In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.
We classify all positive solutions for the following integral system：{ui（x）=∫Rn1/│x-y│^n-α fi（u（y））dy,x∈R^n,i=1,…,m,0〈α〈n,and u（x）=（u1（x）,u2（x）…,um（x））.Here fi（u）, 1 ≤ i ≤m, monotone non...We classify all positive solutions for the following integral system：{ui（x）=∫Rn1/│x-y│^n-α fi（u（y））dy,x∈R^n,i=1,…,m,0〈α〈n,and u（x）=（u1（x）,u2（x）…,um（x））.Here fi（u）, 1 ≤ i ≤m, monotone nondecreasing are real-valued functions of homogeneous degree n＋α/n-α and are monotone nondecreasing with respect to all the independent variables U1, u2, ..., urn.In the special case n ≥ 3 and α = 2. we show that the above system is equivalent to thefollowing elliptic PDE system：This system is closely related to the stationary SchrSdinger system with critical exponents for Bose-Einstein condensate展开更多
In this article, we study positive solutions to the system{Aαu（x） = Cn,αPV∫Rn（a1（x-y）（u（x）-u（y）））/（｜x-y｜n＋α）dy = f（u（x）, Bβv（x） = Cn,βPV ∫Rn（a2（x-y）（v（x）-v（y））/（｜x-y｜n＋β）dy ...In this article, we study positive solutions to the system{Aαu（x） = Cn,αPV∫Rn（a1（x-y）（u（x）-u（y）））/（｜x-y｜n＋α）dy = f（u（x）, Bβv（x） = Cn,βPV ∫Rn（a2（x-y）（v（x）-v（y））/（｜x-y｜n＋β）dy = g（u（x）,v（x））.To reach our aim, by using the method of moving planes, we prove a narrow region principle and a decay at infinity by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.展开更多
基金supported by NSFC(11761082)Yunnan Province,Young Academic and Technical Leaders Program(2015HB028)
文摘In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.
基金supported by NSF Grant DMS-0604638Li partially supported by NSF Grant DMS-0401174
文摘We classify all positive solutions for the following integral system：{ui（x）=∫Rn1/│x-y│^n-α fi（u（y））dy,x∈R^n,i=1,…,m,0〈α〈n,and u（x）=（u1（x）,u2（x）…,um（x））.Here fi（u）, 1 ≤ i ≤m, monotone nondecreasing are real-valued functions of homogeneous degree n＋α/n-α and are monotone nondecreasing with respect to all the independent variables U1, u2, ..., urn.In the special case n ≥ 3 and α = 2. we show that the above system is equivalent to thefollowing elliptic PDE system：This system is closely related to the stationary SchrSdinger system with critical exponents for Bose-Einstein condensate
基金Supported by National Natural Science Foundation of China(11771354)
文摘In this article, we study positive solutions to the system{Aαu（x） = Cn,αPV∫Rn（a1（x-y）（u（x）-u（y）））/（｜x-y｜n＋α）dy = f（u（x）, Bβv（x） = Cn,βPV ∫Rn（a2（x-y）（v（x）-v（y））/（｜x-y｜n＋β）dy = g（u（x）,v（x））.To reach our aim, by using the method of moving planes, we prove a narrow region principle and a decay at infinity by the iteration method. On the basis of these results, we conclude radial symmetry and monotonicity of positive solutions for the problems involving the weighted fractional system on an unit ball and the whole space. Furthermore, non-existence of nonnegative solutions on a half space is given.
