含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性

含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性

The Existence of Infinitely Many Solutions for an Elliptic

Equation Involving Critical Sobolev-Hardy Exponent

with Neumann Boundary Condition

摘要/Abstract

参考文献

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数学物理学报

胡爱莲;张正杰

喀什师范学院数理系新疆喀什 844007

收稿日期:2005-12-14 修回日期:2006-11-15 出版日期:2007-12-25 发布日期:2007-12-25
通讯作者: 胡爱莲
基金资助:
新疆高校科研计划重点项目(XJEDU2004I58)资助

Hu Ailian;Zhang Zhengjie

Department of Mathematics, Kashi Teacher's College, Kashi 844007

Received:2005-12-14 Revised:2006-11-15 Online:2007-12-25 Published:2007-12-25
Contact: Hu Ailian

摘要： 该文研究了如下的奇异椭圆方程Neumann问题

{\begin{cases} \disp - Δ u - \frac{μ u}{| x |^{2}} = \frac{| u |^{2^{*} (s) - 2} u}{| x |^{s}} + λ | u |^{q - 2} u, & x \in Ω, \\ D_{γ} u + α (x) u = 0, & x \in \partial Ω ∖ {0}, \end{cases}

$\left\{ \begin{array}{ll} \disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u, \ \ &x\in\Omega,\\ D_\gamma{u}+\alpha(x)u=0,&x\in\partial\Omega\backslash\{0\}, \end{array}\right.$ 其中

Ω

$\Omega$ 是

R^{N}

$R^N$ 中具有

C^{1}

$C^1$ 边界的有界区域,

0 \in \partial Ω

$0\in\partial\Omega$ ,

N \geq 5

$N\ge5$ .

2^{*} (s) = \frac{2 (N - s)}{N - 2}

$2^{*}(s)=\frac{2(N-s)}{N-2}$ (

0 \leq s \leq 2

$0\leq s\leq 2$ ) 是临界 Sobolev-Hardy 指标, $10$.利用变分方法和对偶喷泉定理, 证明了这个方程无穷多解的存在性.

关键词: Neumann问题, 临界 Sobolev-Hardy 指标, (ps)_c*条件, 对偶喷泉定理

Abstract: This paper deals with the Neumann problem for an elliptic
equation

{\begin{cases} \disp - Δ u - \frac{μ u}{| x |^{2}} = \frac{| u |^{2^{*} (s) - 2} u}{| x |^{s}} + λ | u |^{q - 2} u, & x \in Ω, \\ D_{γ} u + α (x) u = 0, & x \in \partial Ω ∖ {0}, \end{cases}

$\left\{ \begin{array}{ll} \disp -\Delta u-\frac{\mu u}{|x|^2}=\frac{|u|^{2^{*}(s)-2}u}{|x|^s}+\lambda|u|^{q-2}u, \ \ &x\in\Omega,\\ D_\gamma{u}+\alpha(x)u=0, &x\in\partial\Omega\backslash\{0\}, \end{array} \right.$
where

Ω

$\Omega$ is a bounded domain in

R^{N}

$R^N$ with

C^{1}

$C^1$
boundary,

0 \in \partial Ω

$0\in\partial\Omega$ ,

N \geq 5

$N\ge5$ .

2^{*} (s) = \frac{2 (N - s)}{N - 2}

$2^{*}(s)=\frac{2(N-s)}{N-2}$ (

0 \leq s \leq 2

$0\leq s\leq 2$ ) is the critical
Sobolev-Hardy exponent, $1 the unit outward normal to boundary

\partial Ω

$\partial\Omega$ . By
variational method and the dual fountain theorem, the existence of
infinitely many solutions with negative energy is proved.

Key words: Neumann problem, Critical Sobolev-Hardy exponent, (ps)_c*condition, Dual fountain theorem

中图分类号:

35J25

胡爱莲;张正杰. 含有Sobolev-Hardy临界指标的奇异椭圆方程Neumann问题无穷多解的存在性[J]. 数学物理学报, 2007, 27(6): 1025-1034.

Hu Ailian;Zhang Zhengjie.

The Existence of Infinitely Many Solutions for an Elliptic

Equation Involving Critical Sobolev-Hardy Exponent

with Neumann Boundary Condition

[J]. Acta mathematica scientia,Series A, 2007, 27(6): 1025-1034.

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