_n \right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \\ & = \sup_n \sum_{j= j_{\beta}}^{+ \infty} \int_{\mathbb{R}^N} \frac{(p \alpha_{N,\beta}\left|u_n \right|^{\frac{N'}{1- \beta}})^j}{j!}\,{\rm d}x \\ & \leq \sum_{j= j_{\beta}}^{+ \infty} \int_{\mathbb{R}^N} \frac{(p \alpha_{N,\beta}\left|v \right|^{\frac{N'}{1- \beta}})^j}{j!}\,{\rm d}x \\ & = \int_{\mathbb{R}^N} \left({\rm e}^{p \alpha_{N, \beta} \left|v\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(p \alpha_{N, \beta} \left|v\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x <{+} \infty. \end{align*}

   Now, we exhibit a sequence $(\xi _k)_k \subset E_{q,\beta }$ and a function $\xi \in E_{q,\beta }{\setminus} {\left \{0\right \}}$ such that $\left \|\xi _k\right \|_{E_{q,\beta }} = 1,\ \xi _k \rightharpoonup \xi$ weakly in $E_{q,\beta },\ \left \|\xi \right \|_{E_{q,\beta }} < 1,$ and

   \begin{align*} & \int_{\mathbb{R}^N} \left({\rm e}^{\alpha_{N, \beta} p \left|\xi_k\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left( \alpha_{N, \beta} p \left|\xi_k\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x\\ & \quad \to + \infty,\ k \to + \infty,\ \forall\ p > P_{N,\beta}(\xi). \end{align*}
   For $k \in \mathbb {N}\backslash {\left \{0\right \}}$ and $\lambda \in ]0, + \infty [$ to be fixed later, we define the function $\psi _k:\mathbb {R} \to \mathbb {R}$ by
   \[ \psi_k(t) = \lambda \left\{\begin{array}{ll} \dfrac{t^{1- \beta} - (1/2)^{1- \beta}}{\left(k^{1- \beta} - (1/2)^{1- \beta}\right)^{\dfrac{1}{N}}}, & 1/2 \leq t \leq k, \\ \left(k^{1- \beta} - (1/2)^{1- \beta}\right)^{\dfrac{1}{N'}}, & t \geq k, \\ 0, & t \leq 1/2. \end{array} \right. \]
   We also define the function $\psi :[0, + \infty [ \to [0, + \infty [$ by
   \[ \psi(t) = a \left\{\begin{array}{ll} 0, & t \leq 1/4, \\ \dfrac{t^{1- \beta} - (1/4)^{1- \beta}}{\left((1/2)^{1- \beta} - (1/4)^{1- \beta}\right)^{\dfrac{1}{N}}}, & 1/4 \leq t \leq 1/2, \\ \left((1/2)^{1- \beta} - (1/4)^{1- \beta}\right)^{\dfrac{1}{N'}}, & t \geq 1/2, \end{array} \right. \]
   where $a > 0.$ Set $u_k(x) = \omega _{N-1}^{-\frac {1}{N}} \psi _k(t),\ u(x) = \omega _{N-1}^{-\frac {1}{N}} \psi (t),\ \left |x\right | = {\rm e}^{-t},\ t \in \mathbb {R}.$ We have
   \begin{align*} \int_{\mathbb{R}^N} \left| \nabla u_k\right|^N w_{\beta}(x)\,{\rm d}x & = \int_{0}^{+ \infty} t^{\beta(N-1)} \left|\psi_k'(t)\right|^{N}\,{\rm d}t\\ & = \lambda^N \int_{1/2}^k \frac{t^{\beta(N-1)} (1- \beta)^N t^{- \beta N}}{k^{1- \beta} - (1/2)^{1- \beta}}\,{\rm d}t \\ & = \lambda^N (1- \beta)^{N-1}. \end{align*}
   Furthermore,
   \begin{align*} \int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x & = \omega_{N-1}^{1- \frac{q}{N}} \int_{1/2}^k \left|\psi_k'(t)\right|^q\,{\rm e}^{(q-N)t}\,{\rm d}t\\ & = \frac{ \lambda^q\omega_{N-1}^{1- \frac{q}{N}} (1- \beta)^q}{\left(k^{1- \beta} - (1/2)^{1- \beta}\right)^{\frac{q}{N}}} \int_{1/2}^k \frac{{\rm e}^{(q-N)t}}{t^{q \beta}}\,{\rm d}t. \end{align*}
   Clearly,
   \[ \int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x \to 0,\ k \to + \infty. \]
   Set $v_k = u + u_k.$ It yields,
   \begin{align*} \left\|v_k\right\|_{E_{q,\beta}}^N & = \int_{\mathbb{R}^N} \left| \nabla u_k\right|^N w_{\beta}(x)\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla u\right|^N w_{\beta}(x)\, {\rm d}x\\ & \quad + \left( \int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}} \\ & = (1- \beta)^{N-1}(a^N + \lambda^N) + \left( \int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}}. \end{align*}
   Consequently,
   \begin{align*} & \left\|v_k\right\|_{E_{q,\beta}}^N \to (1- \beta)^{N-1}(a^N + \lambda^N) + \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\\ & \quad = (1- \beta)^{N-1} \lambda^N + \left\|u\right\|_{E_{q,\beta}}^N,\ k \to + \infty, \end{align*}
   where we used the fact that
