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

Abstract

We consider a singularly perturbed elliptic equation epsilon(2)Delta u - V(x)u + f(u) = 0, u(x) > 0 on R-N, lim(vertical bar x vertical bar ->infinity) u(x) = 0, where V(x) > 0 for any x is an element of R-N. The singularly perturbed problem has corresponding limiting problems Delta U - cU + f(U) = 0, U(x) > 0 on R-N, lim(vertical bar x vertical bar ->infinity) u(x) = 0, c > 0. Berestycki-Lions [3] found almost necessary and sufficient conditions on the 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 the potential V under possibly general conditions on f. In this paper, we prove that under the optimal conditions of Berestycki-Lions on f is an element of C-1, there exists a solution concentrating around topologically stable positive critical points of V, whose critical values are characterized by minimax methods.