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Research Article | Open Access

Equivalence of solutions for non-homogeneous p ( x )-Laplace equations

María Medina1, Pablo Ochoa2( )
Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain
Facultad de Ingeniería, Universidad Nacional de Cuyo. CONICET. Parque Gral. San Martín 5500. Mendoza, Argentina
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Abstract

We establish the equivalence between weak and viscosity solutions for non-homogeneous p ( x )-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution techniques to state that viscosity solutions are also weak solutions, and comparison principles to prove the converse. The new aspects of the p ( x )-Laplacian compared to the constant case are the presence of log-terms and the lack of the invariance under translations.

Keywords

nonlinear elliptic equationsp(x)-Laplacianviscosity solutionsweak solutionscomparison principle

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Mathematics in Engineering
Volume 5 Issue 2,
April 2023
Pages 1-19
DOI: 10.3934/mine.2023044

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Cite this article:
Medina M, Ochoa P. Equivalence of solutions for non-homogeneous p ( x )-Laplace equations. Mathematics in Engineering, 2023, 5(2): 1-19. https://doi.org/10.3934/mine.2023044

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Received: 06 December 2021
Revised: 23 June 2022
Accepted: 23 June 2022
Published: 15 April 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)