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

Abstract

We consider a singularly perturbed elliptic equation

\[ \e^2\Delta u - V(x) u + f(u)=0, \ u(x) > 0 \textrm{ on } \RN, \, \lim_{|x| \to \infty}u(x) = 0, \]

where $V(x)>0$ for any \( x \in \RN. \) The singularly perturbed problem has corresponding limiting problems

\[ \Delta U - c U + f(U)=0, \ U(x) > 0 \textrm{ on } \ \RN, \, \lim_{|x| \to \infty}U(x) = 0, \ c > 0. \]

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.