Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be poin...Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.展开更多
In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:(A+B∫∫_(R^(2N))|u(x)-u(y)|^(p)/|x-y|^(N+ps)dxdy)^(p-1)(-△...In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:(A+B∫∫_(R^(2N))|u(x)-u(y)|^(p)/|x-y|^(N+ps)dxdy)^(p-1)(-△)_(p)^(s)u+λV(x)|u|^(p-2)u=(∫_(R^(N))|U|^(P_(μ,S)^(*))/|x-y|^(μ)dy)|u|^(P_(μ,S)^(*))^(-2)u,x∈R^(N),where(-△)_(p)^(s)is the fractional p-Laplacian with 0<s<1<p,0<μ<N,N>ps,a,b>0,λ>0 is a parameter,V:R^(N)→R^(+)is a potential function,θ∈[1,2_(μ,s)^(*))and P_(μ,S)^(*)=pN-pμ/2/N-ps is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory.To the best of our knowledge,our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.展开更多
For conformal Hardy-Littlewood-Sobolev(HLS)inequalities[22]and reversed conformal HLS inequalities[8]on S^(n),a new proof is given for the attainability of their sharp constants.Classical methods used in[22]and[8]depe...For conformal Hardy-Littlewood-Sobolev(HLS)inequalities[22]and reversed conformal HLS inequalities[8]on S^(n),a new proof is given for the attainability of their sharp constants.Classical methods used in[22]and[8]depends on rearrangement inequalities.Here,we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method.The merit of the method is that it does not rely on rearrangement inequalities.展开更多
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-...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.展开更多
In this paper, we consider the following integral system： u（x） = R n v q （y） ｜ x y ｜ nα dy, v（x） = R n u p （y） ｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes...In this paper, we consider the following integral system： u（x） = R n v q （y） ｜ x y ｜ nα dy, v（x） = R n u p （y） ｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of （0.1） are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].展开更多
In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/（1 ＋ ...In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/（1 ＋ ｜j｜）^α（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^β,（0.1）vj =∑ k ∈Zn uk^p/（1 ＋ ｜j｜）^β（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^α,where u, v 〉 0, 1 〈 p, q 〈 ∞, 0 〈 λ 〈 n, 0 ≤α ＋ β≤ n- λ,1/p＋1〈λ＋α/n and 1/p＋1＋1/q＋1≤λ＋α＋β/n：=λ^-/n. We first show that positive solutions of（0.1） have the optimal summation interval under assumptions that u ∈ l^p＋1（Z^n） and v ∈ l^q＋1（Z^n）. Then we show that problem（0.1） has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{（n-λ^-）（q＋1）/pq-1,（n-λ^-）（p＋1）/pq-1} ≥λ^-.展开更多
基金supported in part by National Natural Foundation of China (Grant Nos. 11071250 and 11271162)
文摘Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.
文摘In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:(A+B∫∫_(R^(2N))|u(x)-u(y)|^(p)/|x-y|^(N+ps)dxdy)^(p-1)(-△)_(p)^(s)u+λV(x)|u|^(p-2)u=(∫_(R^(N))|U|^(P_(μ,S)^(*))/|x-y|^(μ)dy)|u|^(P_(μ,S)^(*))^(-2)u,x∈R^(N),where(-△)_(p)^(s)is the fractional p-Laplacian with 0<s<1<p,0<μ<N,N>ps,a,b>0,λ>0 is a parameter,V:R^(N)→R^(+)is a potential function,θ∈[1,2_(μ,s)^(*))and P_(μ,S)^(*)=pN-pμ/2/N-ps is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory.To the best of our knowledge,our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.
基金supported by the National Natural Science Foundation of China(Grant No.12071269)Natural Science Foundation of Zhejiang Province(Grant No.LY18A010013).
文摘For conformal Hardy-Littlewood-Sobolev(HLS)inequalities[22]and reversed conformal HLS inequalities[8]on S^(n),a new proof is given for the attainability of their sharp constants.Classical methods used in[22]and[8]depends on rearrangement inequalities.Here,we use the subcritical approach to construct the extremal sequence and circumvent the blow-up phenomenon by renormalization method.The merit of the method is that it does not rely on rearrangement inequalities.
基金supported by National Natural Science Foundation of China(Grant Nos.11831009 and 11571130)
文摘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.
基金Supported by National Natural Science Foundation of China-NSAF (10976026)
文摘In this paper, we consider the following integral system： u（x） = R n v q （y） ｜ x y ｜ nα dy, v（x） = R n u p （y） ｜ x y ｜ nμ dy, （0.1） where 0 〈 α, μ 〈 n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of （0.1） are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].
基金supported by NNSF of China(11261023,11326092),NNSF of China(11271170)Startup Foundation for Doctors of Jiangxi Normal University+1 种基金GAN PO 555 Program of JiangxiNNSF of Jiangxi(20122BAB201008)
文摘In this paper, we are concerned with properties of positive solutions of the following Euler-Lagrange system associated with the weighted Hardy-Littlewood-Sobolev inequality in discrete form{uj =∑ k ∈Zn vk^q/（1 ＋ ｜j｜）^α（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^β,（0.1）vj =∑ k ∈Zn uk^p/（1 ＋ ｜j｜）^β（1 ＋ ｜k- j｜）^λ（1 ＋ ｜k｜）^α,where u, v 〉 0, 1 〈 p, q 〈 ∞, 0 〈 λ 〈 n, 0 ≤α ＋ β≤ n- λ,1/p＋1〈λ＋α/n and 1/p＋1＋1/q＋1≤λ＋α＋β/n：=λ^-/n. We first show that positive solutions of（0.1） have the optimal summation interval under assumptions that u ∈ l^p＋1（Z^n） and v ∈ l^q＋1（Z^n）. Then we show that problem（0.1） has no positive solution if 0 〈λˉ pq ≤ 1 or pq 〉 1 and max{（n-λ^-）（q＋1）/pq-1,（n-λ^-）（p＋1）/pq-1} ≥λ^-.
