数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1495-1524.doi: 10.1007/s10473-020-0519-5

• 论文 • 上一篇    下一篇

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

陈建华1, 黄先玖1, 程毕陶2, 唐先华3   

  1. 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
  • 收稿日期:2018-09-27 修回日期:2020-06-01 出版日期:2020-10-25 发布日期:2020-11-04
  • 通讯作者: Xianjiu HUANG E-mail:xjhuangxwen@163.com
  • 作者简介:Jianhua CHEN,E-mail:cjh19881129@163.com;Bitao CHENG,E-mail:chengbitao2006@126.com;Xianhua TANG,E-mail:tangxh@mail.csu.edu.cn
  • 基金资助:
    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).

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

Jianhua CHEN1, Xianjiu HUANG1, Bitao CHENG2, Xianhua TANG3   

  1. 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:2018-09-27 Revised:2020-06-01 Online:2020-10-25 Published:2020-11-04
  • Contact: Xianjiu HUANG E-mail:xjhuangxwen@163.com
  • 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).

摘要: 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.

关键词: generalized quasilinear Schrödinger equation, ground state solutions, existence, concentration behavior

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

中图分类号: 

  • 35J60