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

### Jaeyoung Byeon

KAIST, Daejeon, South Korea### Kazunaga Tanaka

Waseda University, Tokyo, Japan

## Abstract

We consider a singularly perturbed elliptic equation

where $V(x) > 0$ for any $x \in \RN.$ 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.

## Cite this article

Jaeyoung Byeon, Kazunaga Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1859–1899

DOI 10.4171/JEMS/407