The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

Abstract

In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$\{\begin{array}{ll}g(u)-\mathrm{div}\left(\frac{Du}{\sqrt{1+|Du{|}^2}}\right)=f& \text{in}\Omega ,\\ & \\ u=0& \text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$ ($N\ge 2$) with Lipschitz boundary, $g:\mathbb{R}\to \mathbb{R}$ is a continuous function and $f$ belongs to some Lebesgue spaces. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term $g(u)$ in order to get a solutions for data $f$ merely belonging to $L^1(\Omega )$ and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in $L^N(\Omega )$ as well as uniqueness if $g$ is increasing.