MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

doi:10.1016/S0252-9602(09)60077-1

数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (4): 903-918.doi: 10.1016/S0252-9602(09)60077-1

MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

李工宝,张国

College of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

收稿日期:2008-12-31 出版日期:2009-07-20 发布日期:2009-07-20
基金资助:
Partially supported by NSFC (10571069 and 10631030), and the Lab of Mathematical Sciences, CCNU, Hubei Province, China

MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

LI Gong-Bao, ZHANG Guo

College of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2008-12-31 Online:2009-07-20 Published:2009-07-20
Supported by:
Partially supported by NSFC (10571069 and 10631030), and the Lab of Mathematical Sciences, CCNU, Hubei Province, China

摘要/Abstract

摘要：

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:

−△_pu − △_qu = |u|^p*−2u + μ|u|^r−2u in Ω,
u|∂Ω= 0,

where Ω ⊂ R^N is a bounded domain, N > p, p* = N_p /N−p is the critical Sobolev exponent and μ > 0. We prove that if 1 < r < q < p < N, then there is a μ₀ > 0, such that for any μ ∈ (0, μ₀), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.

关键词: p&q-Laplacian, multiplicity of solutions, critical exponent

Abstract:

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:

−△_pu − △_qu = |u|^p*−2u + μ|u|^r−2u in Ω,
u|∂Ω= 0,

Key words: p&q-Laplacian, multiplicity of solutions, critical exponent

中图分类号:

35J60

李工宝, 张国. MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT[J]. 数学物理学报(英文版), 2009, 29(4): 903-918.

LI Gong-Bao, ZHANG Guo. MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT[J]. Acta mathematica scientia,Series B, 2009, 29(4): 903-918.

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MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT

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