Calculus of Variations and Geometric Measure Theory

D. Giachetti - F. Oliva - F. Petitta

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

created by petitta on 27 Apr 2023
modified on 07 Jun 2024


Accepted Paper

Inserted: 27 apr 2023
Last Updated: 7 jun 2024

Journal: J. Geom. Anal.
Year: 2024

ArXiv: 2304.13611 PDF


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 $\mathbb{R}^N$. 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 $h:\mathbb{R}^+\mapsto \mathbb{R}^+$ 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.