Reverse Stein–Weiss inequalities and existence of their extremal functions

Abstract

In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in R n :We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the ndimensional sphere S n by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities.

Reverse conformally invariant Sobolev inequalities on the sphere

“…The optimal constant H µ,n > 0 and all optimizing functions F, G were obtained in [22]; see also [39,16]. By conformal invariance, (9) has an equivalent version on S n , namely,…”

Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel

“…Some remarkable extensions have already been obtained on the upper half space by Dou and Zhu [31] and on compact Riemannian manifolds by Han and Zhu [42]. The reversed (weighted) Hardy-Littlewood-Sobolev inequalities were derived in [12,32,58,59]. For more results concerning the (weighted) Hardy-Littlewood-Sobolev inequality and Hardy-Sobolev equations, please refer to [2,3,8,11,18,19,20,22,23,24,25,26,28,47,50,53,56,62] and the references therein.…”

Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel

“…The inequality and its extensions have many applications in partial differential equations. Some remarkable extensions have already been obtained on the upper half space by Dou and Zhu [22], on compact Riemannian manifolds by Han and Zhu [35] and the reversed (weighted) Hardy–Littlewood–Sobolev inequality in [10, 23, 48, 49]. For more results about the (weighted) Hardy–Littlewood–Sobolev inequality, the general weighted inequalities and their corresponding Euler–Lagrange equations, refer to e.g.…”