Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials

Zhongwei Tang

doi:10.3934/cpaa.2014.13.237

Article Contents

2014, Volume 13, Issue 1: 237-248. Doi: 10.3934/cpaa.2014.13.237

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Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials

Zhongwei Tang^1,

1.
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received: October 2012

Revised: May 2013

Published: January 2014

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Abstract

In this paper, we consider the following semilinear Schrödinger equations with ciritical growth \begin{eqnarray} -\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u,x\in R^N, \end{eqnarray} where $N\geq 4$, $a(x)\geq 0$ and its zero sets are not empty. $2^*$ is the critical Sobolev exponent. $\delta>0$ is a constant such that the operator $-\Delta +\lambda a(x)-\delta$ might be indefinite but is non-degenerate. We prove the existence of least energy solutions which localize near the potential well $int \{a^{-1}(0)\}$ for $\lambda$ large enough.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J65.

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