We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian(-△)^(s)(0<s<1).First,we obtain a Liouville type theorem in the often-used defining space L_(2s).An interesting ...We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian(-△)^(s)(0<s<1).First,we obtain a Liouville type theorem in the often-used defining space L_(2s).An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases.Considering the anti-symmetric property,one can naturally extend the defining space from L_(2s) to L_(2s+1).Surprisingly,with this extension,we show the existence of non-trivial solutions.This is very different from the previous results of the Lane-Emden system.展开更多
This paper is devoted to studying the existence of positive solutions for the following integral system {u(x)=∫_(R^n)|x-y|~λv-~q(y)dy, ∫_(R^n)|x-y|~λv-~p(y)dy,p,q>0,λ∈(0,∞),n≥1.It is shown that if(u,v) is a...This paper is devoted to studying the existence of positive solutions for the following integral system {u(x)=∫_(R^n)|x-y|~λv-~q(y)dy, ∫_(R^n)|x-y|~λv-~p(y)dy,p,q>0,λ∈(0,∞),n≥1.It is shown that if(u,v) is a pair of positive Lebesgue measurable solutions of this integral system, then 1/(p-1)+1/(q-1)=λ/n, which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.12031012,11831003 and 11701207)Natural Science Foundation of Henan Province of China(Grant No.222300420499)。
文摘We study the anti-symmetric solutions to the Lane-Emden type system involving fractional Laplacian(-△)^(s)(0<s<1).First,we obtain a Liouville type theorem in the often-used defining space L_(2s).An interesting lower bound on the solutions is derived to estimate the Lipschitz coefficient in the sub-linear cases.Considering the anti-symmetric property,one can naturally extend the defining space from L_(2s) to L_(2s+1).Surprisingly,with this extension,we show the existence of non-trivial solutions.This is very different from the previous results of the Lane-Emden system.
基金Supported by National Natural Science Foundation of China(11126148,11501116,11671086,11871208)Natural Science Foundation of Hunan Province of China(2018JJ2159)+1 种基金the Project Supported by Scientific Research Fund of Hunan Provincial Education Department(16C0763)the Education Department of Fujian Province(JA15063)
文摘This paper is devoted to studying the existence of positive solutions for the following integral system {u(x)=∫_(R^n)|x-y|~λv-~q(y)dy, ∫_(R^n)|x-y|~λv-~p(y)dy,p,q>0,λ∈(0,∞),n≥1.It is shown that if(u,v) is a pair of positive Lebesgue measurable solutions of this integral system, then 1/(p-1)+1/(q-1)=λ/n, which is different from the well-known case of the Lane-Emden system and its natural extension, the Hardy-Littlewood-Sobolev type integral equations.
