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

Abstract

In this paper, we will study the existence and qualitative property of standing waves $\psi (x,t)=e^{-\frac{iEt}{\epsilon }}u(x)$ for the nonlinear Schrödinger equation $i\epsilon \frac{\partial \psi }{\partial t}+\frac{{\epsilon }^2}{2m}{\mathrm{\Delta }}_x\psi -(V(x)+E)\psi +K(x)|\psi {|}^{p-1}\psi =0$, $(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^N$. Let $G(x)=[V(x){]}^{\frac{p+1}{p-1}-\frac{N}{2}}[K(x){]}^{-\frac{2}{p-1}}$ and suppose that $G(x)$ has $k$ local minimum points. Then, for any $l\in \{1,\dots ,k\}$, we prove the existence of the standing waves in $H^1({\mathbb{R}}^N)$ having exactly $l$ local maximum points which concentrate near $l$ local minimum points of $G(x)$ respectively as $\epsilon \to 0$. 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].