ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p

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ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p

ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p

ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p

ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p

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doi:10.1016/S0252-9602(06)60043-X

数学物理学报(英文版)

姚仰新; 沈尧天

华南理工大学数学学院, 广州 510640

收稿日期:2003-04-02 修回日期:2003-11-04 出版日期:2006-04-20 发布日期:2006-04-20
通讯作者: 姚仰新
基金资助:
Supported by NSFC(10471047) and NSF Guangdong Province(04020077)

Yao Yangxin; Shen Yaotian

School of Mathematical Sciences, South China University of Technology,
Guangzhou 510640, China

Received:2003-04-02 Revised:2003-11-04 Online:2006-04-20 Published:2006-04-20
Contact: Yao Yangxin

摘要：

This article deals with the problem

- \Lap_{p} u = λ \frac{| u |^{p - 2} u}{\xlnxRt} + f (x, u), x \in Ω; u = 0, x \in \partial \Om,

$-\Lap_p u=\lambda{|u|^{p-2}u\over\xlnxRt}+f(x,u),\quad x\in\Omega;\qquad u=0,x\in\partial\Om,$
where

n = p .

$n = p.$ The authors prove that a Hardy inequality and the constant

(\pp)^{p}

$(\pp)^p$ is optimal. They also prove the existence of a nontrivial solution of the above mentioned problem by using the Mountain Pass Lemma.

关键词: Elliptic equation, Hardy inequality, critical singularity, Mountain Pass Lemma

Abstract:

This article deals with the problem

- \Lap_{p} u = λ \frac{| u |^{p - 2} u}{\xlnxRt} + f (x, u), x \in Ω; u = 0, x \in \partial \Om,

$-\Lap_p u=\lambda{|u|^{p-2}u\over\xlnxRt}+f(x,u),\quad x\in\Omega;\qquad u=0,x\in\partial\Om,$
where

n = p .

$n = p.$ The authors prove that a Hardy inequality and the constant

(\pp)^{p}

$(\pp)^p$ is optimal. They also prove the existence of a nontrivial solution of the above mentioned problem by using the Mountain Pass Lemma.

Key words: Elliptic equation, Hardy inequality, critical singularity, Mountain Pass Lemma

中图分类号:

35J65

姚仰新; 沈尧天. ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p[J]. 数学物理学报(英文版), 2006, 26(2): 209-219.

Yao Yangxin; Shen Yaotian. ON CRITICAL SINGULAR QUASILINEAR ELLIPTIC PROBLEM WHEN n=p[J]. Acta mathematica scientia,Series B, 2006, 26(2): 209-219.

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