   \begin{align*} & \left\|u\right\|_{E_{q,\beta}}^N = \int_{\mathbb{R}^N} \left| \nabla u\right|^N w_{\beta}(x)\,{\rm d}x + \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\\ & \quad = (1- \beta)^{N-1} a^N + \left(\int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}}. \end{align*}
   Choose $\lambda > 0$ and $a > 0$ such that
   (6.12)\begin{equation} (1- \beta)^{N-1} \lambda^N + \left\|u\right\|_{E_{q,\beta}}^N = 1. \end{equation}
   One can easily see that $u_k \rightharpoonup 0$ weakly in $E_{q, \beta }.$ By (6.12), we derive that $\frac {v_k}{ \left \|v_k\right \|_{E_{q,\beta }}}\rightharpoonup u$ weakly in $E_{q,\beta }.$ Let $p > P_{N,\beta }(u).$ Then, there exists $\epsilon > 0$ such that $p = (1+ \epsilon ) P_{N,\beta }(u).$ We have
   (6.13)\begin{align} & \int_{\mathbb{R}^N} \left({\rm e}^{\alpha_{N, \beta} p \left| \frac{v_k}{\left\|v_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p \left| \frac{v_k}{\left\|v_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \nonumber\\ & \geq \omega_{N-1} \int_k^{+ \infty} \exp\left(\frac{\alpha_{N, \beta} p \omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}} \left| \psi(t) + \psi_k(t)\right|^{\frac{N'}{1- \beta}}}{\left\|v_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}}\right) {\rm e}^{{-}Nt}\,{\rm d}t \nonumber\\ & \quad - \sum_{j=0}^{j_{\beta}-1} \frac{( \alpha_{N,\beta} p)^{j}}{j!} \frac{1}{\left\|v_k\right\|_{E_{q\beta}}^{\frac{jN'}{1- \beta}}} \int_{\mathbb{R}^N} \left|v_k\right|^{\frac{jN'}{1- \beta}}\,{\rm d}x \nonumber\\ & = \omega_{N-1} \frac{{\rm e}^{{-}Nk}}{N} \exp\left(\vphantom{\left.\left.\quad + \lambda \left(k^{1- \beta} - (1/2)^{1- \beta}\right)^{\frac{1}{N'}}\right)^{\frac{N'}{1- \beta}}\right)}\theta_k\left(a\left((1/2)^{1- \beta} - (1/4)^{1- \beta}\right)^{\frac{1}{N'}}\right.\right.\nonumber\\ & \left.\left.\quad + \lambda \left(k^{1- \beta} - (1/2)^{1- \beta}\right)^{\frac{1}{N'}}\right)^{\frac{N'}{1- \beta}}\right) \nonumber\\ & \quad -\sum_{j=0}^{j_{\beta}-1} \frac{( \alpha_{N,\beta} p)^{j}}{j!} \frac{1}{\left\|v_k\right\|_{E_{q,\beta}}^{\frac{jN'}{1- \beta}}} \int_{\mathbb{R}^N} \left|v_k\right|^{\frac{jN'}{1- \beta}}\,{\rm d}x, \end{align}
   where
   \begin{align*} \theta_k & = \frac{\alpha_{N, \beta} p \omega_{N-1}^{- \frac{1}{(N-1)(1- \beta)}}}{\left\|v_k\right\|_{E_{q,\beta}}^{\frac{N'}{1- \beta}}} \to N (1- \beta)^{\frac{1}{1- \beta}} p\\ & = N (1- \beta)^{\frac{1}{1- \beta}} (1+\epsilon) P_{N,\beta}(u),\ k \to + \infty. \end{align*}
   On the other hand, by (6.12), we have
   \begin{align*} P_{N,\beta}(u) & = \frac{1}{\left(1- \left\|u\right\|_{E_{q,\beta}}^N\right)^{\frac{1}{(N-1)(1- \beta)}}} = \frac{1}{\left((1- \beta)^{N-1} \lambda^N\right)^{\frac{1}{(N-1)(1- \beta)}}} \\ & = \frac{1}{(1- \beta)^{\frac{1}{1- \beta}} \lambda^{\frac{N'}{1- \beta}}}. \end{align*}
   Thus,
   \[ \theta_k\left(a\left((1/2)^{1- \beta} \!-\! (1/4)^{1- \beta}\right)^{\frac{1}{N'}} \!+\! \lambda \left(k^{1- \beta} \!-\! (1/2)^{1- \beta}\right)^{\frac{1}{N'}}\right)^{\frac{N'}{1- \beta}} \mathop{\sim}\limits_{k \to + \infty} (1\!+\! \epsilon) N k. \]
   For $0 \leq j \leq j_{\beta }-1,$ we have
   \[ \int_{\mathbb{R}^N} \left|v_k\right|^{\frac{jN'}{1- \beta}}\,{\rm d}x = \omega_{N-1}^{1- \frac{j}{(N-1)(1- \beta)}} \int_{1/4}^{+ \infty} \left| \psi(t) + \psi_k(t)\right|^{\frac{jN'}{1- \beta}}\,{\rm e}^{{-}Nt}\,{\rm d}t. \]
   Hence,
   \[ \sup_{k \geq 1} \frac{1}{\left\|v_k\right\|_{E_{q, \beta}}^{\frac{jN'}{1- \beta}}} \int_{\mathbb{R}^N} \left|v_k\right|^{\frac{jN'}{1- \beta}}\,{\rm d}x <{+} \infty. \]
   We finally deduce from (6.13) that
   \begin{align*} & \int_{\mathbb{R}^N} \left({\rm e}^{\alpha_{N, \beta} p \left| \frac{v_k}{\left\|v_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(\alpha_{N, \beta} p \left| \frac{v_k}{\left\|v_k\right\|_{E_{q,\beta}}}\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x \\ & \quad \to +\infty,\ k \to + \infty. \end{align*}

