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

In this paper,we prove that the supremum sup{∫B∫B|u(y)|^p(|y|)|u(x)|^p(|x|)/|x-y|^u dxdy:u∈H0^1,rad(B),‖▽u‖L^2(B)=1}is attained,where B denotes the unit ball in R^N(N≥3),μ∈(0,N),p(r)=2^xμ+r^t,t∈(0,min{N/2-μ/4,N-2})and 2μ^*=(2N-μ)/(N-2)is the critical exponent for the Hardy-Littlewood-Sobolev inequality.

In this paper, we prove that the supremum sup{ ∫B∫B|u（y）|p（|y|）|u（x）|p（|x|）/|x-y|μdxdy : u ∈ H0,rad1（B）, ||?||uL2（B）= 1}is attained, where B denotes the unit ball in RN（N ≥3）, μ ∈（0, N）, p（r） = 2μ*+ rt, t ∈（0, min{N/2-μ/4, N-2}） and 2μ*=（2N-μ）/（N-2） is the critical exponent for the Hardy-Littlewood-Sobolev inequality.