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 $-\Delta u+a(x)u=u^{p-1}+\epsilon u^{2^{\ast }-1}$ in $\Omega \subseteq {\mathbb{R}}^N$, $N\ge 3$, $2^{\ast }=\frac{2N}{N-2}$, $2<p<2^{\ast }$. Here $\Omega$ can be an exterior domain, i.e. ${\mathbb{R}}^N\setminus \Omega$ bounded, or the whole of ${\mathbb{R}}^N$. The potential $a\in L_{\mathrm{loc}}^{N/2}({\mathbb{R}}^N)$ 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 $a\equiv \mathrm{const}$ 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 $\Omega \ne {\mathbb{R}}^N$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $\epsilon$. No hypotheses are assumed on the size of ${\mathbb{R}}^N\setminus \Omega$ and on $\Vert a-a_{\infty }{\Vert }_{L^{N/2}}$.