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

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}$^N$,$N\ge 3$,$2^={2N\over N-2}$,$2<p<2^*$\mathrm{.}Here$\Omega$ can be an exterior domain, i.e. ${\mathbb{R}}^N\setminus \mathrm{\Omega }$ bounded, or the whole of ${\mathbb{R}}^N$. The potential $a\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{N\mathrm{/}2}({\mathbb{R}}^N)$ is assumed to be strictly positive and such that there exists ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}a(x):=a_{\mathrm{\infty }}$, with $a_{\mathrm{\infty }}>0$; in particular $a\equiv {\rm const}$ is allowed. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge a_{\mathrm{\infty }}$ is considered, with $a(x)\equiv ̸a_{\mathrm{\infty }}$ or $\mathrm{\Omega }\mathrm{\ne }{\mathbb{R}}^N$. 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 ${\mathbb{R}}^N\setminus \mathrm{\Omega }$ and on $\mathrm{\parallel }a-a_{\mathrm{\infty }}{\mathrm{\parallel }}_{L^{N\mathrm{/}2}}$.