Global bifurcation diagrams and exact multiplicity of positive solutions for a one-dimensional prescribed mean curvature problem arising in MEMS

Abstract

We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS {−(u′(x)1+(u′(x))2)′=λ(1−u)p,u<1,−L<x<L,u(−L)=u(L)=0, where λ>0 is a bifurcation parameter, and p,L>0 are two evolution parameters. We determine the exact number of positive solutions by the values of p,L and λ. Moreover, for p≥1, the bifurcation diagram undergoes fold and splitting bifurcations. While for 0<p<1, the bifurcation diagram undergoes fold, splitting and segment-shrinking bifurcations. Our results extend and improve those of Brubaker and Pelesko [N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012) 5086–5102] and Pan and Xing [H. Pan, R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to a MEMS model, Nonlinear Anal. RWA 13 (2012) 2432–2445] by generalizing the nonlinearity (1−u)−2 to (1−u)−p with general p∈(1,∞). We also answer an open question raised by Brubaker and Pelesko on the extension of (global) bifurcation diagram results to general p>0. Concerning this open question, we find and prove that global bifurcation diagrams for 0<p<1 are different to and more complicated than those for p≥1.