Global bifurcation curves of a regularized MEMS model (Q2006367)

The two-parameter differential equation \[ u''(x)+\frac{\lambda}{(1-u)^2}-\frac{\lambda\varepsilon^2}{(1-u)^4} \] with the boundary condition \(u(-1)=u(1)=0\) governs the steady-state solutions from a regularized MEMS model, where \(\lambda>0\) and \(\varepsilon\in(0,1)\) are parameters. In this paper, the authors investigate the global bifurcation curve and exact multiplicity of positive solutions for the above problems. They prove that there exist two constants \(\hat{\varepsilon}(\approx0.25458)\) and \(\check{\varepsilon}(\approx 0.29212)\) such that the bifurcation curve is \(S\)-shaped for \(0<\varepsilon\leq\hat{\varepsilon}\) and is strictly increasing for \(\varepsilon\geq \check{\varepsilon}\) in the \((\lambda, \|u\|_\infty)\)-plane. This partly confirms the numerical simulations in [\textit{A. E. Lindsay} et al., Physica D 280--281, 95--108 (2014; Zbl 1349.74127)], and also improves a recent result in [\textit{A. Iuorio} et al., SIAM J. Appl. Dyn. Syst. 18, No. 2, 661--708 (2019; Zbl 1458.34103)], where the \(S\)-shaped curve is proved for sufficiently small \(\varepsilon\).