Positive solutions for quasilinear Schrödinger equationswith critical growth and potential vanishing at infinity

Yinbin Deng; Wei Shuai

doi:10.3934/cpaa.2014.13.2273

Article Contents

2014, Volume 13, Issue 6: 2273-2287. Doi: 10.3934/cpaa.2014.13.2273

This issue Previous Article Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case Next Article Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold

Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity

Yinbin Deng^1, and
Wei Shuai^2,

1.
Department of Mathematics, Huazhong Normal University, Wuhan 430079
2.
Department of Mathematics, Huazhong Normal University, Wuhan, 430079

Received: August 2013

Revised: May 2014

Published online: July 2014

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Abstract

This paper is concerned with the existence of positive solutions for a class of quasilinear Schrödinger equations in $R^N$ with critical growth and potential vanishing at infinity. By using a change of variables, the quasilinear equations are reduced to semilinear one. Since the potential vanish at infinity, the associated functionals are still not well defined in the usual Sobolev space. So we have to work in the weighted Sobolev spaces, by Hardy-type inequality, then the functionals are well defined in the weighted Sobolev space and satisfy the geometric conditions of the Mountain Pass Theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution $v$. In the proof that $v$ is nontrivial, the main tool is classical arguments used by H. Brezis and L. Nirenberg in [13].

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Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35Q55.

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