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

Abstract

We investigate radial solutions for the problem \[ \begin{cases} \displaystyle -ΔU=\frac{λ+δ|\nabla U|^2}{1-U},\; U>0 & \textrm{in}\ B,\\ U=0 & \textrm{on}\ \partial B, \end{cases} \] which is related to the study of Micro-Electromechanical Systems (MEMS). Here, $B\subset {\mathbb{R}}^N$ $(N\ge 2)$ 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.

arXiv comment14 pages