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 $$\Omega \subseteq \mathbb {R}^N$$, $$N\ge 3$$, $$2^*={2N\over N-2}$$, $$2 0$$. 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 $${\varepsilon }$$.