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

TLDR: Pan et al. as discussed by the authors studied the global bifurcation diagrams and exact multiplicity of positive solutions for the 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 .