Remarks on some quasilinear equations with gradient terms and measure data

Contemporary Math., 595, 31-53 (2013)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 $H(x,u,\nabla u)=\pm \left\vert \nabla u\right\vert ^{q}$ ($q>0$). 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. $% H\geqq 0,$ we give two sufficient conditions for existence of a nonnegative solution