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$\begin{cases}-{\triangle u} = (q+1)u^qv^{p+1},~~ u>0~ in~ R^n,\\-{\triangle v} = (p+1)v^pu^{q+1},~~ v>0~in ~R^n,\end{cases}$ |

$ n ≥ 3 $ |

$ p,q>0 $ |

$ \max\{p,q\} ≥ 1 $ |

$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,

$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy. $

We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.

$-Δ u = pu^{p-1}(|x|^{2-n}*u^p),\;\; u>0 \;\;in\;\; R^n, $ |

$n ≥ 3$ |

$p≥ 1$ |

$1 ≤ p <\frac{n+2}{n-2}$ |

$\left\{ \begin{array}{l} - \Delta u = \sqrt p {u^{p - 1}}v,\;\;u > 0\;\;in\;\;{R^n},\\ - \Delta v = \sqrt p {u^p},\;\;v > 0\;\;in\;\;{R^n}.\end{array} \right.$ |

$p = \frac{n+2}{n-2}$ |

$u(x) = c(\frac{t}{t^2+|x-x^*|^2})^{\frac{n-2}{2}}$ |

$c, t$ |

$x^* ∈ R^n$ |

$p>\frac{n+2}{n-2}$ |

$\frac{2}{p-1}$ |

$p ≥ 1+\frac{4}{n-4-2\sqrt{n-1}}$ |

-div$(|\nabla u|^{\gamma-2}\nabla u) =K u^p$,

Here $1<\gamma<2$, $n>\gamma$, $p=\frac{(\gamma-1)(n+\gamma)}{n-\gamma}$ and $K(x)$ is a smooth function bounded by two positive constants. First, the positive solution $u$ of the $\gamma$-Laplace equation above satisfies an integral equation involving a Wolff potential. Based on this, we estimate the decay rate of the positive solutions of the $\gamma$-Laplace equation at infinity. A new method is introduced to fully explore the integrability result established recently by Ma, Chen and Li on Wolff type integral equations to derive the decay estimate.

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