Semi-Classical Bound States for Schrödinger Equations with Potentials Vanishing or Unbounded at Infinity

Abstract

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation . Let . For any integer k ≥ 1, we prove existence of standing wave solutions with u > 0 having k local maximum points and concentrating near a given local maximum point of Γ when ϵ is small. The potentials V and K are allowed to be either vanishing or unbounded at infinity. Existence of solutions concentrating near k distinct non-degenerate critical points of Γ has been proved under the same assumptions on V and K as well.