摘要
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.
基金
supported by National Natural Science Foundation of China(Grant Nos.11831009 and 11571130)