带临界指数的奇异椭圆方程Neumann问题多重解的存在性

华东师范大学学报(自然科学版) ›› 2011, Vol. 2011 ›› Issue (5): 79-87.

带临界指数的奇异椭圆方程Neumann问题多重解的存在性

陈自高

华北水利水电学院数学与信息科学学院, 郑州 450011

收稿日期:2011-03-01 修回日期:2011-06-01 出版日期:2011-09-25 发布日期:2011-11-22

Existence of multiple solutions for singular elliptic equations involving critical exponents with Neumann boundary condition

CHEN Zi-gao

Department of Mathematics and Information Science, North China, University of Water Resources and Electric Power, Zhengzhou 450011, China

Received:2011-03-01 Revised:2011-06-01 Online:2011-09-25 Published:2011-11-22

摘要/Abstract

摘要： 利用变分法, 在n维空间有界区域Ω上, 研究了一类含有Sobolev-Hardy临界指数与Hardy项的奇异椭圆方程Neumann 问题弱解的存在性和多重性. 在f(x,t)满足非二次条件的情况下, 运用对偶喷泉定理与拉直边界的方法, 证明了存在λ*>0使得当λ∈(0,λ*)时, 该问题存在无穷多个具有负能量的弱解{u_k} 被包含于W^{1,2}(Ω)并且当k→∞时, J(u_k)→0.

关键词: Neumann问题, Sobolev-Hardy临界指数, (PS)_c^*条件, 对偶喷泉定理

Abstract: By using variational methods, the existence and multiplicity of weak solutions for Neumann boundary problem for some singular elliptic equations involving critical Sobolev-Hardy exponents and Hardy terms was studied on bounded domain Ω included by R^N. If f(x,t) satisfies the non-quadratic condition, based on the dual fountain theorem and the means of straightening the boundary, we proved that there exists λ*>0 such that for any λ∈(0,λ*), this problem has a sequence of solutions {u_k} W^{1,2}(Ω) such that J(u_k)<0 and J(u_k)→0 as k→∞.

Key words: Neumann problem, critical Sobolev-Hardy exponent, (PS)_c^* condition, dual fountain theorem

中图分类号:

O175.25

陈自高. 带临界指数的奇异椭圆方程Neumann问题多重解的存在性[J]. 华东师范大学学报(自然科学版), 2011, 2011(5): 79-87.

CHEN Zi-gao. Existence of multiple solutions for singular elliptic equations involving critical exponents with Neumann boundary condition[J]. Journal of East China Normal University(Natural Sc, 2011, 2011(5): 79-87.

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