Sharp Hardy-Littlewood-Sobolev inequality on the upper half space

2013/09/09 by Jingbo Dou, Dou, Jingbo, Mei‐Jun Zhu +1 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.1309.2341

Abstract

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent λ=n-α (that is for the case of α>n). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.

Citations

• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• Uniqueness theorems through the method of moving spheres
• Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations
• A relation between pointwise convergence of functions and convergence of functionals
• Logarithmic Sobolev Inequalities

Cited by

• Infinitely many solutions to a conformally invariant elliptic equation with Choquard-type nonlinearity

Related

• Subcritical Approach to Sharp Hardy-Littlewood-Sobolev Type Inequalities on the Upper Half Space
• Reversed Hardy-Littlewood-Sobolev inequalities with vertical weights on the upper half space
• The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space
• Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel
• Affine Hardy--Littlewood--Sobolev inequalities