Abstract
We examine the elliptic system given by1for 1 < p ⩽ θ and the fourth order scalar equation2where 1 < θ. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (1) (respectively, (2)) provided N ⩽ 10 and 2 ⩽ p ⩽ θ (respectively, 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 of3with u = v = 0 on . In particular there is no positive bounded solution of (3) for any 2 ⩽ p ⩽ θ if N ⩽ 11. Higher dimensional results are also obtained.
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