Decay properties of the Hardy-Littlewood-Sobolev systems of the Lane-Emden type

In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\{{array}{l} (-\Delta)^k u=v^q,u>0 \quad in ~R^n, (-\Delta)^k v=u^p,v>0 \quad in ~R^n, {array}. $$ and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations $$ \{{array}{l} u(x)=\displaystyle\int_{R^n}\frac{v^q(y)dy}{|x-y|^{n-\alpha}},u>0 \quad in ~R^n, v(x)=\displaystyle\int_{R^n}\frac{u^p(y)dy}{|x-y|^{n-\alpha}},u>0 \quad in ~R^n. {array}. $$ Such an integral system is related to the study the extremal functions of the HLS inequality. We point out that the bounded solutions $u,v$ converge to zero either with the fast decay rates or with the slow decay rates when $|x| \to \infty$ under some assumptions. In addition, we also find a criterion to distinguish the fast and the slow decay rates: if $u,v$ are the integrable solutions (i.e. $(u,v) \in L^{r_0}(R^n) \times L^{s_0}(R^n)$), then they decay fast; if the bounded solutions $u,v$ are not the integrable solutions (i.e. $(u,v) \not\in L^{r_0}(R^n) \times L^{s_0}(R^n)$), then they decay almost slowly. Here, for the HLS type system, $r_0=\frac{n(pq-1)}{\alpha(q+1)}$, $s_0=\frac{n(pq-1)}{\alpha(p+1)}$; and for the Lane-Emden type system, $r_0,s_0$ are still the forms above where $\alpha$ is replaced by $2k$.