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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A priori estimates for a class of quasi-linear elliptic equations


Authors: Daniel Daners and Pavel Drábek
Journal: Trans. Amer. Math. Soc. 361 (2009), 6475-6500
MSC (2000): Primary 35B45, 35B65, 35J65, 35J70
Published electronically: July 20, 2009
MathSciNet review: 2538601
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Abstract: In this paper we prove a priori estimates for a class of quasi-linear elliptic equations. To make the proofs clear and transparent we concentrate on the $ p$-Laplacian. We focus on $ L_p$-estimates for weak solutions of the problem with all standard boundary conditions on non-smooth domains. As an application we prove existence, continuity and compactness of the resolvent operator. We finally prove estimates for solutions to equations with non-linear source and show that, under suitable growth conditions, all solutions are globally bounded.


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Additional Information

Daniel Daners
Affiliation: School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia
Email: D.Daners@maths.usyd.edu.au

Pavel Drábek
Affiliation: Department of Mathematics, University of West Bohemia, P.O. Box 314, 306 14 Pilsen, Czech Republic
Email: pdrabek@kma.zcu.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04839-9
PII: S 0002-9947(09)04839-9
Keywords: Quasi-linear problems, $p$-Laplacian, $L_p$-estimates, non-smooth domains, Moser iterations
Received by editor(s): November 2, 2007
Published electronically: July 20, 2009
Additional Notes: The second author was supported by Research Plan MSM4977751301 of the Czech Ministry of Education, Youths and Sports
Article copyright: © Copyright 2009 American Mathematical Society