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

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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⊂ⁿ, under Dirichlet boundary conditions u=∂νu=0 on ∂ B. We complete here the results of F.H. Lin and Y.S. Yang LY regarding the identification of a "pull-in voltage" >0 such that a stable classical solution u with 0<u<1 exists for ∈ (0,), while there is none of any kind when >. Our main result asserts that the extremal solution uλ* is regular (uλ* <1) provided N ≤ 8 while uλ* is singular (uλ* =1) for N ≥ 9, in which case 1-C₀|x|⁴/³≤ uλ* (x) ≤ 1-|x|⁴/³ on the unit ball, where C₀:= (λ*/λ)¹/³ and λ:= 8/9 (N-2/3) (N- 8/3).