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)) <sup>2</sup> )′ =λ(1-u) <sup>p</sup> ,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) <sup>-2</sup> to (1-u) <sup>-p</sup> with general pâ̂̂(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. © 2013 Elsevier Ltd. All rights reserved.