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2002 On a nonlinearly Dirichlet problem with a singularity along the boundary
C. Aranda, T. Godoy
Differential Integral Equations 15(11): 1313-1324 (2002).

Abstract

Let $\Omega$ be a $C^{1,1}$ and bounded domain in $R^{n},$ with $n\geq2.$ Let $K$ and $f$ be two nonnegative functions on $\Omega$ belonging to $L^{p} ( \Omega ) $ for some $p>n/2$ and let $\alpha>0.$ In this paper we prove existence and uniqueness of a strong solution $u\in W_{loc}^{2,p} ( \Omega ) \cap C ( \overline{\Omega} ) $ for the elliptic Dirichlet problem $-\Delta u+\lambda u=Ku^{-\alpha}+f$ in $\Omega,$ $u>0$ in $\Omega,$ $u=0$ in $\partial\Omega.$ Moreover, for the case $\lambda=0,$ $f=0,$ we prove, under the weaker hypothesis $p>\frac{ ( \alpha^{2}+1 ) n}{2\alpha^{2}+n},$ the existence of a solution $u\in W_{loc}^{2,p} ( \Omega ) $ for the above problem that satisfies, in a suitable extended sense, the boundary condition $u=0$ on $\partial\Omega.$

Citation

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C. Aranda. T. Godoy. "On a nonlinearly Dirichlet problem with a singularity along the boundary." Differential Integral Equations 15 (11) 1313 - 1324, 2002.

Information

Published: 2002
First available in Project Euclid: 21 December 2012

zbMATH: 1021.35029
MathSciNet: MR1920688

Subjects:
Primary: 35J60
Secondary: 35A05 , 35B50 , 35J25

Rights: Copyright © 2002 Khayyam Publishing, Inc.

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Vol.15 • No. 11 • 2002
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