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

In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$\{\begin{array}{ll}g(u)-\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{Du}{\sqrt{1+\mathrm{\mid }Du{\mathrm{\mid }}^2}}\right)=f& \text{in}\mathrm{\Omega },\\ \\ u=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is an open bounded subset of ${\mathbb{R}}^N$ ($N\ge 2$) with Lipschitz boundary, $g:\mathbb{R}\to \mathbb{R}$ 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 $L^1(\mathrm{\Omega })$ and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in $L^N(\mathrm{\Omega })$ as well as uniqueness if $g$ is increasing.