On the classification and evolution of bifurcation curves for a quasilinear regularized MEMS model

Abstract

The quasilinear equation −(u′(x)1+(u′(x))2)′=λ(1−u)2−λɛ2(1−u)4 with the boundary condition u(−L)=u(L)=0 governs the steady-state solutions from a regularized MEMS model. We prove that for any evolution parameters ɛ∈(0,1) and L>0, the global bifurcation curve of positive solutions is strictly increasing or ⊃-like shaped or S-like shaped in the (λ,‖u‖∞)-plane. The bifurcation curves present a variety of shapes and structures, significantly different from those in non-regularized case (i.e., ɛ=0) and in the simplified semilinear case. The main tools are some new time-map techniques, the total positivity theory, and Sturm’s Theorem.