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2001 Radial solutions for a quasilinear equation via Hardy inequalities
M. García-Huidobro, A. Kufner, R. Manásevich, C. S. Yarur
Adv. Differential Equations 6(12): 1517-1540 (2001).

Abstract

We establish an analogue of the Sobolev critical exponent for the inclusion $V^p(a)\hookrightarrow L^q(b)$, where $a$ and $b$ are weight functions, $V^p(a)$ is a weighted Sobolev space, and $L^q(b)$ is a weighted Lebesgue space. We use this result to study existence of radial solutions to the problem with weights $$(D_w)\quad\quad\begin{cases} -{\rm div}(\tilde{a}(|x|)|\nabla u|^{p-2}\nabla u)= \tilde{b}(|x|)|u|^{q-2}u \qquad \mbox{in} \quad \Omega\subset \mathbb R^N ,\\~~u=0\qquad \mbox{on}\quad \partial\Omega, \end{cases} $$ where $\Omega$ is a ball, $1 <p <q,$ and $\tilde{a}(|x|)=|x|^{1-N}a(|x|),$ $\tilde{b}(|x|)=|x|^{1-N}b(|x|)$. We are interested in the interplay between $q$ and a suitable critical exponent and its consequences for the existence and nonexistence of positive solutions of problem $(D_w).$

Citation

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M. García-Huidobro. A. Kufner. R. Manásevich. C. S. Yarur. "Radial solutions for a quasilinear equation via Hardy inequalities." Adv. Differential Equations 6 (12) 1517 - 1540, 2001.

Information

Published: 2001
First available in Project Euclid: 2 January 2013

zbMATH: 1140.35441
MathSciNet: MR1858431

Subjects:
Primary: 35J60
Secondary: 34B16, 35B33, 35J25

Rights: Copyright © 2001 Khayyam Publishing, Inc.

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Vol.6 • No. 12 • 2001
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