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

Abstract

We consider a singularly perturbed elliptic equation ε 2Δu - V (x)u + f(u) = 0, u(x) > 0 on ℝN, lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ε ℝN: The singularly perturbed problem has corresponding limiting problems ΔU - cU + f(U) = 0, U(x) > 0 on ℝN, lim |x|→∞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 ε C1 In this paper, we prove that under the optimal conditions of Berestycki-Lions on f 2 C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods. © European Mathematical Society 2013.