2. 2. Now, we treat the case $\beta = 1.$ For the first part of the proof (i.e., inequality (1.27)), we can adapt the contradiction argument used for the case $\beta < 1.$ But, due to the existence of some essential technical difference, we give the proof with a minimum of details. Observing that, as previously, the case $\left \|u\right \|_{E_{q,1}} = 1$ can be easily studied using the uniform convexity of the functional space $E_{q,1},$ we can assume that $0 < \left \|u\right \|_{E_{q,1}} < 1.$ Assume by contradiction that there exists $0 < p_1 < \left (\frac {1}{1- \left \|u\right \|_{E_{q,1}}^N}\right )^{\frac {1}{N-1}}$ such that

   \begin{align*} & \sup_n \int_{\mathbb{R}^N} \left({\rm e}^{N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 \left|u_n\right|^{N'}} -1\right)} - S_{j_1-1}\left(N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 \left|u_n\right|^{N'}} -1\right)\right)\right) {\rm d}x \\ & \quad ={+} \infty. \end{align*}
   Arguing as for the case $0 < \beta < 1,$ we can easily find $\epsilon > 0$ small enough and $L$ large enough such that
   (6.14)\begin{align} & \limsup_{n \to + \infty} \left(\int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^N w_{1}(x)\,{\rm d}x+ \left(\int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\right)\nonumber\\ & \quad < \left( \frac{1}{p_1(1+ \epsilon)^2}\right)^{N-1}. \end{align}
   Here, the function $T_L$ is defined by (6.2). Set again $\Omega _{n,L} = \left \{x \in \mathbb {R}^N,\ \left |u_n(x)\right |\right.\left. \geq L\right \}.$ Using (3.3) and Young's inequality, it yields
   (6.15)\begin{align} & \int_{\Omega_{n,L} \cap \mathcal{B}} \left({\rm e}^{N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)} - S_{j_1-1} \left(N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)\right)\right) {\rm d}x \nonumber\\ & \leq \int_{\Omega_{n,L} \cap \mathcal{B}} {\rm e}^{N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)} {\rm d}x \nonumber\\ & \leq {\rm e}^{{-}N} \int_{\Omega_{n,L} \cap \mathcal{B}}\,{\rm e}^{N {\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} (1+ \epsilon)p_1 \left|u_n-L\right|^{N'}}\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 A_1(\epsilon) L^{N'}}}\,{\rm d}x \nonumber\\ & \leq \int_{\Omega_{n,L} \cap \mathcal{B}} \exp \left(N \left[\frac{{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 (1+ \epsilon)^2 \left|T_L(u_n)\right|^{N'}}}{1+ \epsilon} + \frac{\epsilon}{1+ \epsilon}\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 \frac{1+ \epsilon}{\epsilon} A_1( \epsilon) L^{N'}}\right]\right) {\rm d}x \nonumber\\ & \leq {\rm e}^{N\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 \frac{1+ \epsilon}{\epsilon} A_1( \epsilon) L^{N'}}} \int_{\mathcal{B}}\,{\rm e}^{N\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 (1+ \epsilon)^2 \left|T_L(u_n)\right|^{N'}}}\,{\rm d}x, \end{align}
   where $A_1( \epsilon ) = \frac {1+ \epsilon }{((1+ \epsilon )^{\frac {1}{N-1}} -1)^{N-1}}.$ By (6.14), we know that
   \[ \int_{\mathcal{B}} \left| \nabla (p_1(1+ \epsilon)^2)^{\frac{1}{N'}} T_L(u_n)\right|^N w_1(x)\,{\rm d}x < 1. \]
   Using (1.4), we get
   \[ \sup_n \int_{\mathcal{B}}\,{\rm e}^{N\,{\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p_1 (1+ \epsilon)^2 \left|T_L(u_n)\right|^{N'}}}\,{\rm d}x <{+} \infty. \]
   By (6.15), we obtain
   \begin{align*} & \sup_n \int_{\Omega_{n,L} \cap \mathcal{B}} \left({\rm e}^{N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)} \right.\nonumber\\ & \left.\quad -\, S_{j_1-1} \left(N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)\right)\vphantom{\left({\rm e}^{N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)} \right.}\right) {\rm d}x <{+} \infty. \end{align*}
   The boundedness of the sequence
   \[ \int_{\Omega_{n,L}^c} \left({\rm e}^{N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)} - S_{j_1-1} \left(N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}}p_1 \left|u_n\right|^{N'}} -1\right)\right)\right) {\rm d}x, \]
   can be established by proceeding as for the case $0 < \beta < 1$ and the details are omitted. In order to prove (1.28), we consider the sequence of functions $\psi _k: \mathbb {R} \rightarrow \mathbb {R}$ defined by
   (6.16)\begin{equation} \psi_k(t) = \left\{\begin{array}{ll} \dfrac{\log(t+1) - \log (3/2)}{\left(\log(k+1) - \log(3/2)\right)^{\dfrac{1}{N}}}, & 1/2 \leq t \leq k, \\ \left(\log(k+1) -\log (3/2)\right)^{\dfrac{1}{N'}}, & t \geq k, \\ 0, & t \leq 1/2. \end{array} \right. \end{equation}
   We also define the function $\psi :\mathbb {R} \to \mathbb {R}$ by
   \[ \psi(t) = \left\{\begin{array}{ll} 0, & t \leq 1/4, \\ \dfrac{\log(t+1) - \log(5/4)}{\left(\log(3/2) - \log(5/4)\right)^{\dfrac{1}{N}}}, & 1/4 \leq t \leq 1/2, \\ \left(\log(3/2) - \log(5/4)\right)^{\dfrac{1}{N'}}, & t \geq 1/2. \end{array} \right. \]
   Set, as for the first case, $u_k(x) = \omega _{N-1}^{- \frac {1}{N}} \psi _k(t)$ and $u(x) = \omega _{N-1}^{- \frac {1}{N}} \psi (t),\ \left |x\right | = {\rm e}^{-t}.$ Observe that
   \[ \int_{\mathbb{R}^N} \left|\nabla u_k\right|^{q}\,{\rm d}x \to 0,\ k \to + \infty. \]
   Set, $v_k = u + u_k.$ We have,
   \[ \left\|v_k\right\|_{E_{q,1}}^N = \left\|u_k +u\right\|_{E_{q,1}}^N= 2 + \left(\int_{\mathbb{R}^N} \left| \nabla u_k\right|^q\,{\rm d}x + \int_{\mathbb{R}^N} \left|\nabla u\right|^q\,{\rm d}x \right)^{\frac{N}{q}}. \]
   Thus,
   \[ \left\|v_k\right\|_{E_{q,1}}^N \to 2 + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}} =1+ \left\|u\right\|_{E_{q,1}}^N. \]
   Since $u_k \rightharpoonup 0$ weakly in $E_{q,1},$ then
   \[ \frac{v_k}{\left\|v_k\right\|_{E_{q,1}}} \rightharpoonup \widetilde{u} = \frac{u}{\left(1 + \left\|u\right\|_{E_{q,1}}^N\right)^{\frac{1}{N}}}. \]
   Let $\alpha > 0$ and $p > P_{N,1}(\widetilde {u}) = (1 + \left \|u\right \|_{E_{q,1}}^N)^{\frac {1}{N-1}}.$ We have
   (6.17)\begin{align} & \int_{\mathbb{R}^N} \left({\rm e}^{\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p\left| \frac{v_k(x)}{\left\|v_k\right\|_{E_{q,1}}}\right|^{N'}} -1\right)}-S_{j_1-1}\left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p\left| \frac{v_k(x)}{\left\|v_k\right\|_{E_{q,1}}}\right|^{N'}} -1\right)\right)\right) {\rm d}x \nonumber\\ & \geq \omega_{N-1} \int_k^{+ \infty}\,{\rm e}^{\alpha \left({\rm e}^{p\frac{ \log(k+1) - \log(3/2)}{\left(2 + \left(\left|\nabla u\right|_{L^{q}( \mathbb{R}^N)}^{q} + \left|\nabla u_k\right|_{L^q( \mathbb{R}^N)}^{q}\right)^{\frac{N}{q}} \right)^{\frac{1}{N-1}}}}-1\right)}{\rm e}^{{-}Nt} \,{\rm d}t\nonumber\\ & \quad - \int_{\mathbb{R}^N} S_{j_1-1}\left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p\left| \frac{v_k(x)}{\left\|v_k\right\|_{E_{q,1}}}\right|^{N'}} -1\right)\right) {\rm d}x \nonumber\\ & = \omega_{N-1}\,{\rm e}^{\alpha \left({\rm e}^{p\frac{ \log(k+1) - \log(3/2)}{\left(2 + \left(\left|\nabla u\right|_{L^{q}( \mathbb{R}^N)}^{q} + \left|\nabla u_k\right|_{L^q( \mathbb{R}^N)}^{q}\right)^{\frac{N}{q}} \right)^{\frac{1}{N-1}}}}-1\right)}\frac{{\rm e}^{{-}Nk}}{N} \nonumber\\ & \quad - \int_{\mathbb{R}^N} S_{j_1-1}\left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p\left| \frac{v_k(x)}{\left\|v_k\right\|_{E_{q,1}}}\right|^{N'}} -1\right)\right) {\rm d}x. \end{align}
   Since $p > (1 + \left \|u\right \|_{E_{q,1}}^N)^{\frac {1}{N-1}},$ then
   \[ \lim_{k \to + \infty} \left(p\frac{ \log(k+1) - \log(3/2)}{\left(2 + \left(\left|\nabla u\right|_{L^{q}( \mathbb{R}^N)}^{q} + \left|\nabla u_k\right|_{L^q( \mathbb{R}^N)}^{q}\right)^{\frac{N}{q}} \right)^{\frac{1}{N-1}}} -\log k\right) ={+} \infty. \]
   It follows,
   \[ \alpha \left({\rm e}^{p\frac{ \log(k+1) - \log(3/2)}{\left(2 + \left(\left|\nabla u\right|_{L^{q}( \mathbb{R}^N)}^{q} + \left|\nabla u_k\right|_{L^q( \mathbb{R}^N)}^{q}\right)^{\frac{N}{q}} \right)^{\frac{1}{N-1}}}}-1\right) -Nk \to + \infty,\ k \to + \infty. \]
   Finally, one can easily show that
   \[ \sup_{k \geq 1} \int_{\mathbb{R}^N} S_{j_1-1}\left(\alpha \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} p\left| \frac{v_k(x)}{\left\|v_k\right\|_{E_{q,1}}}\right|^{N'}} -1\right)\right) {\rm d}x <{+} \infty. \]
   In view of (6.17), we deduce that (1.28) holds. This ends the proof of theorem 1.5.

