The critical dimension for a 4th order problem with singular nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ∆u = λ (1−u)2 , which models a simple Micro-Electromechanical System (MEMS) device on a ball B ⊂ R , under Dirichlet boundary conditions u = ∂νu = 0 on ∂B. We complete here the results of F.H. Lin and Y.S. Yang [14] regarding the identification of a “pull-in voltage” λ∗ > 0 such that a stable classical solution uλ with 0 < uλ < 1 exists for λ ∈ (0,λ∗), while there is none of any kind when λ > λ∗. Our main result asserts that the extremal solution uλ∗ is regular (supB uλ∗ < 1) provided N ≤ 8 while uλ∗ is singular (supB uλ∗ = 1) for N ≥ 9, in which case 1 − C0|x| ≤ uλ∗(x) ≤ 1 − |x| on the unit ball, where C0 := “ λ∗ λ ” 1 3 and λ̄ := 89 (N − 2 3 )(N − 8 3 ).