On mems equation with fringing field

We consider the MEMS equation with fringing field -Δu = γ(1 + δ/∇u/²)(l - u)^-2 in Ω,u = 0 on ∂Ω, where γ,δ> 0and Ω ⊂ ℝⁿ is a smooth and bounded domain. We show that when the fringing field exists (i.e. δ> 0), given any μ> 0, we have a uniform upper bound of classical solutions u away from the rupture level 1 for all γ ≥ μ. Moreover, there exists γδ̄> 0 such that there are at least two solutions when γ ε (0,γδ̄*); a unique solution exists when γ = γδ̄*; and there is no solution when γ> γδ. This represents a dramatic change of behavior with respect to the zero fringing field case (i.e., δ = 0) and confirms the simulations in a paper by Pelesko and Driscoll as well as a paper by Lindsay and Ward.