7. Proof of theorem 1.6

We start with the case $0 < \beta < 1.$ As in the proof of theorem 1.5, we argue by contradiction. So, assume that there exists $0 < p_1 < (\frac {1}{1- \left \|u\right \|_{q,\beta }})^{\frac {N'}{q( 1- \beta )}}$ such that

\[ \sup_n \int_{\mathbb{R}^N} \left(e^{p_1 \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}} -S_{j_{\beta}-1}\left(p_1 \alpha_{N, \beta} \left|u_n\right|^{\frac{N'}{1- \beta}}\right)\right) {\rm d}x ={+} \infty. \]

The function $T_L$ and $G_L$ being defined by (6.2), it is easy to see that there exist $0 < \epsilon < 1$ small enough and $L > 0$ large enough such that

(7.1)\begin{equation} \left(p_1 (1+ \epsilon)\right)^{\frac{1- \beta}{N'}} < \left(1- \left\|G_L(u)\right\|_{q,\beta}^q\right)^{-\frac{1}{q}}. \end{equation}

The keystone of the proof is to establish the inequality

(7.2)\begin{equation} \limsup_{n \to + \infty} \left\|T_L(u_n)\right\|_{q,\beta} < \left( \frac{1}{p_1(1+ \epsilon)}\right)^{\frac{1- \beta}{N'}}. \end{equation}

For that aim, we argue once again by contradiction. So, we assume that there exists a subsequence of $(u_n)_n,$ still denoted by $(u_n)_n,$ such that

(7.3)\begin{equation} \left\|T_L(u_n)\right\|_{q, \beta} \geq \left( \frac{1}{p_1(1+ \epsilon)}\right)^{\frac{1- \beta}{N'}},\ \forall\ n \geq 0. \end{equation}

First, observe that the general form of (2.8) is given by the following inequality

\[ (a+b)^t \leq ( 1+ \delta) a^t + D_t( \delta) b^t,\ \forall\ t > 1,\ \forall\ a,b \geq 0,\ \forall\ \delta > 0, \]

where $D_t(\delta ) = \frac {1+ \delta }{((1+ \delta )^{\frac {1}{t-1}}-1)^{t-1}}.$ From that last inequality, we can easily deduce another useful inequality, that is

(7.4)\begin{equation} (a^t + b^t)^{\frac{1}{t}} \geq ( 1+ \delta)^{- \frac{1}{t}} a + (D_t( \delta))^{- \frac{1}{t}} b,\ \forall\ t > 1,\ \forall\ a,b \geq 0,\ \forall\ \delta > 0. \end{equation}

Let $\delta > 0.$ Applying (7.4), we obtain

(7.5)\begin{align} \left| \nabla u_n\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)} & = \left( \int_{\mathbb{R}^N} \left| \nabla u_n\right|^N w_{\beta}\,{\rm d}x\right)^{\frac{1}{N}}\nonumber\\ & = \left( \int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^N w_{\beta}\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla G_L(u_n)\right|^N w_{\beta}\,{\rm d}x\right)^{\frac{1}{N}} \nonumber\\ & = \left(\left| \nabla T_L(u_n)\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)}^N + \left| \nabla G_L(u_n)\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)}^N\right)^{\frac{1}{N}} \nonumber\\ & \geq (1+ \delta)^{- \frac{1}{N}} \left| \nabla T_L(u_n)\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)} + (D_N( \delta))^{- \frac{1}{N}} \left| \nabla G_L(u_n)\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)}. \end{align}

In a similar way, we have

(7.6)\begin{align} \left| \nabla u_n\right|_{L^q( \mathbb{R}^N)} & = \left( \int_{\mathbb{R}^N} \left| \nabla u_n\right|^q\,{\rm d}x\right)^{\frac{1}{q}}\nonumber\\ & = \left( \int_{\mathbb{R}^N} \left| \nabla T_L(u_n)\right|^q\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla G_L(u_n)\right|^q\,{\rm d}x\right)^{\frac{1}{q}} \nonumber\\ & = \left(\left| \nabla T_L(u_n)\right|_{L^q( \mathbb{R}^N)}^q + \left| \nabla G_L(u_n)\right|_{L^q( \mathbb{R}^N)}^q\right)^{\frac{1}{q}} \nonumber\\ & \geq (1+ \delta)^{- \frac{1}{q}} \left| \nabla T_L(u_n)\right|_{L^q( \mathbb{R}^N)} + (D_q( \delta))^{- \frac{1}{q}} \left| \nabla G_L(u_n)\right|_{L^q( \mathbb{R}^N)}. \end{align}

A simple analysis shows that the function defined on $]1, + \infty [$ by

\[ t \longmapsto \left(\frac{z}{(z^{\frac{1}{t-1}} -1)^{t-1}}\right)^{-\frac{1}{t}} = \left(1 - z^{ \frac{1}{1-t}}\right)^{\frac{1-t}{t}}, \]

where $z$ is some fixed real number such that $z > 1,$ is nondecreasing. Thus,

\[ (D_N( \delta))^{- \frac{1}{N}} \geq (D_q( \delta))^{- \frac{1}{q}}. \]

From (7.5) and (7.6), we get

\begin{align*} 1 = \left\|u_n\right\|_{q ,\beta} & = \left| \nabla u_n\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)} + \left| \nabla u_n\right|_{L^q( \mathbb{R}^N)} \\ & \geq ( 1+ \delta)^{- \frac{1}{q}} \left(\left| \nabla T_L (u_n)\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)} + \left| \nabla T_L (u_n)\right|_{L^q( \mathbb{R}^N)}\right) \\ & \quad + (D_q( \delta))^{- \frac{1}{q}} \left(\left| \nabla G_L (u_n)\right|_{L^N_{w_{\beta}}( \mathbb{R}^N)} + \left| \nabla G_L (u_n)\right|_{L^q( \mathbb{R}^N)}\right) \\ & \geq (1+ \delta)^{- \frac{1}{q}} \left\|T_L(u_n)\right\|_{q ,\beta} + (D_q( \delta))^{- \frac{1}{q}} \left\|G_L(u_n)\right\|_{q, \beta}. \end{align*}

Putting (7.3) in that last inequality and using the fact that

\[ \liminf_{n \to + \infty} \left\|G_L(u_n)\right\|_{q,\beta} \geq \left\|G_L(u)\right\|_{q , \beta}, \]

we infer

(7.7)\begin{equation} (p_1(1+ \epsilon))^{\frac{1- \beta}{N'}} \geq \frac{(1+ \delta)^{- \frac{1}{q}}}{1- \left\|G_L(u)\right\|_{q ,\beta} (D_q( \delta))^{- \frac{1}{q}}}. \end{equation}

