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 $R^N$, with the Navier boundary condition $ u=\Delta u =0 $ on $ \partial \Omega$. We establish energy estimates which show that for any non-decreasing convex and superlinear nonlinearity $f$ with $f(0)=1$, the extremal solution $ u^*$ is smooth provided $N\leq 5$. If in addition $\liminf_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}>0$, then $u^*$ is regular for $N\leq 7$, while if $\gamma:= \limsup_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}<+\infty$, then the same holds for $N < \frac{8}{\gamma}$. It follows that $u^*$ is smooth if $f(t) = e^t$ and $ N \le 8$, or if $f(t) = (1+t)^p$ and $N< \frac{8p}{p-1}$. We also show that if $ f(t)=(1-t)^{-p}$, $p>1$ and $p\neq 3$, then $u^*$ is smooth for $N \leq \frac{8p}{p+1}$. While these results are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established on radial domains, e.g., $ u^*$ is smooth for $ N \leq 12$ when $f(t)=e^t$ [J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565–592], and for $N\leq8$ when $ f(t)=(1-t)^{-2}$ [C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., 198 (2010), 763–787] (see also [A. Moradifam, On the critical dimension of a fourth order elliptic problem with negative exponent, J. Differential Equations, 248 (2010), 594–616])