In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$ \begin{ cases} g(u) -{ \rm div}\left(\frac{ D u}{ \sqrt{ 1+|D u|^2}}\right) = f & \text{ in}\;\Omega,\\ \newline u=0 & \text{ on}\;\partial\Omega, \end{ cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{ R}^N$ ($N\geq 2$) with Lipschitz boundary, $g:\mathbb{ R}\to\mathbb{ R}$ 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.
Categories: math.AP