Radial regular and rupture solutions for a MEMS model with fringing field

TL;DR

This paper studies radial solutions to a MEMS model involving a nonlinear PDE with fringing field effects, analyzing both regular and rupture solutions and their bifurcation behavior with respect to a key parameter.

Contribution

It introduces a detailed analysis of regular and rupture solutions for a MEMS PDE model, including bifurcation phenomena related to the parameter mbda.

Findings

Identification of conditions for existence of regular solutions.

Characterization of rupture solutions with singular behavior at the origin.

Bifurcation analysis revealing solution branches as mbda varies.

Abstract

We investigate radial solutions for the problem \[ \begin{cases} \displaystyle -\Delta U=\frac{\lambda+\delta|\nabla U|^2}{1-U},\; U>0 & \textrm{in}\ B,\\ U=0 & \textrm{on}\ \partial B, \end{cases} \] which is related to the study of Micro-Electromechanical Systems (MEMS). Here, $B\subset \mathbb{R}^N$ $(N\geq 2)$ denotes the open unit ball and $\lambda, \delta>0$ are real numbers. Two classes of solutions are considered in this work: (i) {\it regular solutions}, which satisfy $0<U<1$ in $B$ and (ii) {\it rupture solutions} which satisfy $U(0)=1$, and thus make the equation singular at the origin. Bifurcation with respect to parameter $\lambda>0$ is also discussed.