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 R²: {(-Δ)12u(x)=up1(x)eq1v(x),x∈R2,(-Δ)v(x)=up2(x)eq2v(x),x∈R2,where 0≤p1<11+K, p₂> 0 , q₁> 0 , q₂≥ 0 , u> 0 and satisfies ∫R2up2(x)eq2v(x)dx<+∞. Under the assumptions, u(x) = O(| x| ^K) at ∞ for some K≥ 1 arbitrarily large and v⁺(x) = O(ln | x|) if q₂> 0 at ∞. 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 | x| → ∞. At last, by using the method of moving spheres in integral form, we give the classification of the classical solutions to (0.1).