Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains

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 $N\leq 5$.