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Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Yutian Lei EMAIL logo
From the journal Forum Mathematicum

Abstract

In this paper, we are concerned with the following quasilinear PDE with a weight:

-divA(x,u)=|x|auq(x),u>0in n,

where n1, q>p-1 with p(1,2] and a0. The positive weak solution u of the quasilinear PDE is 𝒜-superharmonic. We also consider an integral equation involving the Wolff potential

u(x)=R(x)Wβ,p(|y|auq(y))(x),u>0in n,

which the positive solution u of the quasilinear PDE satisfies. Here β>0 and pβ<n. When -a>pβ or 0<q(n+a)(p-1)n-pβ, there does not exist any positive solution to this integral equation. On the other hand, when 0-a<pβ and q>(n+a)(p-1)n-pβ, the positive solution u of the integral equation is bounded and decays with the fast rate n-pβp-1 if and only if it is integrable (i.e., it belongs to Ln(q-p+1)pβ+a(n)). However, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one pβ+aq-p+1. In addition, we also discuss the case of -a=pβ. Thus, all the properties above are still true for the quasilinear PDE.


Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11471164

Funding statement: The research was supported by NSF of China (No. 11471164), and PAPD of Jiangsu Higher Education Institutions.

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Received: 2014-10-2
Revised: 2016-9-5
Published Online: 2016-11-6
Published in Print: 2017-9-1

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