Higher order singular problems of Caffarelli–Kohn–Nirenberg–Lin type

We prove the existence of nontrivial critical points of the functionalJλ(u)=∫RN1p(||x|−a∇ku|p−λh(x)||x|−(a+k)u|p)−1qQ(x)||x|−bu|qdx, related to the Caffarelli–Kohn–Nirenberg inequality and its higher order variant by Lin. As a consequence we obtain nontrivial solutions of the degenerate elliptic equationΔ(|x|−ap|Δu|p−2Δu)−λh(x)|x|−(a+k)p|u|p−2u=Q(x)|x|−bq|u|q−2u.Δ(|x|−ap|Δu|p−2Δu)−λh(x)|x|−(a+k)p|u|p−2u=Q(x)|x|−bq|u|q−2u. We also show that when p=2p=2, JλJλ has infinitely many critical points.