Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-H\'{e}non equations in $\mathbb{R}^{n}$

In this paper, we consider the critical order Hardy-H\'{e}non equations \begin{equation*} (-\Delta)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}}, \,\,\,\,\,\,\,\,\,\,\, x \, \in \,\, \mathbb{R}^{n}, \end{equation*} where $n\geq4$ is even, $-\infty<a<n$, and $1<p<+\infty$. We first prove a Liouville theorem (Theorem \ref{Thm0}), that is, the unique nonnegative solution to this equation is $u\equiv0$. Then as an immediate application, we derive a priori estimates and hence existence of positive solutions to critical order Lane-Emden equations in bounded domains (Theorem \ref{Thm1} and \ref{Thm2}). Our results seem to be the first Liouville theorem, a priori estimates, and existence on the critical order equations in higher dimensions ($n\geq3$). Extensions to super-critical order Hardy-H\'{e}non equations and inequalities will also be included (Theorem \ref{Thm0-sc} and \ref{Thm1-sc}).