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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems


Authors: S. Hill, K. S. Moore and W. Reichel
Journal: Proc. Amer. Math. Soc. 128 (2000), 1673-1683
MSC (2000): Primary 35J65; Secondary 35J70
Published electronically: February 7, 2000
MathSciNet review: 1695131
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Abstract:

On bounded domains $\Omega\subset\mathbb{R} ^2$ we consider the anisotropic problems $u^{-a}u_{xx}+u^{-b}u_{yy}=p(x,y)$ in $\Omega$ with $a,b>1$ and $u=\infty$ on $\partial\Omega$ and $u^cu_{xx}+u^du_{yy}+q(x,y)=0$ in $\Omega$ with $c,d\geq 0$ and $u=0$ on $\partial\Omega$. Moreover, we generalize these boundary value problems to space-dimensions $n>2$. Under geometric conditions on $\Omega$ and monotonicity assumption on $0<p,q\in {\cal C}^\alpha(\overline{\Omega})$ we prove existence and uniqueness of positive solutions.


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Additional Information

S. Hill
Affiliation: Department of Mathematics, Rowan University, Glassboro, New Jersey 08028

K. S. Moore
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Address at time of publication: Department of Mathematics, University of Michigan, 525 East University Ave., Ann Arbor, Michigan 48109-1109
Email: ksmoore@math.lsa.umich.edu

W. Reichel
Affiliation: Mathematisches Institut, Universität zu Köln, 50931 Köln, Germany
Address at time of publication: Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051 Basel, Switzerland
Email: reichel@eichler.math-lab.unibas.ch

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05493-9
PII: S 0002-9939(00)05493-9
Keywords: Anisotropic singular equations, comparison principles
Received by editor(s): July 9, 1998
Published electronically: February 7, 2000
Communicated by: Lesley M. Sibner
Article copyright: © Copyright 2000 American Mathematical Society