一带权多重调和方程正解的存在性及渐近行为

数学物理学报 ›› 2013, Vol. 33 ›› Issue (5): 829-841.

一带权多重调和方程正解的存在性及渐近行为

王中亮

第二军医大学数理教研室上海 200433|华东师范大学数学系上海 200241

收稿日期:2012-06-30 修回日期:2013-05-31 出版日期:2013-10-25 发布日期:2013-10-25
基金资助:
国家自然科学基金(10971067)资助.

Existence and Asymptotic Behaviour of Positive Solutions for a Polyharmonic Equation with Weight

WANG Zhong-Liang

Department of Mathematics and Physics, Second Military Medical University, Shanghai 200433;
Department of Mathematics, East China Normal University, Shanghai 200241

Received:2012-06-30 Revised:2013-05-31 Online:2013-10-25 Published:2013-10-25
Supported by:
国家自然科学基金(10971067)资助.

摘要/Abstract

摘要：

通过建立径向对称解的先验估计, 并运用Pohozaev Pohozaev型恒等式研究多重调和方程
{(-Δ)^mu=|x|^αu^p-1, u>0, 在B中,
Dⁱu|_∂B=0, |i|≤m-1
解的存在性, 其中m ∈N, B是Rⁿ 中单位球, n≥2m+1, α>0 且p>2. 进一步, 运用Blow-up 分析获得上述方程当 α 固定及p→2n/(n-2m) 时, 最小能量解的渐近行为.

关键词: 多重调和方程, Sobolev 临界指数, 径向对称解, 渐近行为, 对称破裂

Abstract:

In this paper, the author establishes a prior estimate of the radially symmetric solutions, and apply a Poho{z}aev-type identity to the research on the existence of solutions for the polyharmonic equation
{(-Δ)^mu=|x|^αu^p-1, u>0, in B,
Dⁱu|_∂B=0, |i|≤m-1
where m ∈N, B is the unit ball in Rⁿ with n≥2m+1, α>0 and p>2. Moreover, by making use of the Blow-up analysis, the asymptotic behaviour of the least energy solutions for above equation is obtained when α is fixed and p→2n/(n-2m) .

Key words: Polyhamonic equation, Critical Sobolev exponent, Radially symmetric solution, Asymptotic behaviour, Symmetry breaking

中图分类号:

35B40

王中亮. 一带权多重调和方程正解的存在性及渐近行为[J]. 数学物理学报, 2013, 33(5): 829-841.

WANG Zhong-Liang. Existence and Asymptotic Behaviour of Positive Solutions for a Polyharmonic Equation with Weight[J]. Acta mathematica scientia,Series A, 2013, 33(5): 829-841.

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