Classification of Solutions for Mixed Order Conformally Invariant Systems Involving Fractional Laplacian

This paper is devoted to establishing the classification of solutions for two mixed order conformally invariant systems involving fractional Laplacian. Firstly, we consider the planar system with exponential nonlinearity: {(-Delta)(s)u=e(pv) in R-2, -Delta v=u(2/1-s) in R-2, u >= 0 in R-2, (0.1) where s is an element of(0,1) and p is an element of(0,infinity). Under the finite total curvature condition integral(2)(R)u(2/1-s> 1 arbitrarily large, we prove the classification of classical solutions to the system (0.1) without imposing any assumptions on v. Secondly, we study the mixed order conformally invariant system involving higher order fractional Laplacian in R-n: {(-Delta)(s)u=v(n+2s/n-2) in R-n, -Delta v=u(n+2/n-2s) in R-n, u >= 0,v >= 0 in R-n, (0.2) where n >= 3 and s is an element of(0,n/2). Classification result of nonnegative classical solutions to the system (0.2) is also proved. Finally, we establish a Liouville theorem for the system (0.2) in both the critical case s=n/2 and the supercritical case s>n/2. Our proofs make use of the method of moving spheres in the corresponding integral systems.