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2020 $L^\infty$-bounds of solutions for a class of strongly nonlinear elliptic equations in Musielak spaces
Mohamed Bourahma, Abdelmoujib Benkirane, Jaouad Bennouna, Mostafa El Moumni
Topol. Methods Nonlinear Anal. 56(1): 129-160 (2020). DOI: 10.12775/TMNA.2020.011

Abstract

In this paper we establish the existence of bounded solutions to a strongly nonlinear elliptic problem of the form $$ -\mathop{\rm div}\mathcal{A}(x,u,{\nabla}u)+g(x,u,\nabla u)= f \quad\text{in }{\Omega}, $$ with $u\in W^1_0L_\varphi({\Omega})\cap L^{\infty}(\Omega)$, where $$ \mathcal{A}(x,s,\xi)\cdot\xi\geq \overline{\varphi}_{x}^{-1} (\varphi(x,h(|s|)))\varphi(x,|\xi|), $$ $h\colon {\mathbb{R}^+} \to \mathopen ]0,1] $ is a continuous decreasing function with unbounded primitive and $g$ is a non-linearity satisfying $|g(x,s,\xi)|\leq\beta(s)\varphi(x,|\xi|)$. We assume the $\Delta_{2}$-condition on the Musielak function $\varphi$.

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Mohamed Bourahma. Abdelmoujib Benkirane. Jaouad Bennouna. Mostafa El Moumni. "$L^\infty$-bounds of solutions for a class of strongly nonlinear elliptic equations in Musielak spaces." Topol. Methods Nonlinear Anal. 56 (1) 129 - 160, 2020. https://doi.org/10.12775/TMNA.2020.011

Information

Published: 2020
First available in Project Euclid: 16 October 2020

MathSciNet: MR4175074
Digital Object Identifier: 10.12775/TMNA.2020.011

Rights: Copyright © 2020 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.56 • No. 1 • 2020
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