Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity

Jianhua Chen; Xianhua Tang; Bitao Cheng

doi:10.3934/cpaa.2019025

Article Contents

2019, Volume 18, Issue 1: 493-517. Doi: 10.3934/cpaa.2019025

This issue Previous Article Subseries and signed series Next Article Semiclassical states for fractional Choquard equations with critical growth

Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity

1.
Department of Mathematics, Nanchang University, Nanchang, 330031 Jiangxi, China
2.
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China
3.
School of Mathematics and Statistics, Qujing Normal University, Qujing, 655011 Yunnan, China

* Corresponding author
* Corresponding author

Received: February 28, 2018

Revised: April 30, 2018

Published: August 2018

This work was supported by National Natural Science Foundation of China (Grant No. 11461043, 11571370 and 11601525), and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009) and the Outstanding Youth Scientist Foundation Plan of Jiangxi (20171BCB23004), Yunnan Local Colleges Applied Basic Research Projects (2017FH001-011) and Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2016B037).

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Abstract

In this paper, we study the following quasilinear Schrödinger equation

$\begin{equation*}-Δ u+V(x)u-Δ(u^2)u = g(u),\,\, x∈\mathbb{R}^N,\end{equation*}$

where $ N>4, 2^* = \frac{2N}{N-2}, V: \mathbb{R}^N \to \mathbb{R}$ satisfies suitable assumptions. Unlike $ g∈ \mathcal{C}^1(\mathbb{R},\mathbb{R})$, we only need to assume that $ g∈ \mathcal{C}(\mathbb{R},\mathbb{R})$. By using a change of variable, we obtain the existence of ground state solutions with general critical growth. Our results extend some known results.

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Mathematics Subject Classification: 35J50, 35J60.

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