Liouville-type theorems for the higher order Hartree type system

In this paper, we are concerned with the non-existence of positive solutions for higher order Hartree type system $$ {\left\{ \begin{array}{ll} \ (-\Delta )^{m} u=(\frac{1}{|x|^\sigma }*v^p)v^{p-1},& x\in \mathbb {R}^{N},\\ \ (-\Delta )^{m} v=(\frac{1}{|x|^\sigma }*u^q)u^{q-1},& x\in \mathbb {R}^{N}, \end{array}\right. } $$ where $$N>2m$$ , $$m\ge 1$$ , $$0<\sigma 1$$ . In the first step, we establish the equivalence between partial differential system and integral system by using super poly-harmonic properties. In addition, we prove that the above system has no positive sup-solution under a Serrin-type condition. By the method of moving sphere, we established the Liouville-type theorem and derive a classification of nonnegative solutions for the integral system in $$\mathbb {R}^{N}$$ . As an application of Liouville-type theorem, through the Doubling Lemma, we obtain the singularity estimates of the nonnegative solutions on a bounded domain.