Remarks on some quasilinear equations with gradient terms and measure data

TL;DR

This paper investigates quasilinear p-Laplacian equations with gradient-dependent terms and measure data, establishing existence, necessary conditions, and removability results, especially focusing on subcritical and supercritical growth cases.

Contribution

It provides new existence results, necessary capacity conditions, and removability theorems for quasilinear equations with measure data and gradient terms, including supercritical cases.

Findings

Existence under subcritical growth assumptions.

Necessary capacity conditions for solutions.

Removability of singularities in solutions.

Abstract

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain, $H$ a Caratheodory function defined in $\Omega \times \mathbb{R\times R}^{N},$ and $\mu $ a bounded Radon measure in $\Omega .$ We study the problem% \begin{equation*} -\Delta_{p}u+H(x,u,\nabla u)=\mu \quad \text{in}\Omega,\qquad u=0\quad \text{on}\partial \Omega, \end{equation*} where $\Delta_{p}$ is the $p$-Laplacian ($p>1$)$,$ and we emphasize the case $H(x,u,\nabla u)=\pm \left\| \nabla u\right\| ^{q}$ ($q>0$). We obtain an existence result under subcritical growth assumptions on $H,$ we give necessary conditions of existence in terms of capacity properties, and we prove removability results of eventual singularities. In the supercritical case, when $\mu \geqq 0$ and $H$ is an absorption term, i.e. $% H\geqq 0,$ we give two sufficient conditions for existence of a nonnegative solution.