Radial single point rupture solutions for a general MEMS model

Abstract

We study the initial value problem $$\begin{aligned} {\left\{ \begin{array}{ll} r^{-(\gamma -1)}\left( r^{\alpha }|u'|^{\beta -1}u'\right) '=\frac{1}{f(u)} &{} \text {for}\ 0<r<r_0,\\ u(r)>0 &{} \text {for}\ 0<r<r_0,\\ u(0)=0, \end{array}\right. } \end{aligned}$$for \(\gamma>\alpha >\beta \ge 1\) and \(f\in C[0,{{\bar{u}}})\cap C^2(0,{{\bar{u}}})\), \(f(0)=0\), \(f(u)>0\) on \((0, {{\bar{u}}})\) and f satisfies certain assumptions which include the standard case of pure power nonlinearities encountered in the study of Micro-Electromechanical Systems (MEMS). We obtain the existence and uniqueness of a solution \(u^*\) to the above problem, the rate at which it approaches the value zero at the origin and the intersection number of points with the corresponding regular solutions \(u(\,\cdot \,,a)\) (with \(u(0,a)=a\)) as \(a\rightarrow 0\). In particular, these results yield the uniqueness of a radial single point rupture solution and other qualitative properties for MEMS models. The bifurcation diagram is also investigated.