Liouville type theorems for higher order elliptic systems

In the first part of this article, by the method of scaling spheres combined with some new iterative approach, we extend the Liouville type theorems [2], [3] for higher order fractional Hénon-Lane-Emden systems to the general higher-order fractional elliptic systems$\{\begin{array}{cc}{(-\Delta )}^{k+\frac{\alpha }{2}}u=|x{|}^a{u}^p{v}^q,& x\in R^n,\\ {(-\Delta )}^{l+\frac{\beta }{2}}v=|x{|}^b{u}^r{v}^s,& x\in R^n.\end{array}$

In the second part of this article, by a new iterative method, we prove that the super-harmonic property for the polyharmonic Hénon-Lane-Emden system$\{\begin{array}{cc}& {(-\Delta )}^{m}u=|x{|}^a{v}^q,x\in R^n,\\ & {(-\Delta )}^{m}v=|x{|}^b{u}^r,x\in R^n\end{array}$ holds for all range $q,r>0$. Therefore, we extends the Liouville type theorem in [24] to full range$\{\begin{array}{ccc}& q,r>0,\frac{n+a}{q+1}+\frac{n+b}{r+1}>n-2m,& \text{if}m<\frac{n}{2},\\ & 0<q,r<+\infty ,& \text{if}m=\frac{n}{2}.\end{array}$