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

The paper concerns with positive solutions of problems of the type $-Delta u+a(x), u=u^{p-1}+arepsilon u^{2^*-1}$ in $OmegasubseteqR^N$, $Nge 3$, $2^*={2Nover N-2}$, $2<2^*$. Here $Omega$ can be an exterior domain, i.e. $R^NsetminusOmega$ is bounded, or the whole of $R^N$. The potential $ain L^{N/2}_{loc}(R^N)$ is assumed to be strictly positive and such that there exists $lim_{|x| oinfty}a(x):=a_infty>0$. First, some existence results of ground state solutions are proved. Then the case $a(x)ge a_infty$ is considered, with $a(x) otequiv a_infty$ or $Omega eqR^N$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $e$.