EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHR&#214;DINGER EQUATIONS IN $\mathbb{R}^N$

Jianhua CHEN; Xianjiu HUANG; Bitao CHENG; Xianhua TANG

doi:10.1007/s10473-020-0519-5

2020 , Vol. 40 >Issue 5: 1495 - 1524

DOI: https://doi.org/10.1007/s10473-020-0519-5

Articles

EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$

Jianhua CHEN ,
Xianjiu HUANG ,
Bitao CHENG ,
Xianhua TANG

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

Received date: 2018-09-27

Revised date: 2020-06-01

Online published: 2020-11-04

Supported by

This work was supported by the National Natural Science Foundation of China (11661053, 11771198, 11901345, 11901276, 11961045 and 11971485), and partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009 and 20181BAB201003), the Outstanding Youth Scientist Foundation Plan of Jiangxi (20171BCB23004), and the Yunnan Local Colleges Applied Basic Research Projects (2017FH001-011).

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Abstract

In this article, we study the generalized quasilinear Schrödinger equation \begin{equation*} -\text{div}(\varepsilon^2g^2(u)\nabla u)+\varepsilon^2g(u)g'(u)|\nabla u|^2+V(x)u=K(x)|u|^{p-2}u,\,\, x\in\mathbb{R}^N, \end{equation*} where $N\geq3$, $\varepsilon>0$, $4 < p < 22^*$, $g\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^{+})$, $V\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global minimum, and $K\in \mathcal{C}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem and establish a phenomenon of exponential decay.

Key words： generalized quasilinear Schrödinger equation; ground state solutions; existence; concentration behavior

Cite this article

Jianhua CHEN , Xianjiu HUANG , Bitao CHENG , Xianhua TANG . EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$[J]. Acta mathematica scientia, Series B, 2020 , 40(5) : 1495 -1524 . DOI: 10.1007/s10473-020-0519-5

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