<div>In this paper, we study the positive solutions of higher order
Lane-Emden system with Navier exterior conditions $$
\begin{cases} \
(-\Delta)^{s}u(x)=v^{p}(x) ,\
u(x)&gt;0,\qquad \
x\in\Omega,\\
\ (-\Delta)^{s}v(x)=u^{q}(x)
,\ v(x)&gt;0,\qquad
\
x\in\Omega,\\
\ u(x)=-\Delta
u(x)=\cdot\cdot\cdot=(-\Delta)^{m}
u(x)=0,\
x\in\Omega^{c},\\
\ v(x)=-\Delta
v(x)=\cdot\cdot\cdot=(-\Delta)^{m}
v(x)=0,\
x\in\Omega^{c},
\end{cases} $$ where $1</div>