Liouville-type theorems and bounds of solutions for Hardy-H\'enon elliptic systems

Abstract

We consider the Hardy-H\'enon system $-\Delta u =|x|^a v^p$, $-\Delta v =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space ${\mathbb R}^N$. In view of known results, it is a natural conjecture that this property should be true if and only if $(N+a)/(p+1)+$ $(N+b)/(q+1)>N-2$. In this paper, we prove the conjecture for dimension N=3 in the case of bounded solutions and in dimensions $N\le 4$ when $a,b\le 0$, among other partial nonexistence results. As far as we know, this is the first optimal Liouville type result for the Hardy-H\'enon system. Next, as applications, we give results on singularity and decay estimates as well as a priori bounds of positive solutions.