Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian
Classification of positive solutions for a static Schrödinger-Maxwell equation with fractional Laplacian
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DOI：
10.1016/j.aml.2014.12.007
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发表时间：
2015-05
期刊：
影响因子：
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通讯作者：
De-Liang Xu;Y. Lei
中科院分区：
文献类型：
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作者：
De-Liang Xu;Y. Lei
本文研究了分数阶非局部静态薛定谔方程（-Δ）α2u=pup-1（|X| α−n = up），u> 0 in Rn，其中n≥3，α∈（1，n）且p>1。它可以被看作是一个包含Riesz势{u（x）=p <$Rnup−1（y）v（y）dy的积分系统|x−y| n−α，u> 0 in Rn，v（x）=p∫Rnup（y）dy| x−y| n−αv>0inRn。首先，p大于Serrin指数nn−α是正解存在的必要条件。在此基础上，我们研究了正解的分类问题。如果系统在Ln（p−1）α（Rn）中有解，则p必须是临界指数n+αn−α，因此所有的正解可以分类为u（x）=v（x）=c（tt 2 +| x−x| 2）n−α2，其中c，t是正常数，x ∈Rn。
In this paper, we study the fractional-order nonlocal static Schrödinger equation (−Δ)α2u=pup−1(|x|α−n∗up),u>0inRn, with n≥3, α∈(1,n) and p>1. It can be viewed as an integral system involving the Riesz potentials {u(x)=p∫Rnup−1(y)v(y)dy|x−y|n−α,u>0inRn,v(x)=p∫Rnup(y)dy|x−y|n−αv>0inRn. First, the fact p is larger than the Serrin exponent nn−α is a necessary condition for the existence of the positive solution. Based on this result, we investigate the classification of the positive solutions. If the system has solutions in Ln(p−1)α(Rn), then p must be the critical exponent n+αn−α, and hence all the positive solutions can be classified as u(x)=v(x)=c(tt2+|x−x∗|2)n−α2, where c,t are positive constants, and x∗∈Rn.