Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:232

Article

Liskevich, Vitali ; Lyakhova, Sofya ; Moroz, Vitaly

关键词: p-Laplacian;    Hardy potential;    Lionville theorems;    improved Hardy inequality;    Phragmen-Lindelof principle;    Prufer transformation;   

DOI  :  10.1016/j.jde.2006.09.001

来源: Elsevier

【 摘 要 】

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equation -Delta(p)u - mu/(vertical bar x vertical bar p)u(p-1) = C/(vertical bar x vertical bar sigma)u(q) in exterior domains of R-N (N >= 2). Here p epsilon (1, + infinity) and mu <= C-H, where C-H is the critical Hardy constant. We provide a sharp characterization of the set of (q, sigma) epsilon R-2 such that the equation has no positive (super)solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the p-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the p-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Prufer transformation. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

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