On the classification and evolution of bifurcation curves for a quasilinear regularized MEMS model. (October 2022) - Liang, Yuhan; Pan, Hongjing
Abstract: The quasilinear equation − ( u ′ ( x ) 1 + ( u ′ ( x ) ) 2 ) ′ = λ ( 1 − u ) 2 − λ ɛ 2 ( 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.