The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation Delta(2)u = lambda/(1-u)(2), which models a simple micro-electromechanical system (MEMS) device on a ball B subset of R(N), under Dirichlet boundary conditions u = partial derivative(v)u = 0 on partial derivative B. We complete here the results of Lin and Yang [14] regarding the identification of a "pull-in voltage" lambda* > 0 such that a stable classical solution u(lambda) with 0 < u(lambda) < 1 exists for lambda is an element of (0, lambda*), while there is none of any kind when lambda > lambda*. Our main result asserts that the extremal solution u(lambda*) is regular (supB u(lambda*) < 1) provided N <= 8 while u(lambda*) is singular (sup(B) u(lambda*) = 1) for N >= 9, in which case 1 - C(0)vertical bar x vertical bar(4/3) <= u(lambda*)(x) <= 1 - vertical bar x vertical bar(4/3) on the unit ball, where C(0) := (lambda*/<(lambda)over bar>)(1/3) and (lambda)over bar> := 8/9 (N - 2/3) (N - 8/3).