Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

Guofa Li; Yisheng Huang

doi:10.3934/dcdsb.2021214

Article Contents

2022, Volume 27, Issue 7: 3971-3989. Doi: 10.3934/dcdsb.2021214

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Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

Guofa Li^1, , and
Yisheng Huang^2,

1.
College of Mathematics and Statistics, Qujing Normal University, Qujing 655011, Yunnan, China
2.
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China

^* Corresponding author: Guofa Li
^* Corresponding author: Guofa Li

Received: March 31, 2021

Revised: May 31, 2021

Published: July 2022

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971339, 11901345), the Yunnan Local Colleges Applied Basic Research Projects (Grant No. 202001BA070001-032) and Scientific and Technological Innovation Team of Nonlinear Analysis and Algebra with Their Applications in Universities of Yunnan Province, China (Grant No. 2020CXTD25)

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Abstract

In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations

$ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $

where $ \kappa>0 $, $ \lambda>0, \mu>0, h\in C(\mathbb{R}, \mathbb{R}) $ is superlinear at infinity, the potentials $ V(x) $ and $ K(x) $ are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful $ L^{\infty} $-estimates. For the subcritical case ($ \mu = 0 $) we can deal with large $ \kappa>0 $. For the critical case we treat that $ \kappa>0 $ is small.

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

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