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

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