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

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.