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)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=fracl(1-u)²{{Delta^2} u=frac{lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì mathbbR^N{Bsubset{mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=partial_nu u=0} on ?B{partial 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 < u _λ < 1 exists for l ? (0,l^*){lambdain (0,lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{lambda^*}} is regular (sup_B u_l^* < 1 ){({rm sup}_B u_{lambda^*} <1 )} provided N leqq 8{N leqq 8} while u_l^*{u_{lambda^*}} is singular (sup_B u_l^* = 1){({rm sup}_B u_{lambda^*} =1)} for N geqq 9{N geqq 9}, in which case 1-C₀|x|^4/3 leqq u_l^* (x) leqq 1-|x|^4/3{1-C_0|x|^{4/3} leqq u_{lambda^*} (x) leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(fracl^*[`(l)])<sup>frac</sup>13{C_0:=left(frac{lambda^*}{overline{lambda}}right)^frac{1}{3}} and [`(l)]: = frac89(N-frac23)(N- frac83){bar{lambda}:= frac{8}{9}left(N-frac{2}{3}right)left(N- frac{8}{3}right)}.