具有临界增长及Hardy项的半线性椭圆方程多解的存在性

数学物理学报 ›› 2016, Vol. 36 ›› Issue (6): 1137-1144.

具有临界增长及Hardy项的半线性椭圆方程多解的存在性

袁海华¹, 张正杰¹, 徐国进²

1. 华中师范大学数学与统计学学院武汉 430079;
2. 湖北工程学院数学与统计学院湖北孝感 432000

收稿日期:2016-03-23 修回日期:2016-10-18 出版日期:2016-12-26 发布日期:2016-12-26
通讯作者: 张正杰 E-mail:zjz@mail.ccnu.edu.cn
基金资助:
国家自然科学基金（11371159）和教育部高校长江创新团队（IRT13066）资助

Multiple Solutions for Semilinear Elliptic Equations with Critical Exponents and Hardy Potential

Yuan Haihua¹, Zhang Zhengjie¹, Xu Guojin²

1. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079;
2. School of Mathematics and Statistics, Hubei Engineering University, Hubei Xiaogan 432000

Received:2016-03-23 Revised:2016-10-18 Online:2016-12-26 Published:2016-12-26
Supported by:
Supported by the NSFC (11371159) and the Yangtze River Innovation Team of the Ministry Education in Colleges and Universities (IRT13066)

摘要/Abstract

摘要：

该文研究如下问题
-△u+（u）/（|x|²）=|u|^{2^*-2}u+g（x），x∈R^N，（0.1）
u（x）→0（|x|→∞），u∈D^1，2（R^N）
多解的存在性，这里g（x）≥0，g（x）≠0，g（x）∈L^{（2N）/（N+2）}（R^N）.证明了：存在常数C（适当小），如果‖g‖_{L^{（2N）/（N+2）}（R^N）}}≤C，则上述问题至少有两个解存在.

关键词: 半线性, Hardy项, 山路引理

Abstract:

In this paper, we consider the following problem
-△u+(u)/(|x|²)=|u|^{2^*-2}u+g(x),x∈R^N,
u(x)→0(|x|→∞),u∈D^1,2(R^N)
where g(x)≥0,g(x)≠0, 且g(x)∈L^(2N)/(N+2)(R^N). We can prove that there exists a constant C, which is small enough,such that ‖g‖_{L^(2N)/(N+2)(R^N)}≤C, then there are at least two solutions for the above problem.

Key words: Semi-linear, Hardy potential, Mountain pass lemma

中图分类号:

O175.23

袁海华, 张正杰, 徐国进. 具有临界增长及Hardy项的半线性椭圆方程多解的存在性[J]. 数学物理学报, 2016, 36(6): 1137-1144.

Yuan Haihua, Zhang Zhengjie, Xu Guojin. Multiple Solutions for Semilinear Elliptic Equations with Critical Exponents and Hardy Potential[J]. Acta mathematica scientia,Series A, 2016, 36(6): 1137-1144.

参考文献

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