On MEMS equation with fringing field

where λ, δ > 0 and 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 uniform upper bound of classical solutions u away from the rupture level 1 for all λ ≥ μ. Moreover, there exists λδ > 0 such that there are at least two solutions when λ ∈ (0, λ ∗ δ); a unique solution exists when λ = λδ ; and there is no solution when λ > λ ∗ δ . This represents a dramatic change of behavior with respect to the zero fringing field case (i.e. δ = 0) and confirms the simulations in [14, 11].