Symmetry and classification of positive standing waves of nonlinear Hartree type equations

Abstract

This paper presents some qualitative properties of positive solutions to the strongly coupled system \[ \begin{cases} \displaystyle - \Delta u + \tau u = \frac{2 p}{p + q} \left( I_\alpha \ast |v|^q \right) |u|^{p - 2} u &\text{in} ~ \mathbb{R}^N, \\ \\ \displaystyle - \Delta v + \eta v = \frac{2 q}{p + q} \left( I_\alpha \ast |u|^p \right) |v|^{q - 2} v &\text{in} ~ \mathbb{R}^N, \end{cases} \] with $\tau, \eta > 0$, $N \in \mathbb{N}$, $0 < \alpha < N$, \[ \max \left\{1, \frac{2 \alpha}{N}\right\} < p, q < 2^* \quad \text{and} \quad \frac{2 (N + \alpha)}{N} < p + q < 2_\alpha^*, \] where $I_\alpha$ denotes the Riesz potential, \[ 2^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 N}{N - 2}, &\text{if} ~ N \geq 3, \end{cases} \quad \text{and} \quad 2_\alpha^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 (N + \alpha)}{N - 2}, &\text{if} ~ N \geq 3. \end{cases} \] More precisely, by means of the moving planes method, we prove that positive solutions to this system are radially symmetric and strictly radially decreasing when $p, q \geq 2$, and we obtain a classification result for positive ground states in the case $p = q$ and $\tau = \eta$.