Classification of positive solutions to an integral system with the poly-harmonic extension operator

• 来源期刊: 中国科学（数学）
• 年，卷（期）: 2018,（9）
• 页码范围: 1603-1616
• 页数: 14页
• 语种: 中文

摘要

In this paper,we investigate the positive solutions to the following integral system with a polyharmonic extension operator on IRn+:{u(x)=cn,a ∫(ε)Rn+x1n-avk(y)/|x-y|n-ady,x∈Rn+,u(x)=cn,a ∫Rn+x1-anuθ(x)/|x-y|n-adx,x∈(ε)Rn+.where n ≥ 2,2-n ＜ a ＜ 1,k,θ ＞ 0.This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen (2014).The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case k =n-2+a/n-a,θ =n+2-a/n-2+a.Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.