Global structure of radial sign-changing solutions for the prescribed mean curvature problem in a ball

TL;DR

This paper investigates the global structure of radial solutions with prescribed nodal properties for a mean curvature boundary value problem in a ball, using bifurcation techniques to analyze solutions depending on the nonlinear term's behavior.

Contribution

It provides a comprehensive analysis of the global bifurcation structure of radial solutions for the prescribed mean curvature problem, considering various behaviors of the nonlinear term near zero.

Findings

Characterization of solution branches depending on nonlinear term behavior

Existence of multiple solutions with prescribed nodal properties

Use of bifurcation techniques to analyze solution structure

Abstract

In this paper, we are concerned with the global structure of radial solutions, with prescribed nodal properties, to the 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.