Standing waves for nonlinear Schrödinger equations involving critical growth

Abstract

We consider the following singularly perturbed nonlinear elliptic problem: - ε 2 Δ u + V ( x ) u = f ( u ) , u ∈ H 1 ( R N ) , where N ⩾ 3 and f is the nonlinearity of critical growth. In this paper, we construct a solution u ε of the above problem, which concentrates at an isolated component of the positive local minimum points of V as ε → 0 under certain conditions on f. Our result completes the study made in some very recent works in the sense that, in those papers, only the subcritical growth was considered.