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), p. 1205-1226 / Harvested from Numdam

Résumé

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].

Détail

Publié le : 2010-01-01

Zbl 1200.35282

DOI : https://doi.org/10.1016/j.anihpc.2010.05.003
Classification:  35J20,  35J60

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