  In this paper, we are mainly concerned with the Dirichlet problems in
exterior domains for the following elliptic equations:
\begin{equation}\label{GPDE0}
  (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u) \,\,\,\,\,\,\,\,\,\,\,\, \text{in}
\,\,\,\, \Omega_{r}:=\{x\in\mathbb{R}^{n}\,|\,|x|>r\} \end{equation} with
arbitrary $r>0$, where $n\geq2$, $0<\alpha\leq 2$ and $f(x,u)$ satisfies some
assumptions. A typical case is the Hardy-H\'{e}non type equations in exterior
domains. We first derive the equivalence between \eqref{GPDE0} and the
corresponding integral equations \begin{equation}\label{GIE0}
  u(x)=\int_{\Omega_{r}}G_{\alpha}(x,y)f(y,u(y))dy, \end{equation} where
$G_{\alpha}(x,y)$ denotes the Green's function for
$(-\Delta)^{\frac{\alpha}{2}}$ in $\Omega_{r}$ with Dirichlet boundary
conditions. Then, we establish Liouville theorems for \eqref{GIE0} via the
method of scaling spheres developed in \cite{DQ0} by Dai and Qin, and hence
obtain the Liouville theorems for \eqref{GPDE0}. Liouville theorems for
integral equations related to higher order Navier problems in $\Omega_{r}$ are
also derived.
