Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

We examine the fourth order problem $\Delta^2 u = \lambda f(u)$ in $\Omega$ with $\Delta u = u =0$ on $\partial \Omega$, where $\lambda > 0$ is a parameter, $\Omega$ is a bounded domain in $R^N$ and where $f$ is one of the following nonlinearities: $f(u)=e^u$, $f(u)=(1+u)^p$ or $f(u)= \frac{1}{(1-u)^p}$ where $p>1$. We show the regularity of all semi-stable solutions and hence of the extremal solutions, provided [N < 2 + 4 \sqrt{2} + 4 \sqrt{2 - \sqrt{2}} \approx 10.718 when $f(u)=e^u$,] and [\frac{N}{4} < \frac{p}{p-1} + \frac{p+1}{p-1} (\sqrt{\frac{2p}{p+1}} + \sqrt{\frac{2p}{p+1} - \sqrt{\frac{2p}{p+1}}} - 1/2)] when $f(u)=(u+1)^p$. New results are also obtained in the case where $f(u)=(1-u)^{-p}$. These are substantial improvements to various results on critical dimensions obtained recently by various authors. We view the equation as a system and then derive a new stability inequality, valid for minimal solutions, which allows a method of proof which is reminiscent of the second order case.