Remarks on some quasilinear equations with gradient terms and measure data

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ be a smooth bounded domain, $H$ a Caratheodory function defined in $\Omega \times {\mathbb{R}\times \mathbb{R}}^N,$ and $\mu$ a bounded Radon measure in $\Omega .$ We study the problem% \begin{equation*} -Δ_{p}u+H(x,u,\nabla u)=μ\quad \text{in}Ω,\qquad u=0\quad \text{on}\partial Ω, \end{equation*} where ${\Delta }_p$ is the $p$-Laplacian ($p>1$)$,$ and we emphasize the case $H(x,u,\nabla u)=\pm {\Vert \nabla u\Vert }^q$ ($q>0$). 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. we give two sufficient conditions for existence of a nonnegative solution.

To appear in Contemporary Mathematics