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A p(x)-Laplacian extension of the Díaz-Saa inequality and some applications

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  24 January 2019

Peter Takáč  and
Jacques Giacomoni
Peter Takáč
Affiliation:
Institut für Mathematik, Universität Rostock Ulmenstraße 69, Haus 3 D-18055 Rostock, Germany (peter.takac@uni-rostock.de)
Jacques Giacomoni
Affiliation:
LMAP (UMR 5142) Université de Pau et des Pays de l'Adour Avenue de l'Université, F-64013 Pau cedex, France (jacques.giacomoni@univ-pau.fr)
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Abstract

The main result of this work is a new extension of the well-known inequality by Díaz and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian, $\Delta _{p(x)}u\equiv {\rm div}( \vert \nabla u \vert ^{p(x)-2}\nabla u)$. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.

Keywords

p(x)-Laplacianquasilinear Dirichlet problem with variable exponentsray-strictly convex energy functionaluniqueness and comparison principles

MSC classification

Primary: 35J62: Quasilinear elliptic equations 35J92: Quasilinear elliptic equations with $p$-Laplacian
Secondary: 35B09: Positive solutions 35A02: Uniqueness problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 205 - 232
Copyright
Copyright © Royal Society of Edinburgh 2019

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