临界增长的分数阶Hartree方程的Delaunay解
Delaunay solutions to the fractional Hartree equation with critical growth
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中文总结 AI 辅助
本文研究带原点不可移除孤立奇点的临界分数阶Hartree方程，结合Caffarelli–Silvestre延拓与移动球方法证明其正奇异解径向对称，并通过最小化Rayleigh型商构造出所有足够大周期的Delaunay型周期奇异解。
中文摘要 AI 辅助
我们研究原点处带有不可移除孤立奇点的临界分数阶Hartree方程的正解。该方程是双重非局部的，同时包含分数阶拉普拉斯算子和Riesz卷积势。我们首先结合Caffarelli–Silvestre延拓与移动球方法，证明所有正奇异解关于原点径向对称。随后建立Delaunay型周期奇异解的存在性。经Emden–Fowler变换后，Hartree卷积作为真正的非局部积分项保留，所得周期方程无法简化为常微分方程。我们通过在周期分数阶Sobolev空间中最小化Rayleigh型商，对所有足够大的周期构造出非常数周期解。
英文摘要
We study positive solutions of the critical fractional Hartree equation with a non-removable isolated singularity at the origin. This equation is doubly nonlocal, involving both the fractional Laplacian and a Riesz convolution potential. We first prove that every positive singular solution is radially symmetric about the origin, by combining the Caffarelli--Silvestre extension with the method of moving spheres. We then establish the existence of Delaunay-type periodic singular solutions. After the Emden--Fowler transformation, the Hartree convolution survives as a genuinely nonlocal integral term, so that the resulting periodic equation cannot be reduced to an ordinary differential equation. We construct nonconstant periodic solutions for all sufficiently large periods by minimizing a Rayleigh-type quotient in a periodic fractional Sobolev space.
