On the existence of positive solutions for some nonlinear boundary value problemsand applications to MEMS models

Motivated by some nonlinear models recently arising in Micro-Electro-Mechanical System (MEMS) and new progress on one-dimensional mean curvature type problems, we investigate the existence and exact numbers of positive solutions for a class of boundary value problems with $\varphi$-Laplacian $$ -(\varphi(u'))'=\lambda f(u)\; on (-L, L),\quad u(-L)=u(L)=0, $$ when the parameters $\lambda$ and $L$ vary. Various exact multiplicity results as well as global bifurcation diagrams are obtained. These results include the applications to one-dimensional MEMS equations with fringing field as well as mean curvature type problems. We also extend and improve one of the main results of Korman and Li [ Proc. Roy. Soc. Edinburgh Sect. A, 140(6):1197--1215, 2010 ] (Theorem 3.4). With the aid of numerical simulations, we find many interesting new examples, which reveal the striking complexity of bifurcation patterns for the problem.