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 $\Delta^2 u = \frac{\lambda}{(1-u)^}$, which models a simple micro-electromechanical system (MEMS) device on a ball $B \subset R^N$, under Dirichlet boundary conditions $u = \partial_\nu u =0$ on ∂B. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” $λ∗ > 0$ such that a stable classical solution $u_λ$ with $0 λ∗. Our main result asserts that the extremal solution u_λ∗ is regular (sup_B u_λ∗ < 1) provided N\leq 8 while u_λ∗ is singular (sup_B u_λ∗ = 1) for N\geq 9, in which case $1−C_0 |x|^{4/3}\leq u_λ∗(x)\leq 1 − |x|^{4/3}$ on the unit ball, where $C 0= (λ∗/\bar λ)^{1/3} and $\bar λ=8/9 (N−2/3)(N−8/3)$