An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term

Abstract

In this paper we study a Dirichlet problem for an elliptic equation withdegenerate coercivity and a singular lower order term with natural growth with respect to the gradient.The model problem is$$\begin{equation}\left\{\begin{array}{11}-div\left(\frac{\nabla u}{(1+|u|)^p}\right) + \frac{|\nabla u|^{2}}{|u|^{\theta}} = f & \mbox{in $\Omega$,} \\u = 0 & \mbox{on $\partial\Omega$,}\end{array}\right.\end{equation}$$where $\Omega$ is an open bounded set of $\mathbb{R}^N$, $N\geq 3$ and $p, \theta>0$.The source $f$ is a positive function belonging to some Lebesgue space.We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions, when $p>\theta-1$ and $\theta<2$.