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