Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

Yanfang Xue; Chunlei Tang

doi:10.3934/cpaa.2018054

Article Contents

2018, Volume 17, Issue 3: 1121-1145. Doi: 10.3934/cpaa.2018054

This issue Previous Article A nonlocal concave-convex problem with nonlocal mixed boundary data Next Article Sign-changing solutions for non-local elliptic equations with asymptotically linear term

Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

Yanfang Xue^1,2, and
Chunlei Tang^1, ,

1.
School of Mathematics and Statistics, Southwest University, Chongqing, 400700, China
2.
School of Mathematics and Statistics, Xin-Yang Normal University, Xinyang, 464000, China

* Corresponding author: Chunlei Tang
* Corresponding author: Chunlei Tang

Received: January 2017

Revised: November 2017

Published: May 2018

The research is supported by National Natural Science Foundation of China(No. 11471267,11601438) and Chongqing Research Program of Basic Research and Frontier Technology(No.cstc2017jcyjAX0331)

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Abstract

In this paper, we are concerned with the existence of ground state solutions for the following quasilinear Schrödinger equation:

$-Δ u+V(x)u-Δ (u^2)u = K(x)|u|^{22^*-2}u+g(x,u), \ \ x∈ \mathbb{R}^N\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$

where $N≥ 3$, $V, \ g$ are asymptotically periodic functions in $x$. By combining variational methods and the concentration-compactness principle, we obtain a ground state solution for equation (1) under a new reformative condition which unify the asymptotic processes of $V, g $ at infinity.

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Mathematics Subject Classification: Primary: 35A15, 35B33, 35J60.

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