Now, consider the function defined on $]1, + \infty [$ by

\[ x \longmapsto \frac{1}{x - \left\|G_L(u)\right\|_{q , \beta} \left(x^{\frac{q}{q-1}} - 1\right)^{\frac{q-1}{q}}}. \]

A quick analysis of this function shows that it attains its maximum at the point

\[ x_0 = \frac{1}{\left(1- \left\|G_L(u)\right\|_{q , \beta}^q\right)^{\frac{q-1}{q}}} \]

and this maximum is

\[ \frac{1}{\left(1- \left\|G_L(u)\right\|_{q, \beta}^q\right)^{\frac{1}{q}}}. \]

Hence,

\begin{align*} & \max_{ x > 1} \frac{x^{- \frac{1}{q}}}{1-\left\|G_L(u)\right\|_{q, \beta} \left(\frac{x}{(x^{\frac{1}{q-1}} -1)^{q-1}}\right)^{-\frac{1}{q}}} \\ & = \max_{x > 1} \frac{1}{x^{\frac{1}{q}} - \left\|G_L(u)\right\|_{q ,\beta} (x^{\frac{1}{q-1}} -1)^{\frac{q-1}{q}}} \\ & = \max_{x > 1} \frac{1}{x - \left\|G_L(u)\right\|_{q , \beta} \left(x^{\frac{q}{q-1}} - 1\right)^{\frac{q-1}{q}}} \\ & = \frac{1}{\left(1- \left\|G_L(u)\right\|_{q, \beta}^q\right)^{\frac{1}{q}}}. \end{align*}

Consequently, the function defined on $]0, + \infty [$ by

\[ \delta \longmapsto \frac{(1+ \delta)^{- \frac{1}{q}}}{1-\left\|G_L(u)\right\|_{q, \beta} (D_q( \delta))^{\frac{-1}{q}} } \]

attains its maximum at the point $\delta _0 > 0$ given by the identity

\[ ( 1 + \delta_0)^{\frac{1}{q}} = \frac{1}{\left(1- \left\|G_L(u)\right\|_{q, \beta}^q\right)^{\frac{q-1}{q}}}, \]

and this maximum is

\[ \frac{1}{\left(1- \left\|G_L(u)\right\|_{q, \beta}^q\right)^{\frac{1}{q}}}. \]

Thus, choosing $\delta = \delta _0$ in (7.7), it comes

\[ (p_1( 1 +\epsilon))^{\frac{1- \beta}{N'}} \geq \frac{1}{\left(1- \left\|G_L(u)\right\|_{q, \beta}^q\right)^{\frac{1}{q}}}, \]

which is in contradiction with (7.1). Therefore, (7.2) holds. The rest of the proof is similar to what has been done in the proof of theorem 1.5 (with suitable adaptation) and, in order to avoid redundancy, the details will be omitted. For the case $\beta = 1,$ we can easily adapt the same arguments used previously for the case $0 < \beta < 1$ to prove that there exists $\epsilon > 0$ small enough and $L > 0$ large enough such that

\[ \limsup_{n \to + \infty} \left\|T_L(u_n)\right\|_{q,1} < \left( \frac{1}{p_1(1+ \epsilon)^2}\right)^{\frac{1}{N'}}. \]

The rest of the proof of (1.31) is similar to the proof of (1.27) and will be omitted. The same can be said concerning (1.32) whose proof is similar to (1.28). This ends the proof of theorem 1.6.

8. Applications to some elliptic equations

In this section, we deal with the following elliptic equation:

(8.1)\begin{equation} - \text{div}\left(w_1(x) \left| \nabla u\right|^{N-2} \nabla u\right) - \Delta_q u = f(u),\ \text{in}\ \mathbb{R}^N,\ N \geq 2,\ 1 < q < N, \end{equation}

where $f : \mathbb {R} \rightarrow \mathbb {R}$ is a continuous function such that $f(s) = 0,\ \forall \ s \leq 0.$ Here, we assume that the weight given by (1.13) satisfies

\[ \inf_{x \in \mathbb{R}^N} w_1(x) > 0. \]

By this assumption, it yields $E_{q, 1} \hookrightarrow E_r^{N,q}$ with continuous embedding where $E^{N,q}_r$ is defined in [Reference Carvalho, Figueiredo, Furtado and Medeiros19] as the subspace of radial functions of the completion of $C_0^{\infty }( \mathbb {R}^N)$ with respect to the norm

\[ u \longmapsto \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^N\,{\rm d}x + \left( \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x\right)^{\frac{N}{q}}\right)^{\frac{1}{N}},\ u \in E^{N,q}_r. \]

By [Reference Carvalho, Figueiredo, Furtado and Medeiros19, proposition 2.1], we know that $E_r^{N,q}$ is continuously (resp. compactly) embedded into $L^t( \mathbb {R}^N),\ \forall \ q^* \leq t < + \infty$ (resp. $\forall \ q^* < t < + \infty ).$ Consequently, the embedding $E_{q,1} \hookrightarrow L^t( \mathbb {R}^N)$ is continuous for $q^* \leq t < + \infty$ and compact for $q^* < t < + \infty.$ We assume that $f$ has a critical double exponential growth at infinity, that is there exists a constant $\alpha _0 > 0$ such that

(8.2)\begin{equation} \lim_{s \to + \infty} \frac{f(s)}{{\rm e}^{N {\rm e}^{\alpha s^{N'}}}} = \left\{\begin{array}{@{}ccc} 0,\ \text{if}\ \alpha > \alpha_0,\\ + \infty,\ \text{if}\ \alpha < \alpha_0. \end{array} \right. \end{equation}

We also assume that: $(F_1)$ There exists $\theta > N$ such that

\[ 0 < \theta F(s) = \theta \int_0^s f(t)\,{\rm d}t \leq f(s) s,\ \forall\ s > 0. \]

$(F_2)$ There exist $C > 0,\ s_1 > 0$ and $p > \max \left \{q^*,N\right \}$ such that

\[ f(s) \leq C s^{p-1},\ \forall\ 0 \leq s \leq s_1. \]

$(F_3)$ There exist $A > 0$ and $r > q$ such that

\[ F(s) \geq A s^r,\ \forall\ s \geq 0. \]

Example An example of a function $f$ satisfying the conditions $(F_1)-(F_3)$ is given by: $f(s) = F'(s),$ where

\[ F(s) = s^m\left({\rm e}^{N\left({\rm e}^{\alpha s^{N'}} -1\right)} -1\right),\ \forall\ s \geq 0,\ F(s) = 0,\ \forall\ s \leq 0, \]

with $\alpha > 0,\ m > \max \left \{N,q^*\right \}$. A radial weak solution of the equation (8.1) is a function $u \in E_{q,1}$ such that

\[ \int_{\mathbb{R}^N} w_{1}(x) \left| \nabla u\right|^{N-2} \nabla u \nabla v\,{\rm d}x + \int_{\mathbb{R}^N} \left| \nabla u\right|^{q-2} \nabla u \nabla v\,{\rm d}x = \int_{\mathbb{R}^N} f(u) v\,{\rm d}x,\ \forall\ v \in E_{q,1}. \]

