Touchdown solutions in general MEMS models

Abstract

We study general equations modeling electrostatic MEMS devices \begin{equation} \begin{cases} \label{P} φ\big(r,- u'(r)\big)=λ\int_0^r\frac{f(s)}{g(u(s))}\,\mathrm{d}s, & r\in(0,1), \\ 0 < u(r) < 1, & r\in(0,1), \\ u(1) = 0, \tag{$P_{\lambda }$} \end{cases} \end{equation} where $\phi$, $g$, $f$ are some functions on $[0,1]$ and $\lambda >0$ is a parameter. We obtain results on the existence and regularity of a touchdown solution to \eqref{P} and find upper and lower bounds on the respective pull-in voltage. In the particular case, when $\phi (r,v)=r^{\alpha }|v{|}^{\beta }v$, i.e., when the associated differential equation involves the operator $r^{-\gamma }(r^{\alpha }|u^{\prime }{|}^{\beta }{u}^{\prime }{)}^{\prime }$, we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.