Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded 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. $\mathbb {R}^N{\setminus }\Omega$ is bounded, or the whole of $\mathbb {R}^N$. The potential $a\in L^{N/2}_{\mathop {\mathrm{loc}}\nolimits }(\mathbb {R}^N)$ is assumed to be strictly positive and such that there exists $\lim _{|x|\rightarrow \infty }a(x):=a_\infty >0$. First, some existence results of ground state solutions are proved. Then the case $a(x)\ge a_\infty$ is considered, with $a(x)\not \equiv a_\infty$ 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 ${\varepsilon }$.