Bounded Solutions for Non-parametric Mean Curvature Problems with Nonlinear Terms

Abstract

In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain \(\Omega \) of \({{\,\mathrm{\mathbb {R}}\,}}^N\). The mean curvature, that depends 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 \(h:{{\,\mathrm{\mathbb {R}}\,}}^+\mapsto {{\,\mathrm{\mathbb {R}}\,}}^+\) is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, 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 is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples.