一类带临界指数的Kirchhoff型问题正基态解的存在性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (2): 418-426.

一类带临界指数的Kirchhoff型问题正基态解的存在性

吉蕾¹(),廖家锋^2,*()

¹ 晋中学院数学系山西晋中 030600
² 西华师范大学公共数学学院四川南充 637002

收稿日期:2020-10-28 出版日期:2022-04-26 发布日期:2022-04-18
通讯作者: 廖家锋 E-mail:jilei@jzxy.edu.cn;liaojiafeng@163.com
作者简介:吉蕾, E-mail: jilei@jzxy.edu.cn
基金资助:
四川省教育厅自然科学重点资助科研项目(18ZA0471);西华师范大学国家一般培育项目(18B015);西华师范大学创新团队项目(CXTD2018-8)

Existence of Positive Ground State Solutions for a Class of Kirchhoff Type Problems with Critical Exponent

Lei Ji¹(),Jiafeng Liao^2,*()

¹ Department of Mathematics, Jingzhong University, Shanxi Jingzhong 030600
² College of Mathematics Education, China West Normal University, Sichuan Nanchong 637002

Received:2020-10-28 Online:2022-04-26 Published:2022-04-18
Contact: Jiafeng Liao E-mail:jilei@jzxy.edu.cn;liaojiafeng@163.com
Supported by:
the Scientific Research Fund of Sichuan Provincial Education Department(18ZA0471);the Fundamental Research Funds of China West Normal University(18B015);the Innovative Research Team of China West Normal University(CXTD2018-8)

摘要/Abstract

摘要：

研究了一类带临界指数的Kirchhoff型方程 $\left\{\begin{array}{ll}\displaystyle-\Big[a+b\Big(\int_{\Omega}|\nabla u|^2{\rm d}x\Big)^{m}\Big]\Delta u= |u|^{2^{\ast}-2}u+\lambda|u|^{q-2}u, \ & x \in \Omega, \\u=0, & x\in\partial\Omega,\end{array}\right.$ 其中 $\Omega\subset \mathbb{R} ^{N}(N\geq3)$ 是一个具有光滑边界 $\partial\Omega$ 的有界区域, $a, \lambda>0, b\geq0, 0 < m < \frac{2}{N-2}, 2 < q < 2m+2, 2^{\ast}=\frac{2N}{N-2}$ 为参量. 利用山路引理获得了该问题正解的存在性. 进一步, 利用Nehari方法获得该问题正基态解的存在性. 该结果补充并完善了近期相关结果.

关键词: Kirchhoff型问题, 临界指数, 山路引理, Nehari方法, 正基态解

Abstract:

The following Kirchhoff type equations with critical exponent $\begin{eqnarray*}\left\{\begin{array}{ll}\displaystyle-\Big[a+b\Big(\int_{\Omega}|\nabla u|^2{\rm d}x\Big)^{m}\Big]\Delta u= |u|^{2^{\ast}-2}u+\lambda|u|^{q-2}u,\ & x \in \Omega, \\u=0, & x\in\partial\Omega,\end{array}\right.\end{eqnarray*}$ where $\Omega\subset\mathbb{R} ^{N}(N\geq3)$ is a smooth bounded domain with smooth boundary $\partial\Omega,$ $a, \lambda>0,$ $b\geq0,$ $0 < m < \frac{2}{N-2}, 2 < q < 2m+2, 2^{\ast}=\frac{2N}{N-2}$ are parameters, are considered. By using the Moutain-Pass Theorem, the existence positive solutions is obtained. Moreover, the existence positive ground state solutions is obtained by the Nehari method. Our result completes and improves the recent corresponding results in the literature.

Key words: Kirchhoff type problem, Critical exponent, Mountain-Pass Theorem, Nehari method, Positive ground state solution

中图分类号:

O177.91

吉蕾,廖家锋. 一类带临界指数的Kirchhoff型问题正基态解的存在性[J]. 数学物理学报, 2022, 42(2): 418-426.

Lei Ji,Jiafeng Liao. Existence of Positive Ground State Solutions for a Class of Kirchhoff Type Problems with Critical Exponent[J]. Acta mathematica scientia,Series A, 2022, 42(2): 418-426.

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