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

Published online by Cambridge University Press:  24 January 2019

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)

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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