Radial single point rupture solutions for a general MEMS model

We study the initial value problem $$\{\begin{array}{ll}{r}^{-(\gamma -1)}{\left(r^{\alpha }\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}^{\beta -1}{u}^{\mathrm{\prime }}\right)}^{\mathrm{\prime }}=\frac{1}{f(u)}& \text{for }0<r<r_0,\\ u(r)>0& \text{for }0<r<r_0,\\ u(0)=0,\end{array}$$ 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 $u^{\ast }$ 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 $a\to 0$. 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.