Uniqueness and nondegeneracy of ground states for the Schr\"{o}dinger-Newton equation with power nonlinearity

Abstract

In this article, we study the Schr\"{o}dinger-Newton equation \begin{equation} -\Delta u+\lambda u=\frac{1}{4\pi}\left(\frac{1}{|x|}\star u^{2}\right)u+|u|^{q-2}u \quad \text{in}~\mathbb{R}^3, \end{equation} where $\lambda\in\mathbb{R}_+$, $q\in (2,3)\cup(3, 6)$. By investigating limit profiles of ground states as $\lambda\to0^+$ or $\lambda\to+\infty$, we prove the uniqueness of ground states. By the action of the linearized eqaution with respect to decomposition into spherical harmonics, we obtain the nondegeneracy of ground states.