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In this paper we prove existence of nonnegative bounded solutions for thenon-autonomous prescribed mean curvature problem in non-parametric form on anopen bounded domain $\Omega$ of $\mathbb{R}^N$. The mean curvature, thatdepends on the location of the solution $u$ itself, is asked to be of the form$f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(\Omega)$ and$h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possiblyunbounded near zero. As a preparatory tool for our analysis we propose a purelyPDE approach to the prescribed mean curvature problem not depending on thesolution, i.e. $h\equiv 1$. This part, which has its own independent interest,aims to represent a modern and up-to-date account on the subject. Uniqueness isalso handled in presence of a decreasing nonlinearity. The sharpness of theresults is highlighted by mean of explicit examples.

Bounded solutions for non-parametric mean curvature problems with nonlinear terms