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

Abstract

We consider a singularly perturbed elliptic equation \varepsilon^2\Delta u - V(x) u + f(u)=0, \ u(x) > 0 \text{ on } \mathbb R^N, \, \lim_{|x| \to \infty}u(x) = 0, where V(x) > 0 for any x \in \mathbb R^N. The singularly perturbed problem has corresponding limiting problems \Delta U - c U + f(U)=0, \ U(x) > 0 \text{ on } \ \mathbb R^N, \, \lim_{|x| \to \infty}U(x) = 0, \ c > 0. 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.