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

Abstract

We consider the Hardy–Hénon equation − Δ u = | x | a u p 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 R N . It has been conjectured that this property is true if (and only if) p < p S ( a ) , where p S ( a ) is the Hardy–Sobolev exponent, given by ( N + 2 + 2 a ) / ( N − 2 ) . However, when N ⩾ 3 , the conjecture had up to now been proved only for a ⩽ 0 . Indeed the case a > 0 seems more difficult, due to p S ( 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.