Advances in Differential Equations

Radial solutions for a quasilinear equation via Hardy inequalities

M. García-Huidobro, A. Kufner, R. Manásevich, and C. S. Yarur

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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 $(D_w).$

Article information

Adv. Differential Equations, Volume 6, Number 12 (2001), 1517-1540.

First available in Project Euclid: 2 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 34B16: Singular nonlinear boundary value problems 35B33: Critical exponents 35J25: Boundary value problems for second-order elliptic equations


García-Huidobro, M.; Kufner, A.; Manásevich, R.; Yarur, C. S. Radial solutions for a quasilinear equation via Hardy inequalities. Adv. Differential Equations 6 (2001), no. 12, 1517--1540.

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