一类带临界指数的非齐次Kirchhoff型问题第二个正解的存在性

数学物理学报 ›› 2019, Vol. 39 ›› Issue (5): 1094-1101.

一类带临界指数的非齐次Kirchhoff型问题第二个正解的存在性

吉蕾¹(),廖家锋^2,*()

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

收稿日期:2018-08-29 出版日期:2019-10-26 发布日期:2019-11-08
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);西华师范大学英才科研项目(17YC383);西华师范大学基本科研项目(18B015);西华师范大学创新团队项目(CXTD2018-8)

Existence of the Second Positive Solution for a Class of Nonhomogeneous Kirchhoff Type Problems with Critical Exponent

Lei Ji¹(),Jiafeng Liao^2,*()

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

Received:2018-08-29 Online:2019-10-26 Published:2019-11-08
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 Meritocracy Research Funds of China West Normal University(17YC383);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非齐次Kirchhoff型方程

$\left\{\begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2{\rm d}x)\Delta u=|u|^{4}u+\lambda f(x), & \ \ x \in \Omega, \\u=0, & \ \ x\in\partial\Omega, \end{array}\right.$

其中$\Omega\subset {\Bbb R} ^{3}$是一个非空有界开集, $a\geq0, b, \lambda>0$为参量, $f\in L^{\frac{6}{5}}(\Omega)$是个非零非负函数.利用变分方法获得了该问题第二个正解,补充并完善了文献[3]和[5]的结果.

关键词: Kirchhoff型问题, 临界指数, 变分法, 正解

Abstract:

The following nonhomogeneous Kirchhoff type equations with critical exponent

$\begin{eqnarray*}\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2{\rm d}x)\Delta u=|u|^{4}u+\lambda f(x), & \ \ x \in \Omega, \\u=0, & \ \ x\in\partial\Omega, \end{array}\right. \end{eqnarray*} $

where $\Omega$ is a smooth bounded domain in ${\Bbb R} ^{3}, $ $a, b, \lambda>0$ are parameters and $f\in L^{\frac{6}{5}}(\Omega)$ is nonzero and nonnegative, are considered. By the variational method, the second positive solution is obtained which completes and improves the resulsts of [3] and [5].

Key words: Kirchhoff equation, Critical exponent, Variational method, Positive solution

中图分类号:

O177.25

吉蕾,廖家锋. 一类带临界指数的非齐次Kirchhoff型问题第二个正解的存在性[J]. 数学物理学报, 2019, 39(5): 1094-1101.

Lei Ji,Jiafeng Liao. Existence of the Second Positive Solution for a Class of Nonhomogeneous Kirchhoff Type Problems with Critical Exponent[J]. Acta mathematica scientia,Series A, 2019, 39(5): 1094-1101.

参考文献 18

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