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

TL;DR

This paper investigates the existence and regularity of solutions to a nonlinear elliptic PDE modeling transparent media, demonstrating that the degenerate coefficient induces regularization effects and allows solutions without small data restrictions.

Contribution

It establishes the existence and regularity of solutions for a class of nonlinear elliptic equations with degenerate coefficients, expanding understanding of transparent media models.

Findings

Solutions exist under minimal data size assumptions.

Degenerate coefficient provides a regularizing effect.

Solutions are non-trivial and bounded.

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.