Global bifurcation curves of a regularized MEMS model

Abstract

The two-parameter differential equation u′′(x)+λ(1−u)2−λε2(1−u)4=0 with the boundary condition u(−1)=u(1)=0 governs the steady-state solutions from a regularized MEMS model. We prove that there exist two constants εˆ(≈0.25458) and εˇ(≈0.29212) such that the bifurcation curve is S-shaped for 0<ε⩽εˆ and is strictly increasing for ε⩾εˇ in the λ,‖u‖∞-plane. This partly confirms the numerical simulations in Lindsay et al. (2014), and also improves a recent result in Iuorio et al. (2019), where the S-shaped curve is proved for sufficiently small ε.