A SINGULARITY ANALYSIS OF POSITIVE SOLUTIONS TO AN EULER-LAGRANGE INTEGRAL SYSTEM

In this paper we study the asymptotic behavior of the positive solutions of the following system of Euler-Lagrange equations of Hardy-Littlewood-Sobolev type in R-n u(x) = 1/vertical bar x vertical bar(alpha) integral(n)(R) v(y)(q)/vertical bar y vertical bar(beta)vertical bar x - y vertical bar(lambda) dy, v(x) = 1/vertical bar x vertical bar(beta) integral(n)(R) u(y)(p)/vertical bar y vertical bar(alpha)vertical bar x - y vertical bar(lambda) dy. We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of Li and Lim [17] are studied here. In [15], the authors obtained the asymptotic estimates of solutions for the case alpha,beta >= 0. In this paper, we extend the case alpha,beta >= 0 to alpha + beta >= 0 with some restriction, and we obtain asymptotic estimates for the solutions.