Uniqueness for Neumann problems for nonlinear elliptic equations

Maria Francesca Betta; Olivier Guibé; Anna Mercaldo

doi:10.3934/cpaa.2019050

Article Contents

2019, Volume 18, Issue 3: 1023-1048. Doi: 10.3934/cpaa.2019050

This issue Previous Article Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator Next Article Entire solutions in nonlocal monostable equations: Asymmetric case

Uniqueness for Neumann problems for nonlinear elliptic equations

1.
Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Centro Direzionale, Isola C4 80143 Napoli, Italy
2.
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l'Université, BP.12 76801 Saint-Étienne-du-Rouvray, France
3.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy

* Corresponding author
* Corresponding author

Received: September 30, 2017

Revised: August 31, 2018

Published: May 2019

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Abstract

In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is

$\left\{ \begin{align} & -\text{div}({{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u)-\text{div}(c(x)|u{{|}^{p-2}}u)=f\ \ \ \text{in}\ \Omega , \\ & \left( {{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u+c(x)|u{{|}^{p-2}}u \right)\cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}=0\ \ \ \text{on}\ \partial \Omega , \\ \end{align} \right.$

where $Ω$ is a bounded domain of $\mathbb{R}^{N}$, $N≥ 2$, with Lipschitz boundary, $ 1 < p < N$, $\underline n$ is the outer unit normal to $\partial Ω$, the datum $f$ belongs to $L^{(p^{*})'}(Ω)$ or to $L^{1}(Ω)$ and satisfies the compatibility condition $\int{{}}_Ω f \, dx = 0$. Finally the coefficient $c(x)$ belongs to an appropriate Lebesgue space.

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

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