On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type

Abstract

We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation −u′1+u′2′=λu1+up,−L0L,p\gt 0 are two evolution parameters. We prove that on the (λ,‖u‖∞)\left(\lambda ,\Vert u{\Vert }_{\infty })-plane, for 01p\gt 1, the bifurcation curve is reversed ε\varepsilon -like shaped bifurcation if L>L∗L\gt {L}^{\ast }, and is exactly decreasing for λ>λ∗\lambda \gt {\lambda }^{\ast } if 0<L<L∗0\lt L\lt {L}_{\ast }.

Keywords

• mean curvature
• bifurcation curve
• logistic function
• 34b18
• 34c23
• 34b15