Symmetry Of Positive Solutions For A Class Of Multi-component Fractional Laplace Systems

In this paper,we study the following multi-component fractional systemwhere 0<?i<2,i = 1,…,m,n>2.We will prove that u1(x),…,vm(x)are constant,or u1(x),…,um(x)are radially symmetric about some point in Rn,or for each i?{1,…,m},[fi(ti(x))+gi(x/|x|2)]/ti(x)Pi is a const,ant in a fixed set Ei when f1,…,fm:R+? and g1,…,gm:R+?R satisfy some monotonicity conditions and decay conditions,ti(x)= |x|n-?i+1ui+1(x)(i=1,…,m-1),tm(x)=|x|n-?1u1(x),ui(x)is the Kelvin transform of ui(x).we also establish the equivalence between the above mentioned differential system and the following integral systemwhere f1,…,fm:R+ ? R are homogeneous and g1,…,gm:R+?R satisfy the monotonicity condition.Moreover,we prove the symmetry of positive solutions in the critical case and the non-existence of positive solutions in the sub-critical case by using moving plane method in integral form.Our main results generalizes the main results of Yan LI,Pei MA?Science China Mathematics?1-20(2017)and Li D,Li Z?Front.Math.China?389-402(2017).