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 -Δu+a(x)u=up-1+εu2∗-1 in Ω ⊆ RN, N≥ 3 , 2∗=2NN-2, 2 < p< 2 ∗. Here Ω can be an exterior domain, i.e. RN Ω is bounded, or the whole of RN. The potential a∈LlocN/2(RN) is assumed to be strictly positive and such that there exists lim |x|→∞a(x) : = a∞> 0. First, some existence results of ground state solutions are proved. Then the case a(x) ≥ a∞ is considered, with a(x) ≢ a∞ or Ω ≠ RN. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small ε.