In this paper we prove that the best constant in the Sobolev trace embedding $H^1(\varOmega)\hookrightarrow L^q(\partial\varOmega)$ in a bounded smooth domain can be obtained as the limit as $\varepsilon\to0$ of the best constant of the usual Sobolev embedding $H^1(\varOmega) \hookrightarrow L^q(\omega_\varepsilon,\mathrm{d} x/\varepsilon)$, where $\omega_\varepsilon=\{x\in\varOmega:\mathrm{dist}(x,\partial\varOmega)<\varepsilon\}$ is a small neighbourhood of the boundary. We also analyse symmetry properties of extremals of the latter embedding when $\varOmega$ is a ball.

We present some transference results for a convolution operator with kernel $K$ which is bounded from $L^{p_0}(w_0)$ into $L^{p_1}(w_1)$. Our results are a natural extension of the classical results of Coifman and Weiss and their further extensions. We shall give several applications in the setting of restriction of Fourier multipliers.

We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow–fast two-saddle cycle, formed by a regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral.

This paper is contributed to the inhomogeneous semilinear elliptic equation

\begin{equation} \Delta u+K(|x|)u^{p}+f(x)=0\quad\text{in }\mathbb{R}^n, \tag{$*$} \label{*} \end{equation}

where

$$ \Delta=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}} $$

is the Laplacian operator, $n\geq3$, $p>1$, $f(x)\geq0$ and $K(|x|)>0$ is a given locally Hölder continuous function in $\mathbb{R}^n\setminus\{0\}$. The existence, non-existence and decay properties of positive solutions for \eqref{*} are obtained under some assumptions on $f(x)$ and $K(|x|)$ satisfying the slow-decay condition, i.e. $K(|x|)\geq C|x|^{l}$ at infinity for some constants $C>0$ and $l>-2$. The decay properties of positive solutions for $(\ast)$ are also discussed for the critical decay case on $K(|x|)$ with $l=-2$.

We investigate the behaviour of the eigenvalues of a self-adjoint Sturm–Liouville problem with a separated boundary condition when the interval of the problem shrinks to an end point. It is shown that all the eigenvalues, except possibly the first, approach $+\infty$. The choices of the boundary condition are found for which the first eigenvalue tends to $+\infty$, independent of the coefficient functions, and the same is done for the $-\infty$ limit. For the remaining choices of the boundary condition, several types of condition on the coefficient functions are given, so that the first eigenvalue has a finite or infinite limit and, when the limit is finite, an explicit expression for the limit is obtained. Moreover, numerous examples are presented to illustrate these results, and a construction is given to perturb the finite-limit case to the no-limit case.

By using the moving plane method combined with integral inequalities and Hardy's inequality, some new Liouville-type theorems for semilinear polyharmonic equations in $\mathbb{R}^N$ and in $\mathbb{R}^N_+$ are proved.

A physically transparent and mathematically streamlined derivation is presented for a third-order nonlinear dynamical system that describes the curious chiral reversals of a celt (rattleback). The system is integrable, and its solutions are periodic, showing an infinite succession of spin reversals. Inclusion of linear dissipation allows any given number of reversals, and a typical celt's observed behaviour is well captured by tuning the dissipation parameters.

We prove the existence of the global smooth solutions to $p$-systems with damping under weaker conditions than those given by Lin and Zheng. The analysis is based on several key a priori estimates, which are obtained via the maximum principle.

We consider the Cauchy problem for quasilinear parabolic equations $u_t=\Delta\phi(u)+f(u)$, with the bounded non-negative initial data $u_0(x)$ ($u_0(x)\not\equiv0$), where $f(\xi)$ is a positive function in $\xi>0$ satisfying a blow-up condition $\int_1^{\infty}1/f(\xi)\,\mathrm{d}\xi<\infty$. We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation $\mathrm{d} v/\mathrm{d} t=f(v)$ with the initial data $\|u_0\|_{L^{\infty}(\mathbb{R}^N)}>0$. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on $u_0$ for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on $u_0$ for blow-up with the least blow-up time, provided that $f(\xi)$ grows more rapidly than $\phi(\xi)$.

The global entropy solution to the radial motion of isothermal self-gravitating gases is studied. A transformation of variables is applied. The global approximate solutions are constructed by a generalized shock-capturing scheme. The source terms include an integral which is non-local and is carefully estimated. The uniform estimate of the approximate solutions is obtained. The convergence of the approximate solutions is proved by the compensated compactness method. The global existence of weak entropy solution with large $L^{\infty}$ initial data is established. The solution allows a possible vacuum state.

We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.