Jaeyoung Byeon

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

We consider a singularly perturbed elliptic equation ε21u− V (x)u+ f (u) = 0, u(x) > 0 on R , lim |x|→∞ u(x) = 0, where V (x) > 0 for any x ∈ RN . The singularly perturbed problem has corresponding limiting problems 1U − cU + f (U) = 0, U(x) > 0 on R , 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 . In this paper, we prove that under the optimal conditions of Berestycki–Lions on f ∈ C1, there exists a solution concentrating around topologically stable positive critical points of V , whose critical values are characterized by minimax methods.