Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential

Hongxia Shi; Haibo Chen

doi:10.3934/cpaa.2018004

Article Contents

2018, Volume 17, Issue 1: 53-66. Doi: 10.3934/cpaa.2018004

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Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential

Hongxia Shi^1, and
Haibo Chen^2, ,

1.
School of Mathematics and Computational Science, Hunan First Normal University, Changsha, 410205 Hunan, China
2.
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

* Corresponding author
* Corresponding author

Received: October 31, 2016

Revised: June 30, 2017

Published: December 2017

This research was supported by National Natural Science Foundation of China 11671403 and by the Mathematics and Interdisciplinary Sciences project of CSU.

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Abstract

We investigate a class of generalized quasilinear Schrödinger equations

$ -{\rm div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+V(x)u=f(x,u) \mbox{ in }\mathbb{R}^{N},$

where $ g(u):\mathbb{R}\to\mathbb{R}^{+} $ is a nondecreasing function with respect to $ |u| $, the potential $ V(x) $ and the primitive of the nonlinearity $ f(x,u) $ are allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of infinitely many nontrivial solutions. The proof is based on a change of variable as well as symmetric Mountain Pass Theorem.

Keywords:

Mathematics Subject Classification: 35J20, 35J60.

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