Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains

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.