The energy functional associated to (8.1) is

\[ I(u) = \frac{1}{N} \int_{\mathbb{R}^N} \left| \nabla u\right|^N w_1(x)\,{\rm d}x + \frac{1}{q} \int_{\mathbb{R}^N} \left| \nabla u\right|^q\,{\rm d}x- \int_{\mathbb{R}^N} F(u)\,{\rm d}x,\ u \in E_{q,1}. \]

Proof. For the simplicity in notation, set

\[ \Phi_1(s) = e^s - S_{j_1-1}(s),\ \forall\ s \geq 0. \]

Let $(u_n)_n \subset E_{q,1}$ be a (PS) sequence of $I$ at a level $c < \left (\frac {1}{N} - \frac {1}{\theta }\right )\min \left \{1, \frac {\omega _{N-1}}{\alpha _0^{N-1}2^{N-1}}\right \}.$ It yields

\begin{align*} & \theta I(u_n) - \left\langle I'(u_n), u_n\right\rangle = \theta c + o_n(1) \left\|u_n\right\|_{q,1}\\ & \Updownarrow \\ & \left(\frac{\theta}{N} - 1\right) \int_{\mathbb{R}^N} \left| \nabla u_n\right|^N w_1(x)\,{\rm d}x + \left(\frac{\theta}{q} - 1\right) \int_{\mathbb{R}^N} \left| \nabla u_n\right|^q\,{\rm d}x \\ & \quad+ \int_{\mathbb{R}^N} \left(f(u_n)u_n - \theta F(u_n)\right) {\rm d}x = \theta c + o_n(1) \left\|u_n\right\|_{q,1}, \end{align*}

where $o_n(1)$ stands for any sequence of nonnegative real numbers converging to zero when $n$ tends to $+ \infty.$ Since $\theta > N > q,$ from $(F_1)$ we can immediately deduce that $(u_n)_n$ is bounded. Moreover,

(8.3)\begin{equation} \limsup_{n \to + \infty} \left( \left| \nabla u_n\right|^N_{L^N_{w_1}( \mathbb{R}^N)} + \left| \nabla u_n\right|_{L^q( \mathbb{R}^N)}^q\right) \leq \frac{\theta c}{\frac{\theta}{N}-1} = \frac{N \theta c}{\theta -N}. \end{equation}

Taking into account that $\frac {N \theta c}{\theta -N} < 1,$ we deduce from (8.3) that, up to a subsequence,

\[ \left| \nabla u_n\right|^N_{L^N_{w_1}( \mathbb{R}^N)} + \left| \nabla u_n\right|_{L^q( \mathbb{R}^N)}^q < 1. \]

Thus,

\begin{align*} \left| \nabla u_n\right|^N_{L^N_{w_1}( \mathbb{R}^N)} + \left| \nabla u_n\right|_{L^q( \mathbb{R}^N)}^q & \geq \left| \nabla u_n\right|^N_{L^N_{w_1}( \mathbb{R}^N)} + \left| \nabla u_n\right|_{L^q( \mathbb{R}^N)}^N \\ & \geq 2^{1-N} \left(\left| \nabla u_n\right|_{L^N_{w_1}( \mathbb{R}^N)} + \left| \nabla u_n\right|_{L^q( \mathbb{R}^N)}\right)^N \\ & \geq 2^{1-N} \left\|u_n\right\|_{q,1}^N. \end{align*}

Putting that last inequality in (8.3), we get

\[ \limsup_{n \to + \infty} \left\|u_n\right\|_{q,1}^N \leq \frac{N \theta 2^{N-1} c}{\theta -N}. \]

Then, there exist $\epsilon _0,\ \epsilon _1 > 0$ and a subsequence of $(u_n)_n,$ still denoted by $(u_n)_n,$ such that

\[ \left\|u_n\right\|_{q,1} < \left(\frac{\omega_{N-1}^{\frac{1}{N-1}}}{(1+\epsilon_0)(1+ \epsilon_1) \alpha_0}\right)^{\frac{1}{N'}},\ \forall\ n. \]

Let $u \in E_{q,1}$ be the weak limit of $(u_n)_n$ in $E_{q,1}.$ We claim that, up to a subsequence, $u_n \to u$ strongly in $E_{q,1}.$ Let $t > q^*-1.$ By (8.2) and $(F_2),$ we have

(8.4)\begin{align} & \left| \int_{\mathbb{R}^N} f(u_n) (u_n-u)\,{\rm d}x\right| \nonumber\\ & \leq c_{6} \int_{\mathbb{R}^N} \left|u_n\right|^{p-1} \left|u_n-u\right| {\rm d}x\nonumber\\ & \quad + c_{6} \int_{\mathbb{R}^N} \left|u_n\right|^{t-1} \left|u_n-u\right| \Phi_1\left(N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right)\right) {\rm d}x \nonumber\\ & \leq c_{6} \left(\int_{\mathbb{R}^N} \left|u_n\right|^p\,{\rm d}x\right)^{\frac{p-1}{p}} \left(\int_{\mathbb{R}^N} \left|u_n -u\right|^p\,{\rm d}x\right)^{\frac{1}{p}} \nonumber\\ & \quad + c_{7} \left(\int_{\mathbb{R}^N} \left|u_n\right|^{t+1}\right)^{\frac{t-1}{t+1}} \left(\int_{\mathbb{R}^N} \left|u_n -u\right|^{t+1}\,{\rm d}x\right)^{\frac{1}{t+1}} \nonumber\\ & \quad \times \left(\int_{\mathbb{R}^N} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right)\right){\rm d}x\right)^{\frac{1}{t+1}},\ \forall\ n. \end{align}

Clearly, there exist $c_{8} > 0$ and $s_2 > 0$ such that

\begin{align*} & \Phi_1\left((t+1) N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|s\right|^{N'}} -1\right)\right)\\ & \quad \leq c_{8} \Phi_1\left(N \left({\rm e}^{(1+ \epsilon_0)(1+ \epsilon_1)\alpha_0 \left|s\right|^{N'}} -1\right)\right),\ \forall\ \left|s\right| \geq s_2. \end{align*}

Hence,

(8.5)\begin{align} & \int_{\mathbb{R}^N} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x \nonumber\\ & = \int_{\left\{\left|u_n\right| \geq s_2\right\}} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x \nonumber\\ & \quad + \int_{\left\{\left|u_n\right| < s_2\right\}} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x\nonumber\\ & \leq c_8 \int_{\mathbb{R}^N} \Phi_1\left(N \left({\rm e}^{(1+ \epsilon_0)(1+ \epsilon_1)\alpha_0 \left|u_n\right|^{N'}} -1\right)\right) {\rm d}x \nonumber\\ & \quad+ \int_{\left\{\left|u_n\right| < s_2\right\}} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x. \end{align}

Plainly, there exists a positive constant $\kappa$ such that

\[ {\rm e}^{(1+ \epsilon_0)\alpha_0 s^{N'}} -1 \leq \kappa s^{N'},\ \forall\ 0 \leq s \leq s_2. \]

