Existence and multiplicity of semiclassical states for a  quasilinear Schrödinger equation in $\mathbb{R}^N$

Minbo Yang; Yanheng Ding

doi:10.3934/cpaa.2013.12.429

Article Contents

2013, Volume 12, Issue 1: 429-449. Doi: 10.3934/cpaa.2013.12.429

This issue Previous Article A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth Next Article Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros

Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$

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: April 30, 2011

Revised: February 29, 2012

Published: December 2012

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Abstract

In this paper we consider the following modified version of nonlinear Schrödinger equation:

$-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=g(x,u) $

in $\mathbb{R}^N$, $N\geq 3$ and $g(x,u)$ is a superlinear but subcritical function. Applying variational methods we show the existence and multiplicity of solutions provided $\varepsilon$ is sufficiently small enough.

Keywords:

Mathematics Subject Classification: Primary: 35J20, 35J60, 35Q55.

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