CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

doi:10.1016/S0252-9602(14)60107-7

数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (5): 1603-1618.doi: 10.1016/S0252-9602(14)60107-7

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

王莉|汪继秀*

School of Basic Science, East China Jiaotong University, Nanchang 330013, China； School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang 441053, China

收稿日期:2012-10-09 修回日期:2013-11-22 出版日期:2014-09-20 发布日期:2014-09-20
通讯作者: 汪继秀，wangjixiu127@aliyun.com E-mail:wangli.423@163.com；wangjixiu127@aliyun.com
基金资助:
This work was supported by the National Natural Science Foundation of China (11326139, 11326145), Tian Yuan Foundation (KJLD12067), and Hubei Provincial Department of Education (Q20122504).

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

WANG Li, WANG Ji-Xiu*

School of Basic Science, East China Jiaotong University, Nanchang 330013, China； School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang 441053, China

Received:2012-10-09 Revised:2013-11-22 Online:2014-09-20 Published:2014-09-20
Contact: WANG Ji-Xiu，wangjixiu127@aliyun.com E-mail:wangli.423@163.com；wangjixiu127@aliyun.com
Supported by:
This work was supported by the National Natural Science Foundation of China (11326139, 11326145), Tian Yuan Foundation (KJLD12067), and Hubei Provincial Department of Education (Q20122504).

摘要/Abstract

摘要：

Let B₁ ⊂ R^N be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:
{−div(|∇u|^p−2∇u) = |x|^s|u|^p*(s)−2u + λ|x|^t|u|^p−2u, x ∈ B₁,
u|_∂B1 = 0,
where t, s > −p, 2 ≤ p < N, p*(s) = (N+s)p/N−p and λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N >p(p − 1)t + p(p²− p + 1) and λ ∈ (0, λ₁,t), where λ₁, t is the first eigenvalue of −Δp with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(p^s+p)min{1, p+t/p+s }+p²/p−(p−1)min{1, p+t/p+s } and λ > 0 is small.

关键词: singular coefficients, radial solution, critical exponent, p-Laplace equations

Abstract:

Key words: singular coefficients, radial solution, critical exponent, p-Laplace equations

中图分类号:

35B33

王莉, 汪继秀. CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS[J]. 数学物理学报(英文版), 2014, 34(5): 1603-1618.

WANG Li, WANG Ji-Xiu. CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS[J]. Acta mathematica scientia,Series B, 2014, 34(5): 1603-1618.

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CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

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