On MEMS equation with fringing field

We consider the MEMS equation with fringing field

$$-\Delta u = \lambda(1 + \delta\vert\nabla u\vert^2)(1 - u)^{-2}$$    in $$\Omega, u=0$$    on $$\partial \Omega,$$

where $\lambda, \delta >0$ and $\Omega \subset \mathbb{R}^n$ is a smooth and bounded domain. We show that when the fringing field exists (i.e. $\delta > 0$), given any $\mu > 0$, we have a uniform upper bound of classical solutions $u$ away from the rupture level 1 for all $\lambda \geq \mu$. Moreover, there exists $\overline \lambda_{\delta}^{*}>0$ such that there are at least two solutions when $\lambda \in (0, \overline \lambda_{\delta}^{*})$; a unique solution exists when $\lambda= \overline \lambda_{\delta}^{*}$; and there is no solution when $\lambda >\overline \lambda_{\delta}^{*}$. This represents a dramatic change of behavior with respect to the zero fringing field case (i.e., $\delta =0$) and confirms the simulations in a paper by Pelesko and Driscoll as well as a paper by Lindsay and Ward.