Comparison result for quasi-linear elliptic equations with general growth in the gradient

Angelo Alvino; Maria Francesca Betta; Anna Mercaldo; Roberta Volpicelli

doi:10.3934/cpaa.2024013

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2024, Volume 23, Issue 3: 339-355. Doi: 10.3934/cpaa.2024013

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Comparison result for quasi-linear elliptic equations with general growth in the gradient

1.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università di Napoli Federico Ⅱ, Napoli, Italy
2.
Dipartimento di Ingegneria, Università di Napoli Parthenope, Napoli, Italy

^*Corresponding author: Anna Mercaldo
^*Corresponding author: Anna Mercaldo

Received: October 2023

Early access: February 28, 2024

Published online: February 28, 2024

This work is partially supported by a MIUR-PRIN 2017 grant "Direct and inverse problems for partial differential equations: theoretical aspects and applications", 201758MTR2-003, MIUR-PRIN 2022 grant "Partial differential equations and related geometric-functional inequalities" and PRIN PNRR 2022 - P2022YFAJH - Linear and Nonlinear PDE's: New directions and Applications.

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Abstract

In this paper we prove a comparison result for a class of Dirichlet boundary problems whose model is

$ \left\{ \begin{array}{ll} -\Delta u = {\beta}|\nabla u|^q +c u + f &\rm{in\;} \Omega \\ u = 0&\rm{su\;} \partial \Omega \,, \end{array} \right. $

where $ \Omega $ is an open bounded subset of $ {{\mathbb R}^N} $, $ N>2 $. We also prove an existence and uniqueness result for weak solution to these problems.

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Mathematics Subject Classification: Primary: 35J25, 35J62; Secondary: 35J60.

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