2012/01/25 by Nicholas D. Brubaker, Brubaker, Nicholas D., John A. Pelesko +1
Mathematics · #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Differential Equations Analysis #math.CA
paper · pdf · doi:10.48550/arxiv.1201.5432
20 pages, 5 figures
openalex publication_date 2012/01/25 · arxiv created 2012/04/19 · arxiv updated 2012/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we analyze the classical solution set (λ,u), for λ>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, λ and L, and undergoes two bifurcations. The first is a standard saddle node bifurcation, which happens for all L at λ = λ*(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.