Then, we can derive that

(8.6)\begin{align} & \int_{\left\{\left|u_n\right| < s_2\right\}} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x\nonumber\\ & = \sum_{j= j_1}^{+ \infty} \frac{(t+1)^j N^j}{j!}\int_{\left|u_n\right| < s_2} \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}}-1\right)^j\,{\rm d}x \nonumber\\ & \leq \sum_{j= j_1}^{+ \infty} \frac{(t+1)^j N^j \kappa^j}{j!}\int_{\left|u_n\right| < s_2} \left|u_n\right|^{N'j}\,{\rm d}x. \end{align}

Having in mind that if $j \geq j_1,$ then $N'j \geq q^*,$ it follows from (8.6) that

\begin{align*} & \int_{\left\{\left|u_n\right| < s_2\right\}} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x \\ & \leq \sum_{j= j_1}^{+ \infty} \frac{(t+1)^j N^j \kappa^j s_2^{N'j}}{j!}\int_{\left|u_n\right| < s_2} \left|\frac{u_n}{s_2}\right|^{N'j}\,{\rm d}x \\ & \leq \sum_{j= j_1}^{+ \infty} \frac{(t+1)^j N^j \kappa^j s_2^{N'j}}{j!}\int_{\left|u_n\right| < s_2} \left|\frac{u_n}{s_2}\right|^{q^*}\,{\rm d}x \\ & \leq \frac{e^{(t+1)N\kappa s_2^{N'}}}{s_2^{q^*}} \int_{\mathbb{R}^N} \left|u_n\right|^{q^*}\,{\rm d}x. \end{align*}

Hence, there exists a positive constant $c_9 > 0$ such that

\[ \int_{\left\{\left|u_n\right| < s_2\right\}} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x \leq c_9 \int_{\mathbb{R}^N} \left|u_n\right|^{q^*}\,{\rm d}x. \]

Putting that last inequality in (8.5), we get

\begin{align*} & \int_{\mathbb{R}^N} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x \\ & \leq c_{8} \int_{\mathbb{R}^N} \Phi_1\left(N \left({\rm e}^{(1+ \epsilon_0)(1+ \epsilon_1)\alpha_0 \left|u_n\right|^{N'}} -1\right) \right) {\rm d}x + c_{9} \int_{\mathbb{R}^N} \left|u_n\right|^{q^*}\,{\rm d}x \\ & = c_{8} \int_{\mathbb{R}^N} \Phi_1\left(N \left({\rm e}^{(1+ \epsilon_0)(1+ \epsilon_1)\alpha_0 \left\|u_n\right\|_{q,1}^{N'} \left|\frac{u_n}{\left\|u_n\right\|_{q,1}}\right|^{N'}} -1\right) \right) {\rm d}x + c_{9} \int_{\mathbb{R}^N} \left|u_n\right|^{q^*}\,{\rm d}x \\ & \leq c_{8} \int_{\mathbb{R}^N} \Phi_1\left(N \left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} \left|\frac{u_n}{\left\|u_n\right\|_{q,1}}\right|^{N'}}-1\right)\right) {\rm d}x + c_{9} \int_{\mathbb{R}^N} \left|u_n\right|^{q^*}\,{\rm d}x. \end{align*}

Using (1.22), it yields

\[ \sup_{n} \int_{\mathbb{R}^N} \Phi_1\left((t+1)N \left({\rm e}^{(1+ \epsilon_0)\alpha_0 \left|u_n\right|^{N'}} -1\right)\right){\rm d}x <{+} \infty. \]

Now, since $t+1 > q^*$ and $p > q^*,$ then the embeddings $E_{q,1} \hookrightarrow L^{t+1}( \mathbb {R}^N)$ and $E_{q,1} \hookrightarrow L^p(\mathbb {R}^N)$ are compact. It follows from (8.4) that

\[ \int_{\mathbb{R}^N} f(u_n) (u_n-u)\,{\rm d}x \to 0,\ n \to + \infty. \]

Taking into account that

\begin{align*} \left\langle I'(u_n), u_n-u\right\rangle & = \int_{\mathbb{R}^N} \left| \nabla u_n\right|^{N-2} \nabla u_n \nabla (u_n -u) w_1(x)\,{\rm d}x\\ & \quad + \int_{\mathbb{R}^N} \left| \nabla u_n\right|^{q-2} \nabla u_n \nabla (u_n -u)\,{\rm d}x \\ & \quad - \int_{\mathbb{R}^N} f(u_n) (u_n-u)\,{\rm d}x \\ & \to 0,\ n \to + \infty, \end{align*}

we infer $u_n \to 0$ strongly in $E_{q,1}.$ This ends the proof of lemma 8.2.

Now, we claim that there exist $\rho _0 > 0,\ \rho _1 > 0,\ e_0 \in E_{q,1}$ such that $\left \|e_0\right \|_{q,1} > \rho _0 ,\ I(e_0) < 0,$ and

\[ I(u) \geq \rho_1,\ \forall\ u \in E_{q,1},\ \left\|u\right\|_{q,1} = \rho_0. \]

Given $u \in E_{q,1}$ and $t > p.$ By (8.2) and $(F_2),$ we have

(8.7)\begin{align} \int_{\mathbb{R}^N} \left|F(u)\right| {\rm d}x & \leq c_{10} \int_{\mathbb{R}^N} \left|u\right|^p\,{\rm d}x + c_{10} \int_{\mathbb{R}^N} \left|u\right|^t \Phi_1\left(N\left({\rm e}^{2 \alpha_0 \left|u\right|^{N'}} -1\right)\right) {\rm d}x\nonumber\\ & \leq c_{10} \int_{\mathbb{R}^N} \left|u\right|^p\,{\rm d}x + c_{11} \left(\int_{\mathbb{R}^N} \left|u\right|^{t+1}\,{\rm d}x\right)^{\frac{t}{t+1}}\nonumber\\ & \quad\int_{\mathbb{R}^N} \Phi_1\left((t+1)N\left({\rm e}^{2 \alpha_0 \left|u\right|^{N'}} -1\right)\right) {\rm d}x \nonumber\\ & \leq c_{12} \left\|u\right\|_{q,1}^p + c_{12} \left\|u\right\|_{q,1}^t \int_{\mathbb{R}^N} \Phi_1\left((t+1)N\left({\rm e}^{2 \alpha_0 \left|u\right|^{N'}} -1\right)\right) {\rm d}x \nonumber\\ & \leq c_{12} \left\|u\right\|_{q,1}^p + c_{13} \left\|u\right\|_{q,1}^t \int_{\mathbb{R}^N} \Phi_1\left(N\left({\rm e}^{3 \alpha_0 \left|u\right|^{N'}} -1\right)\right) {\rm d}x. \end{align}

For $u \in E_{q,1}$ such that $\left \|u\right \|_{q,1} = \rho _0 < \min \left \{1,\ \left (\frac {\omega _{N-1}^{\frac {1}{N-1}}}{3 \alpha _0}\right )^{\frac {1}{N'}}\right \},$ by (1.22), there exists a constant $c_{14} > 0$ such that

\[ \int_{\mathbb{R}^N} \Phi_1\left(N\left({\rm e}^{3 \alpha_0 \left|u\right|^{N'}} -1\right)\right) {\rm d}x \leq \int_{\mathbb{R}^N} \Phi_1\left(N\left({\rm e}^{\omega_{N-1}^{\frac{1}{N-1}} \left|\frac{u}{\left\|u\right\|_{q,1}}\right|^{N'}} -1\right)\right) {\rm d}x \leq c_{14}. \]

