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

In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: $$ \begin{cases} \ (-\Delta)^{\frac{\alpha}{2}}u=|x|^{\gamma}u^{p}v^{q+1},\\ \ (-\Delta)^{\frac{\beta}{2}}v=|x|^{\tau}u^{p+1}v^{q}, \end{cases} $$ where $0<\alpha,\beta<2,$ $p,q>0\ and\ \max\{p,q\}\geq1,$ $\alpha+\gamma>0,\beta+\tau>0,$ $n\geq2$. First of all, while in the subcritical case, i.e. $n+\alpha+\gamma-p(n-\alpha)-(q+1)(n-\beta)>0$, $n+\beta+\tau-(p+1)(n-\alpha)-q(n-\beta)>0$, we prove the nonexistence of positive solution for the above system in $\mathbb{R}^{n}$. Moreover, though $Doubling\ Lemma$ to obtain the singularity estimates of the positive solution on bounded domain $\Omega$. In addition, while in the critical case, i.e. $n+\alpha+\gamma-p(n-\alpha)-(q+1)(n-\beta)=0$, $n+\beta+\tau-(p+1)(n-\alpha)-q(n-\beta)=0$, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of $Moving\ planes$ in $\mathbb{R}^{n}$.

In this paper, we are concerned with the positive continuous entire solutions of the Wolff-type integral system \begin{equation*} \left\{ \begin{array}{ll} &u(x) =C_{1}(x)W_{\beta,\gamma} (v^{-q})(x), \\[3mm] &v(x) =C_{2}(x)W_{\beta,\gamma} (u^{-p})(x), \end{array} \right. \end{equation*} where $n\geq1$, $\min\{p,q\}>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma\neq n$. In addition, $C_{i}(x) \ (i=1,2)$ are some double bounded functions. If $\beta\gamma\in (0,n)$, the Serrin-type condition is critical for existence of the positive solutions for some double bounded functions $C_{i}(x)$ $(i=1,2)$. Such an integral equation system is related to the study of the $\gamma$-Laplace system and $k$-Hessian system with negative exponents. Estimated by the integral of the Wolff type potential, we obtain the asymptotic rates and the integrability of positive solutions, and studied whether the radial solutions exist.