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 (The equation is abbreviated) where λ＞ 0 is a bifurcation parameter, and p,L ＞ 0 are two evolution parameters. We are able to determine the exact number of positive solutions by the positive 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 [2] and Pan and Xing [16] 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 diagrams results to general p ＞ 0. To 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.