Putting that inequality in (8.7), for $u \in E_{q,1}$ such that $\left \|u\right \|_{q,1} = \rho _0,$ it yields

(8.8)\begin{align} I(u) & \geq \frac{1}{N} \left| \nabla u\right|_{L^N_{w_1}( \mathbb{R}^N)}^N + \frac{1}{q} \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^q - c_{15} \left(\left\|u\right\|_{q,1}^p + \left\|u\right\|_{q,1}^t\right)\nonumber\\ & \geq \frac{1}{N} \left(\left| \nabla u\right|_{L^N_{w_1}( \mathbb{R}^N)}^N + \left| \nabla u\right|_{L^q( \mathbb{R}^N)}^N\right) -c_{15} \left(\left\|u\right\|_{q,1}^p + \left\|u\right\|_{q,1}^t\right) \nonumber\\ & \geq \frac{2^{1-N}}{N} \left(\left| \nabla u\right|_{L^N_{w_1}( \mathbb{R}^N)} + \left| \nabla u\right|_{L^q( \mathbb{R}^N)}\right)^N-c_{15} \left(\left\|u\right\|_{q,1}^p + \left\|u\right\|_{q,1}^t\right) \nonumber\\ & \geq \frac{2^{1-N}}{N} \left\|u\right\|_{q,1}^N-c_{15} \left(\left\|u\right\|_{q,1}^p + \left\|u\right\|_{q,1}^t\right) \nonumber\\ & \geq \frac{2^{1-N}}{N} \rho_0^N - 2c_{15} \rho_0^p. \end{align}

Plainly, one could choose $\rho _0$ small enough such that $\rho _0 < \left (\frac {2^{1-N}}{2Nc_{15}}\right )^{\frac {1}{p-N}}.$ By (8.8), we deduce that

\[ I(u) \geq \frac{2^{1-N}}{N} \rho_0^N - 2c_{15} \rho_0^p = \rho_1 > 0,\ \forall\ u \in E_{q,1},\ \left\|u\right\|_{q,1} = \rho_0. \]

Now, for a fixed $\phi \in C_{0,rad}^{\infty }( \mathbb {R}^N) \backslash {\left \{0\right \}}$ such that $\phi \geq 0,$ we have

\[ I(t \phi) \to - \infty,\ \text{as}\ t \to + \infty. \]

In fact, from the hypothesis $(F_1),$ one can see that there exist two positive constants $c_{16}$ and $c_{17}$ such that

\[ F(s) \geq c_{16} s^{\theta} -c_{17},\ \forall\ s \geq 0. \]

Let

\[ c \!=\! \inf_{\gamma \in \Gamma} \max_{0 \leq t \leq 1} I( \gamma(t)) \!\geq\! \rho_1 \!>\! 0,\ \text{where}\ \Gamma\!=\! \left\{ \gamma \in C([0,1], E_{q,1}),\ \gamma(0) \!=\! 0,\ I(\gamma(1)) \!<\! 0\right\}, \]

the mountain pass level of the functional $I.$ In order to complete the proof of theorem 8.1, it suffices to show that there exists $A_0 > 0$ such that

(8.9)\begin{equation} c < \frac{\theta -N}{2^{N-1} N \theta} \min\left\{2^{N-1}, \frac{\omega_{N-1}}{\alpha_0^{N-1}}\right\},\ \forall\ A > A_0. \end{equation}

For that aim, fix a function $\phi _0 \in C_{0,rad}^{\infty }( \mathbb {R}^N)$ such that $\phi _0 \neq 0$ and $\phi _0 \geq 0.$ By $(F_3),$ we have

\begin{align*} I( \phi_0) & = \frac{\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N}{N} + \frac{\left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{q} - \int_{\mathbb{R}^N} F( \phi_0) {\rm d}x \\ & \quad \leq \frac{\left| \nabla \phi_0\right|_{L^N_{w_1}(\mathbb{R}^N)}^N}{N} + \frac{\left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{q} -A \left|\phi_0\right|_{L^r( \mathbb{R}^N)}^r. \end{align*}

We infer,

\[ I(\phi_0) < 0 ,\ \forall\ A > \frac{\frac{\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N}{N} + \frac{\left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{q}}{\left|\phi_0\right|_{L^r( \mathbb{R}^N)}^r}. \]

Hence, for $A > \frac {\frac {\left | \nabla \phi _0\right |_{L^N_{w_1}( \mathbb {R}^N)}^N}{N} + \frac {\left |\nabla \phi _0\right |_{L^q( \mathbb {R}^N)}^q}{q}}{\left |\phi _0\right |_{L^r( \mathbb {R}^N)}^r},$ the function $\gamma _0:[0,1] \rightarrow E_{q,1}$ defined by

\[ \gamma_0(t) = t \phi_0 \]

belongs to $\Gamma.$ Consequently,

\begin{align*} c & \leq \max_{0 \leq t \leq 1} I( t \phi_0) \\ & \leq \max_{0 \leq t \leq 1} \left(\frac{t^N\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N}{N} + \frac{t^q\left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{q} -A t^r\left|\phi_0\right|_{L^r( \mathbb{R}^N)}^r\right) \\ & \leq \max_{0 \leq t \leq 1} \left(\left(\frac{\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N}{N} + \frac{\left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{q}\right)t^q -A t^r\left|\phi_0\right|_{L^r( \mathbb{R}^N)}^r\right) \\ & =\left(\frac{\frac{q}{N}\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N + \left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{Ar\left|\phi_0\right|_{L^r( \mathbb{R}^N)}^r} \right)^{\frac{q}{r-q}} \left(\frac{\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N}{N} + \frac{\left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{q}\right)\left(1- \frac{q}{r}\right). \end{align*}

Clearly, there exists $A_0 > \frac {\frac {\left | \nabla \phi _0\right |_{L^N_{w_1}( \mathbb {R}^N)}^N}{N} + \frac {\left |\nabla \phi _0\right |_{L^q( \mathbb {R}^N)}^q}{q}}{\left |\phi _0\right |_{L^r( \mathbb {R}^N)}^r}$ large enough such that

\begin{align*} & \left(\frac{\frac{q}{N}\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N + \left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{Ar\left|\phi_0\right|_{L^r( \mathbb{R}^N)}^r} \right)^{\frac{q}{r-q}} \left(\frac{\left| \nabla \phi_0\right|_{L^N_{w_1}( \mathbb{R}^N)}^N}{N} + \frac{\left|\nabla \phi_0\right|_{L^q( \mathbb{R}^N)}^q}{q}\right)\left(1- \frac{q}{r}\right) \\ & < \frac{\theta -N}{2^{N-1} N \theta} \min\left\{2^{N-1}, \frac{\omega_{N-1}}{\alpha_0^{N-1}}\right\},\ \forall\ A > A_0. \end{align*}

This ends the proof of theorem 8.1.