On the critical dimension of a fourth order elliptic problem with negative exponent

We study the regularity of the extremal solution of the semilinear biharmonic equation on a ball B⊂RN, under Navier boundary conditions u=Δu=0 on ∂B, where λ>0 is a parameter, while τ>0, β>0 are fixed constants. It is known that there exists λ∗ such that for λ>λ∗ there is no solution while for λ<λ∗ there is a branch of minimal solutions. Our main result asserts that the extremal solution u∗ is regular (supBu∗<1) for N⩽8 and β,τ>0 and it is singular (supBu∗=1) for N⩾9, β>0, and τ>0 with small. Our proof for the singularity of extremal solutions in dimensions N⩾9 is based on certain improved Hardy–Rellich inequalities.