We examine the elliptic system given byfor 1 < p ≤ θ and the fourth order scalar equationwhere 1 < θ. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (1) (resp. (2)) provided N ≤ 10 and 2 ≤ p ≤ θ (resp. N ≤ 10 and 1 < θ). Results for higher dimensions are also obtained. These results regarding stable solutions on the full space imply various Liouville theorems for positive (possibly unstable) bounded solutions ofwith u = v = 0 on ∂R N + . In particular there is no positive bounded solution of (3) for any 2 ≤ p ≤ θ if N ≤ 11. Higher dimensional results are also obtained.

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 $R^N$,\ud with the Navier boundary condition $ u=\Delta u =0 $ on $ \partial \Omega$.\ud 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 $N\leq 5$.\ud 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 $N < \frac{8}{\gamma}$.\ud It follows that $u^*$ is smooth if $f(t) = e^t$ and $ N \le 8$, or if $f(t) = (1+t)^p$ and\ud $N< \frac{8p}{p-1}$.\ud 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 \ud 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\ud 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]

We study the regularity of the extremal solution of the semilinear biharmonic equation ∆ 2 u = λ (1−u) 2 , which models a simple Micro-Electromechanical System (MEMS) device on a ball B ⊂ R N , under Dirichlet boundary conditions u = ∂ν u = 0 on ∂B. We complete here the results of

We examine the fourth order problem ∆ 2 u = λf (u) in Ω with ∆u = u = 0 on ∂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) = 2010 Mathematics Subject Classification.

Abstract. We are interested in the existence versus non-existence of nontrivial stable sub-and super-solutions ofwith positive smooth weights ω 1 (x), ω 2 (x). We consider the cases f (u) = e u , u p where p > 1 and −u −p where p > 0. We obtain various non-existence results which depend on the dimension N and also on p and the behaviour of ω 1 , ω 2 near infinity. Also the monotonicity of ω 1 is involved in some results. Our methods here are the methods developed by Farina, [9]. We examine a specific class of weights ω 1 (x) = (|x| 2 + 1) α 2 and ω 2 (x) = (|x| 2 + 1)is a positive function with a finite limit at ∞. For this class of weights non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.