Weighted gradient estimates and refined existence results for a Hardy-type elliptic equation

We review Dirichlet-type problems of the form -Δ u=λ u/|x|²+ g(· ,∇ u) in Ω , u=0\ on ∂ Ω , - Δ u = λ u | x | 2 + g ( · , ∇ u ) in Ω , u = 0 on ∂ Ω , where N 3 N ⩾ 3 , Ω ⊂ RN Ω ⊂ R N is a bounded regular domain containing the origin, and 0<λ < Λ N=( N-2/2) ² 0 < λ < Λ N = N - 2 2 2 , the optimal Hardy constant. We first consider the Hardy–Poisson problem, i.e., the case g(· ,z)=f g ( · , z ) = f with f in a weighted Lebesgue space, and we provide sharp weighted regularity estimates for the SOLA (solution obtained as the limit of approximations) of the linear problem. These results allow us to extend and refine earlier existence results for a KPZ-type problem. In particular, for nonlinearities of the form g(· ,z)=|z|q+ρ f g ( · , z ) = | z | q + ρ f , with f belonging to a suitable weighted Lᵐ-space f belonging to a suitable weighted L m -space , we establish the existence of a weak solution under suitable conditions on m and q , for ρ >0 ρ > 0