Global bifurcation curves of a regularized MEMS model

Abstract The two-parameter differential equation u ′ ′ ( x ) + λ ( 1 − u ) 2 − λ e 2 ( 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 e ˆ ( ≈ 0 . 25458 ) and e ˇ ( ≈ 0 . 29212 ) such that the bifurcation curve is S-shaped for 0 e ⩽ e ˆ and is strictly increasing for e ⩾ e ˇ 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 e .