Classification of Solutions to Mixed Order Conformally Invariant Systems in R2

In this paper, we are concerned with the following mixed-order conformally invariant system with coupled nonlinearity in R-2: {(-Delta)(1/2)(x) = u(p1) (x)e(q1v(x)), x is an element of R-2, (0.1) (-Delta)v(x) = u(p2 )(x)e(q2v(x)), x is an element of R-2, where 0 <= p(1) < 1/1+K, P-2 > 0, q(1) > 0, q(2) >= 0, u > 0 and satisfies integral(R2) u(p2 )(x)e(q2v(x)) dx < infinity. Under the assumptions, u(x) = O(vertical bar x vertical bar(K)) at infinity for some K >= 1 arbitrarily large and v(+)(x) = O(ln vertical bar x vertical bar) if q(2) > 0 at infinity. We firstly derived the equivalent integral representation formula for (0.1). Then we discuss the exact asymptotic behavior of the solutions to system (0.1) as vertical bar x vertical bar -> infinity. At last, by using the method of moving spheres in integral form, we give the classification of the classical solutions to (0.1).