Liouville type theorems for stable solutions of the weighted elliptic system

Abstract We examine the weighted elliptic system { − Δ u = ( 1 + | x | 2 ) α 2 v , − Δ v = ( 1 + | x | 2 ) α 2 u p , in R N , and prove Liouville type theorems for the classical positive and nonnegative stable solutions in higher dimension. In particular, there are no positive stable solutions for any N ≤ 12 + 5 α , p > 1 and α > 0 . Our proof is based on a combination of the bootstrap argument, Souplet's inequality and intermediate stability criterion, which is used to obtain sharp results.