Oscillating solutions for prescribed mean curvature equations: Euclidean and Lorentz-Minkowski cases

TL;DR

This paper investigates oscillating solutions to prescribed mean curvature equations in Euclidean and Lorentz-Minkowski spaces, demonstrating existence and characterizing their behavior depending on the spatial dimension.

Contribution

It establishes the existence of oscillating solutions with unbounded zeros for these equations, including their periodicity in one dimension and decay in higher dimensions.

Findings

Existence of oscillating solutions with unbounded zeros.

Periodic solutions in one-dimensional case.

Radial solutions decaying at infinity in higher dimensions.

Abstract

This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.