Liouville theorems for stable Lane-Emden systems and biharmonic problems

We examine the elliptic system given by −∆u = v, −∆v = u, in R , (1) for 1 < p ≤ θ and the fourth order scalar equation ∆u = u, in R , (2) where 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 of −∆u = v, −∆v = u, in RN−1, (3) with u = v = 0 on ∂R+ . In particular there is no positive bounded solution of (3) for any 2 ≤ p ≤ θ if N ≤ 11. Higher dimensional results are also obtained. 2010 Mathematics Subject Classification: 35J61, 35J47.