Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

Abstract

We consider a singularly perturbed elliptic equation

where $V(x)>0$ for any $x\in {\mathbb{R}}^N.$ The singularly perturbed problem has corresponding limiting problems

Berestycki–Lions found almost necessary and sufficient conditions on nonlinearity $f$ for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential $V$ under possibly general conditions on $f$. In this paper, we prove that under the optimal conditions of Berestycki–Lions on $f\in C^1$, there exists a solution concentrating around topologically stable positive critical points of $V$, whose critical values are characterized by minimax methods.