Achievability of a supremum for the Hardy-Littlewood-Sobolev inequality with supercritical exponent

Abstract

In this paper, we prove that the supremum

$$\sup \left\{{\int_B {\int_B {{{{{\left| {u(y)} \right|}^{p(\left| y \right|)}}{{\left| {u(x)} \right|}^{p(\left| x \right|)}}} \over {{{\left| {x - y} \right|}^\mu}}}}} dxdy:u \in H_{0,{\rm{rad}}}^1(B),\;\;{{\left\| {\nabla u} \right\|}_{{L^2}(B)}} = 1} \right\}$$

is attained, where B denotes the unit ball in ℝ^N (N ⩾ 3), μ ∈ (0, N), p(r) = 2^*_μ + r^t, t ∈ (0, min{N/2 − μ/4, N − 2}) and 2^*_μ = (2N − μ)/(N − 2) is the critical exponent for the Hardy-Littlewood-Sobolev inequality.