Remarks on some quasilinear equations with gradient terms and measure data

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be a smooth bounded domain, $H$ a Caratheodory function defined in $\mathrm{\Omega }\times {\mathbb{R}\times \mathbb{R}}^N,$ and $\mu$ a bounded Radon measure in $\mathrm{\Omega }\mathrm{.}$ 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 ${\mathrm{\Delta }}_p$ 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. $$ we give two sufficient conditions for existence of a nonnegative solution.