一类带有Neumann边值条件的拟线性椭圆外部问题的多解性

数学物理学报 ›› 2015, Vol. 35 ›› Issue (5): 845-854.

一类带有Neumann边值条件的拟线性椭圆外部问题的多解性

宋洪雪^1,2, 闫庆伦¹

1 南京邮电大学理学院南京 210023;
2 河海大学理学院南京 210098

收稿日期:2014-05-19 修回日期:2015-03-04 出版日期:2015-10-25 发布日期:2015-10-25
作者简介:宋洪雪,songhx@njupt.edu.cn
基金资助:
国家自然科学基金(11201241,11401165,11201240)和江苏省高校"青蓝工程"项目(QL2014)资助

Multiple Solutions for Quasilinear Elliptic Exterior Problem with Neumann Boundary Conditions

Song Hongxue^1,2, Yan Qinglun¹

1 College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023;
2 College of Science, Hohai University, Nanjing 210098

Received:2014-05-19 Revised:2015-03-04 Online:2015-10-25 Published:2015-10-25

摘要/Abstract

摘要：

该文考虑下面的带有Neumann边值条件的拟线性椭圆外部问题

Ω,其中1< p< N, 1< q< p< r< p^*, p^*=Np/(N-p), Ω是欧几里德空间(R^N,|·|)(N ≥ 3)中的光滑外部区域,也就是说, Ω是某个带有C^1,δ(0< δ< 1)边界的有界区域Ω'的补集, n是其边界∂Ω的单位外法向量, λ是一个正参数.由山路引理和Ekeland变分原理,我们得出:当函数a(x), b(x), h₁(x), h₂(x)和g(x)满足一定的条件时,该方程至少有两个非平凡弱解.

关键词: 山路引理, 变分法, Neumann边值条件

Abstract:

In this paper, we consider the following quasi-linear elliptic exterior problem with Neumann boundary value conditions

where Ω is a smooth exterior domain of the Euclidean space(R^N,|·|)(N ≥ 3), and n is the unit vector of the outward normal on the boundary ∂Ω. λ is a positive parameter, 1< p< N, 1< q< p< r< p^*, p^*=Np/(N-p). By the mountain-pass theorem and Ekeland's variational principle, we establish the existence of two solutions for this problem when functions a(x), b(x), h₁(x), h₂(x) and g(x) satisfy certain conditions.

Key words: Mountain-pass Theorem, Variational methods, Neumann boundary conditions

中图分类号:

O175.23

宋洪雪, 闫庆伦. 一类带有Neumann边值条件的拟线性椭圆外部问题的多解性[J]. 数学物理学报, 2015, 35(5): 845-854.

Song Hongxue, Yan Qinglun. Multiple Solutions for Quasilinear Elliptic Exterior Problem with Neumann Boundary Conditions[J]. Acta mathematica scientia,Series A, 2015, 35(5): 845-854.

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