Integral equations on compact manifold with boundary

Let (M n ,g,) be a smooth compact Riemannian manifold with boundary and n 3 .This paper is devoted to studying a class of integral systemSince the system is the Euler-Lagrange equations of extremal problemwe will study the existence of the system by concentration-compactness principle.Firstly, we get N K (,M) C e (n,, p ) , where C e (n,, p ) is the best constant of Hardy-Littlewood-Sobolev inequalities on the upper half space established by Dou and Zhu [6] and equals to (n,, p ) , we prove that N K (,M) is attained.Namely, under the criterion N K (,M) > C e (n,, p ) , we get the existence of the system.Lastly, a concrete example satisfying the criterion is given.The example is closely related to the conformal problems studied by Escobar [9,10].