Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials

In this paper, we will study the existence and qualitative property of standing waves for the nonlinear Schrödinger equation , (t,x)∈R+×RN. Let and suppose that G(x) has k local minimum points. Then, for any l∈{1,…,k}, we prove the existence of the standing waves in H1(RN) having exactly l local maximum points which concentrate near l local minimum points of G(x) respectively as ε→0. The potentials V(x) and K(x) are allowed to be either compactly supported or unbounded at infinity. Therefore, we give a positive answer to a problem proposed by Ambrosetti and Malchiodi (2007) [2].