论文标题
一种通用和统一的方法，用于证明基态解决方案的唯一性和存在/不存在的独特性，以及与各种NLS应用的归一化解决方案的多样性
A General and Unified Method to prove the Uniqueness of Ground State Solutions and the Existence/Non-existence, and Multiplicity of Normalized Solutions with applications to various NLS
论文作者
论文摘要
我们首先提供一个抽象框架，以显示大量PDE的基态解决方案（GSS）的独特性。据我们所知，文献中的所有现有结果仅针对特定情况。此外，我们的独立方法提供了一个一般框架来研究归一化解决方案的存在/不存在和多样性。我们将展示我们方法应用的具体示例，并验证我们需要的所有假设。我们的方法适用于广泛的操作员和域，只要我们的假设得到验证。此外，我们证明了有关在一般环境中阳性GSS的非分类和唯一性的新结果。 我们的发现适用于具有非自治非线性的分数非线性schrodinger方程。特别是，我们能够将[13，14]的主要结果扩展到一般的非自治和混合非线性。通过使用上述突破性论文的作者开发的方法，这似乎是不可能的。得益于非统一的轨道稳定性/不稳定将得到解决。
We first give an abstract framework to show the uniqueness of Ground State Solutions (GSS) for a large class of PDEs. To the best of our knowledge, all the existing results in the literature only addressed particular cases. Moreover, our self-contained approach offers a general framework to study the existence/non-existence and multiplicity of normalized solutions. We will exhibit concrete examples to which our method applies, and verify all the assumptions we need. Our approach is applicable to a wide range of operators and domains provided that our hypotheses are verified. Additionally, we prove new results about the non-degeneracy and uniqueness of positive GSS in a general setting. Our findings are applicable to fractional nonlinear Schrodinger equations with non-autonomous nonlinearities. In particular, we were able to extend the main results of [13, 14] to general non-autonomous and mixed nonlinearities. This does not seem possible by using the approach developed by the authors of the above breakthrough papers. The orbital stability/instability of the standing waves will be addressed thanks to the non-degeneracy.
