On Liouville Theorems of a Hartree–Poisson system

Abstract

Abstract In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system: \begin{equation*} \left\{ \begin{aligned} &-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\ &-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n}, \end{aligned} \right. \end{equation*} where $n \geq3$ and $\min\{p,q\} \gt 1$ . We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.