  We examine the regularity of the extremal solution of the nonlinear
eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain
$\Omega$ in $ \IR^N$, with the Navier boundary condition $ u=\Delta u =0 $ on $
\pOm$. Here $ \lambda$ is a positive parameter and $f$ is a non-decreasing
nonlinearity with $f(0)=1$. We give general pointwise bounds and energy
estimates which show that for any convex and superlinear nonlinearity $f$, the
extremal solution $ u^*$ is smooth provided $N\leq 5$.
