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

Abstract

In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type $2k$-order equations $$\{arrayl(-\Delta {)}^{k}u=v^q,u>0in{R}^n,(-\Delta {)}^{k}v=u^p,v>0in{R}^n,array.$$ and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations $$\{arraylu(x)={\int }_{R^n}\frac{v^q(y)dy}{|x-y{|}^{n-\alpha }},u>0in{R}^n,v(x)={\int }_{R^n}\frac{u^p(y)dy}{|x-y{|}^{n-\alpha }},u>0in{R}^n.array.$$ 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. $(u,v)\notin L^{r_0}(R^n)\times L^{s_0}(R^n)$), 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 $2k$.

24 pages

References in corpus (2)

• Sharp criteria of Liouville type for some nonlinear systems
• A degree theory approach for the shooting method

Cited by in corpus (2)

• Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality
• A characterization of fast decaying solutions for quasilinear and Wolff type systems with singular coefficients