Uniqueness and nondegeneracy of ground states for the Schrödinger-Newton equation with power nonlinearity

Abstract

In this article, we study the Schrödinger-Newton equation \begin{equation} -Δu+λu=\frac{1}{4π}\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 $\lambda \to 0^{+}$ 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.

arXiv commentarXiv admin note: text overlap with arXiv:2112.05869 by other authors