We establish multiplicity results for the following class of quasilinear problems
P
\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*} where
$\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$ for a generalized N-function
$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$. We consider
$\Omega \subset \mathbb {R}^{N}$ to be a smooth bounded domain that contains two disjoint open regions
$\Omega _N$ and
$\Omega _p$ such that
$\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$. The main feature of the problem
$(P)$ is that the operator
$-\Delta _{\Phi }$ behaves like
$-\Delta _N$ on
$\Omega _N$ and
$-\Delta _p$ on
$\Omega _p$. We assume the nonlinearity
$f:\Omega \times \mathbb {R}\to \mathbb {R}$ of two different types, but both behave like
$e^{\alpha |t|^{\frac {N}{N-1}}}$ on
$\Omega _N$ and
$|t|^{p^{*}-2}t$ on
$\Omega _p$ as
$|t|$ is large enough, for some
$\alpha >0$ and
$p^{*}=\frac {Np}{N-p}$ being the critical Sobolev exponent for
$1< p< N$. In this context, for one type of nonlinearity
$f$, we provide a multiplicity of solutions in a general smooth bounded domain and for another type of nonlinearity
$f$, in an annular domain
$\Omega$, we establish existence of multiple solutions for the problem
$(P)$ that are non-radial and rotationally non-equivalent.