Sharp reversed Hardy–Littlewood–Sobolev inequality with extension kernel

In this paper, we prove the following reversed Hardy–Littlewood–Sobolev inequality with extension kernel: x^β_n |x − _y|n−_α f(y)g(x) dy dx ≥ C_n,α,β,p∥f∥_Lp₍∂_Rn₊₎∥g∥_Lq′_(Rn₊₎ Rⁿ₊ ∂Rⁿ₊ for any nonnegative functions f ∈ L^p(∂Rⁿ₊) and g ∈ L^q′(Rⁿ₊), where n ≥ 2, p, q^′ ∈ (0, 1), α > n, 0 ≤ β < ^α_n⁻₋ⁿ₁ , p > _α−1−ⁿ⁻_(n¹₋1)_β are such that ⁿ⁻_n¹_p¹ + _q¹_′ − α+β⁻¹ = 1. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal n invariant case, we classify all the extremal functions and hence derive the best constant via the method of moving spheres. It is quite surprising that the extremal functions do not depend on β. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler–Lagrange equations by using Pohozaev identities.