Liouville-type theorems and bounds of solutions of Hardy–Hénon equations

We consider the Hardy–Hénon equation −Δu=a|x|up with p>1 and a∈R and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space RN. It has been conjectured that this property is true if (and only if) p0 seems more difficult, due to pS(a)>(N+2)/(N−2).In this paper, we prove the conjecture for a>0 in dimension N=3, in the case of bounded solutions. Next, for the conjecture in the case a<0, and for related estimates near isolated singularities and at infinity, we give new proofs – based in particular on doubling-rescaling arguments – and we provide some extensions of these estimates. These proofs are significantly simpler than the previously known ones. Finally, we clarify some of the previous results on a priori estimates for the related Dirichlet problem.