Global structure of radial positive solutions for a prescribed mean curvature problem in a ball

TL;DR

This paper investigates the global structure of positive radial solutions to a prescribed mean curvature boundary value problem in a ball, using bifurcation techniques to analyze how solutions depend on a parameter.

Contribution

It provides a comprehensive analysis of the solution structure for a mean curvature problem, considering the nonlinear term's behavior near zero, which is a novel application of bifurcation methods.

Findings

Characterization of solution branches depending on nonlinear term behavior

Existence of multiple positive solutions under certain conditions

Global bifurcation diagrams for the problem

Abstract

In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem$$\text{div}\big(\phi_{N}(\nabla v)\big)+\lambda f(|x|, v)=0 \text{in} B(R), v=0 \text{on} \partial B(R), $$where $\phi_{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}}, y\in \mathbb{R}^{N}$, $\lambda$ is a positive parameter, $B(R)=\{x\in \mathbb{R}^{N} :|x|<R\}$, and $|\cdot|$ denote the Euclidean norm in $\mathbb{R}^{N}$. All results, depending on the behavior of nonlinear term $f$ near 0, are obtained by using global bifurcation techniques.