Multiple bounded variation solutions for a prescribed mean curvature equation with Neumann boundary conditions

TL;DR

This paper establishes the existence of multiple positive bounded variation solutions for a prescribed mean curvature equation with Neumann boundary conditions, using approximation and shooting methods, and explores conditions for classical solutions.

Contribution

It introduces a novel approach to find multiple solutions for a mean curvature problem with Neumann conditions, including criteria for classical solutions.

Findings

Multiple positive BV-solutions exist for the prescribed mean curvature equation.

Solutions are characterized by their intersections with a constant solution.

Conditions are provided for solutions to be classical and have continuous energy.

Abstract

We prove the existence of multiple positive BV-solutions of the Neumann problem $$ \begin{cases} \displaystyle -\left(\frac{u'}{\sqrt{1+u'^2}}\right)'=a(x)f(u)\quad&\mbox{in }(0,1), u'(0)=u'(1)=0,& {cases} $$ where $a(x) > 0$ and $f$ belongs to a class of nonlinear functions whose prototype example is given by $f(u) = -\lambda u + u^p$, for $\lambda > 0$ and $p > 1$. In particular, $f(0)=0$ and $f$ has a unique positive zero, denoted by $u_0$. Solutions are distinguished by the number of intersections (in a generalized sense) with the constant solution $u = u_0$. We further prove that the solutions found have continuous energy and we also give sufficient conditions on the nonlinearity to get classical solutions. The analysis is performed using an approximation of the mean curvature operator and the shooting method.