标题： 归一化基态在势场下的临界Choquard方程的轨道稳定性 显示英文标题

标题： Orbital stability of normalized ground states for critical Choquard equation with potential

摘要： 在本文中，我们研究了非线性临界Choquard方程 \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u -V(x)u+(I_{\alpha}\ast|u|^{q})|u|^{q-2}u+(I_{\alpha}\ast|u|^{2_{\alpha}^*})|u|^{2_{\alpha}^*-2}u=0,\ (x, t) \in \mathbb{R}^d \times \mathbb{R}, \\ \left.u\right|_{t=0}=\varphi \in H ^1(\mathbb{R}^d), \end{array}\right. \end{equation*} 的规定质量的基态驻波和轨道稳定性，其中 $I_{\alpha}$ 是阶为 $\alpha\in(0,d),\ d\geq3,\ 2_{\alpha}^*=\frac{2d-\alpha}{d-2}$ 的Riesz势，是由于Hardy-Littlewood-Sobolev不等式产生的上临界指数，$\frac{2d-\alpha}{d}<q<\frac{2d-\alpha+2}{d}$。在适当的势条件下，我们获得了新的Strichartz估计并构造了新的空间以获得归一化基态的轨道稳定性。据我们所知，这是该模型的第一个轨道稳定性结果。我们的方法也适用于其他带有势的混合非线性方程。 显示英文摘要

摘要： In this paper, we study the existence of ground state standing waves and orbital stability, of prescribed mass, for the nonlinear critical Choquard equation \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u -V(x)u+(I_{\alpha}\ast|u|^{q})|u|^{q-2}u+(I_{\alpha}\ast|u|^{2_{\alpha}^*})|u|^{2_{\alpha}^*-2}u=0,\ (x, t) \in \mathbb{R}^d \times \mathbb{R}, \\ \left.u\right|_{t=0}=\varphi \in H ^1(\mathbb{R}^d), \end{array}\right. \end{equation*} where $I_{\alpha}$ is a Riesz potential of order $\alpha\in(0,d),\ d\geq3,\ 2_{\alpha}^*=\frac{2d-\alpha}{d-2}$ is the upper critical exponent due to Hardy-Littlewood-Sobolev inequality, $\frac{2d-\alpha}{d}<q<\frac{2d-\alpha+2}{d}$. Under appropriate potential conditions, we obtain new Strichartz estimates and construct the new space to get orbital stability of normalized ground state. To our best knowledge, this is the first orbital stability result for this model. Our method is also applicable to other mixed nonlinear equations with potential.