Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part

Minbo Yang; Yanheng Ding

doi:10.3934/cpaa.2013.12.771

Article Contents

2013, Volume 12, Issue 2: 771-783. Doi: 10.3934/cpaa.2013.12.771

This issue Previous Article The Riemann problem of conservation laws in magnetogasdynamics Next Article Multiplicity results for a class of elliptic problems with nonlinear boundary condition

Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part

Minbo Yang^1, and
Yanheng Ding^2,

1.
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004
2.
Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080

Received: June 30, 2011

Revised: October 31, 2011

Published: February 2013

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Abstract

In the present paper we study the existence of solutions for a nonlocal Schrödinger equation \begin{eqnarray*} -\varepsilon^2\Delta u +V(x)u =(\int_{R^3} \frac{|u|^{p}}{|x-y|^{\mu}}dy)|u|^{p-2}u, \end{eqnarray*} where $0 < \mu < 3$ and $\frac{6-\mu}{3} < p < {6-\mu}$. Under suitable assumptions on the potential $V(x)$, if the parameter $\varepsilon$ is small enough, we prove the existence of solutions by using Mountain-Pass Theorem.

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Mathematics Subject Classification: Primary: 35J50, 35J60, 35J25.

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