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

We examine the fourth order problem $Δ^2 u = λf(u) $ in $ Ω$ with $ Δu = u =0 $ on $ \partial Ω$, where $ λ> 0$ is a parameter, $ Ω$ 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.