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

Ba, Na; Deng, Yinbin; Peng, Shuangjie

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1205-1226.

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

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