We study the regularity of the extremal solution of the semilinear biharmonic
equation βΔ2u−τΔu=(1−u)2λ 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 a 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 βτ small. Our proof for the
singularity of extremal solutions in dimensions N≥9 is based on certain
improved Hardy-Rellich inequalities