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

Abstract: We consider the Hardy–Hénon equation with and and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the whole space . It has been conjectured that this property is true if (and only if) , where is the Hardy–Sobolev exponent, given by . However, when , the conjecture had up to now been proved only for . Indeed the case seems more difficult, due to . In this paper, we prove the conjecture for in dimension , in the case of bounded solutions. Next, for the conjecture in the case , 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. [Copyright &y& Elsevier]