Singular elliptic equations having a gradient term with natural growth

Abstract

We study a class of Dirichlet boundary value problems whose prototype is \begin{equation}\label{1.2abs} \left\{\begin{array}{ll} -Δ_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, &\quad\hbox{in } \ Ω\,,\\ u\ge 0\,,&{\quad\hbox{in } \ Ω}\\ u = 0\,&\quad\hbox{on }\partial Ω\,,\end{array}\right. \end{equation} where $\Omega$ an open bounded subset of ${\mathbb{R}}^N$, $0<q<1$, $1<p<N$, $h$ is a continuous function and $f$ belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term and a first order term with natural growth in the gradient. A priori estimates and existence results are proved depending on the summability of the datum $f$.