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 $$\{\begin{array}{ll}-\mathrm{div}\left(|u{|}^m\frac{Du}{|Du|}\right)=f& \text{in}\Omega ,\\ & \\ u=0& \text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$ ($N\ge 2$) 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.