Properties Of Positive Solutions For A System Of Integral Equation On Half Space

In this paper, we consider properties of positive solutions for a system of integral equation on the half space where R+n is the upper half Euclidian space, x* is the reflection of the point x about the plane xn=0. We assume that v∈Lq+1(R+n),u∈Lp+1(R+n) with We obtain the following theorems:Theorem 3.1 Assume that(u(x), v(x)) is a pair of positive solution of (0-1), u∈LP+1(R+n), v∈Lq+1(R+n). Then (u(x), v(x)) is uniformly bounded in R+n. Moreover (u(x), v(x)) is con-tinuous.Theorem 3.2 Let p, q> 1, then every positive solution (u(x), v(x)) of (0-1) is rotationally symmetric about some line parallel to xn-axis.Theorem 3.3 Let (u(x),v(x)) be a smooth solution of Then u(x) satisfies k=0,1,…,α/2-1. Let Rn be the n-dimensional Euclidean space, and let a be a real number satisfying 0<α<n. Consider the integral equations where 1/(q+1)+1/(q+1)=(n-α)/n When p=q and u(x)= v(x),(0-3) reduces to the following integral equation equation (0-4) arises as an Euler-Lagrange equation for a functional under a constraint in the context of the Hardy-Littlewood-Sobolev inequality (See[17]). In [17], Lieb classified the maximizers of the functional, and obtained the best constant of the Hardy-Littlewood-Sobolev inequality. He then posed the classification of all the critical points of the functional-the solutions of the integral equation (0-4) as an open problem.In [11], Chen, Li, and Ou solved Lieb's open problem by using the method of moving planes. They proved the following proposition:Proposition Every positive regular solution u(x) of (0-4) is radially symmetric and decreasing about some point x0 and therefore assumes the form with some positive constants c and t.They also established the equivalence between the integral equations and the following well-known family of semi-linear partial differential equations In the special caseα=2, there have been a series of results concerning the classification of the solutions (cf. [2], [5],[16], and [18]). Recently, Wei and Xu [29] generalized these results to the cases that a is any even number between 0 and n. For any real value of a between 0 and n, equation (0-6) is also of practical interest and importance. It can be proved that integral system (0-3) is equivalent to the PDE system In the special caseα=2, it reduces to the well-known Lane-Emden system.On the upper half Euclidian space the same equation naturally arises with Navier boundary conditions. In particular, when a is an even number, we have k=0,1,…,α/2-1...