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

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,−L0 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 00. Concerning this open question, we find and prove that global bifurcation diagrams for 0