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July  2010, 28(3): 1033-1050. doi: 10.3934/dcds.2010.28.1033

Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains

Craig Cowan 1, , Pierpaolo Esposito 2, and  Nassif Ghoussoub 3,

1. 

Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2
2. 

Dipartimento di Matematica, Università degli Studi "Roma Tre”, 00146 Roma, Italy
3. 

Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2

Received  March 2010 Revised  April 2010 Published  April 2010

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 δΩ. 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 $\lim$i$nf_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}>0$, then u * is regular for $N\leq 7$, while if $\gamma$:$= \lim$s$up_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}<+\infty$, 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 $p\ne 3$, 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 \le 12$ when $ f(t) = e^t$ [11], and for $ N \le 8$ when $ f(t) = (1-t)^{-2}$ [9] (see also [22]).
Keywords: Stable solution, Superlinear and Singular nonlinearities., Extremal solution.
Mathematics Subject Classification: Primary: 58E30, 58J05, 35J35; Secondary: 34B1.
Citation: Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033
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