2013/02/22 by Yutian Lei, Congming Li, Lei, Yutian +1
Mathematics · #35J48 #45E10 #45G05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP #msc:35J48 #msc:45E10 #msc:45G05
paper · pdf · doi:10.48550/arxiv.1302.5567
24 pages
arxiv created 2013/02/22 · openalex publication_date 2013/02/22 · arxiv updated 2013/02/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper, we study the asymptotic behavior of positive solutions of the nonlinear differential systems of Lane-Emden type 2k-order equations \arrayl (-Δ)k u=vq,u>0 in ~Rn, (-Δ)k v=up,v>0 in ~Rn, array. and the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations \arrayl u(x)=∫Rn\fracvq(y)dy|x-y|n-α,u>0 in ~Rn, v(x)=∫Rn\fracup(y)dy|x-y|n-α,u>0 in ~Rn. 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| → ∞ 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) ∈ Lr0(Rn) × Ls0(Rn)), then they decay fast; if the bounded solutions u,v are not the integrable solutions (i.e. (u,v) \not∈ Lr0(Rn) × Ls0(Rn)), then they decay almost slowly. Here, for the HLS type system, r0=(n(pq-1))/(α(q+1)), s0=(n(pq-1))/(α(p+1)); and for the Lane-Emden type system, r0,s0 are still the forms above where α is replaced by 2k.