Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in exterior domains

Abstract

The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e. $\mathbb{R}^N\setminus\Omega$ bounded, or the whole of $\mathbb{R}^N$. The potential $a\in L^{N/2}_{\rm loc}(\mathbb{R}^N)$ is assumed to be strictly positive and such that there exists $\lim_{|x|\to\infty}a(x):=a_\infty$, with $a_\infty>0$; in particular $a\equiv {\rm const}$ is allowed. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge a_\infty$ is considered, with $a(x)\not\equiv a_\infty$ or $\Omega\neq\mathbb{R}^N$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $\varepsilon$. No hypotheses are assumed on the size of $\mathbb{R}^N\setminus\Omega$ and on $\|a-a_\infty\|_{L^{N/2}}$.