Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity

TL;DR

This paper investigates how solutions to a one-dimensional prescribed mean curvature equation with singular nonlinearity depend on parameters, revealing two types of bifurcations and highlighting differences from the semilinear case.

Contribution

It identifies and characterizes a splitting bifurcation unique to the mean curvature operator with singular nonlinearity, contrasting it with the semilinear case.

Findings

Solution set depends on parameters {} and L

Two bifurcations are identified: saddle node and splitting bifurcation

Splitting bifurcation occurs only in the mean curvature case, not in the semilinear case

Abstract

In this paper we analyze the classical solution set ({\lambda},u), for {\lambda}>0, of a one-dimensional prescribed mean curvature equation on the interval [-L,L]. It is shown that the solution set depends on the two parameters, {\lambda} and L, and undergoes two bifurcations. The first is a standard saddle node bifurcation, which happens for all L at {\lambda} = {\lambda}*(L). The second is a splitting bifurcation; specifically, there exists a value L* such that as L transitions from greater than or equal L* to less than L* the upper branch of the bifurcation diagram splits into two parts. In contrast, the solution set of the semilinear version of the prescribed mean curvature equation is independent of L and exhibits only a saddle node bifurcation. Therefore, as this analysis suggests, the splitting bifurcation is a byproduct of the mean curvature operator coupled with the singular nonlinearity.