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

In this paper, the classical solution set (λ,u)(λ,u) of the one-dimensional prescribed mean curvature equation equation(⋆⋆)−(u′1+(u′)2)′=λ(1−u)2,−L0λ>0 and L>0L>0, is analyzed via a time map. It is shown that the solution set depends on both parameters λλ and LL and undergoes two bifurcations. The first is a standard saddle node bifurcation, which happens for all LL at λ=λ∗(L)λ=λ∗(L). The second is a splitting   bifurcation, namely, there exists a value L∗L∗ such that as LL transitions from greater than or equal to L∗L∗ to less than L∗L∗ the upper branch of the bifurcation diagram of problem (⋆⋆) splits into two parts. In contrast, the solution set of the semilinear version of problem (⋆⋆) is independent of LL 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 forcing.