AI 中文总结
本文研究分数阶 Choquard 方程，利用尖锐不等式确定临界质量阈值，证明阈值下无归一化解，阈值上能量无下界。
AI 中文摘要
对于 \\(\frac12\le s<1\\)，我们研究具有规定质量和下 Hardy-Littlewood-Sobolev 及 \\(L^2\\) 临界指数非线性项的分数阶 Choquard 方程。尖锐的 Hardy-Littlewood-Sobolev 和 Choquard Gagliardo-Nirenberg 不等式确定了一个显式临界质量 \\(a_*\\)。对于 \\(0<a\le a_*\\)，我们计算了约束能量的精确下确界，并证明该下确界无法达到，且不存在归一化解。对于 \\(a>a_*\\)，能量在质量球面上无下界，而 Pohozaev 集非空。
英文摘要
For \(\frac12\le s<1\), we study a fractional Choquard equation with prescribed mass and nonlinearities at the lower Hardy-Littlewood-Sobolev and \(L^2\)-critical exponents. The sharp Hardy-Littlewood-Sobolev and Choquard Gagliardo-Nirenberg inequalities determine an explicit critical mass \(a_*\). For \(0<a\le a_*\), we compute the exact infimum of the constrained energy and prove that it is not attained and that no normalized solution exists. For \(a>a_*\), the energy is unbounded from below on the mass sphere, while the Pohozaev set is nonempty.
Comments24 pages, comments are welcome
