Liouville type results for semi-stable solutions of the weighted Lane–Emden system

We examine the weighted Lane–Emden system{−Δu=(1+|x|2)α2vp,−Δv=(1+|x|2)α2uq,inRN,where 10α>0, and the weighted Lane–Emden equation−Δu=(1+|x|2)α2up,inRN,where p>43 and α>0α>0. We prove that the system (equation) does not have the classical positive semi-stable solution in dimension N<2+(4+2α)(q+1)pq−1τ0+ (N<2+2(2+α)p−1(p+p2−p)). In particular, there is no positive semi-stable solution of the weighted Lane–Emden system if N≤10+4αN≤10+4α and 2≤p≤q2≤p≤q.