Existence and regularity of solutions for the elliptic nonlinear transparent media equation

Abstract

In this paper we study existence and regularity of solutions to Dirichlet problems as $$ \begin{cases} - {\rm div}\left(|u|^m\frac{D u}{|D u|}\right) = f & \text{in}\;\Omega,\\ \newline u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary, $m>0$, and $f$ belongs to the Lorentz space $L^{N,\infty}(\Omega)$. In particular, we explore the regularizing effect given by the degenerate coefficient $|u|^m$ in order to get non-trivial and bounded solutions with no smallness assumptions on the size of the data.