Liouville theorems for Choquard-Pekar equations on the half space

We study the following Dirichlet problem to the Choquard-Pekar equation$\{\begin{array}{cc}& -\Delta u=\left(\underset{\Omega }{\int }\frac{u^p\left(y\right)}{|x-y{|}^{n-\beta }}dy\right)u^q,x\in \Omega ,\\ & u\left(x\right)\equiv 0,x\notin \Omega .\end{array}$ For $1<p\le \frac{n+\beta }{n-2}$, $1\le q\le \frac{2+\beta }{n-2}$ and $\Omega =R_{+}^n$, we prove the non-existence of non-negative solutions by the method of moving planes. As an application of the Liouville theorem in half space and the Liouville theorem in whole space obtained in [13], [28], we carry on blowing-up and rescaling argument on the Choquard-Pekar equation in a bounded domain $\Omega \subset R^n$, and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray-Schauder degree theory, we establish the existence of positive solutions.