# Symmetry of Positive Solutions for the Fractional Schr$ \ddot{\textrm{o}}$dinger Equations with Choquard-type Nonlinearities

## TL;DR

This paper proves that positive solutions to certain fractional Schrödinger equations with Choquard-type nonlinearities are radially symmetric, using decay estimates, a narrow region principle, and the method of moving planes.

## Contribution

It introduces a generalized direct method of moving planes to establish symmetry of solutions for fractional Schrödinger equations with nonlocal nonlinearities.

## Key findings

- Positive solutions decay at infinity.
- Solutions are radially symmetric.
- Method applicable to nonlocal nonlinear equations.

## Abstract

This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dinger equations with Choquard-type nonlinearities \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} (-\Delta)^{\frac{\alpha}{2}}u + u - C_{n,-\beta} \,(|x|^{\beta-n}\ast u^{p})\, u^{p-1}& = &0 & \mbox{in}\ \ \mathbb{R}^{n}\,, \\[0.05cm] u & > & 0 & \mbox{on}\ \ \mathbb{R}^{n}, \end{array}\right. \end{equation*} where $ 0< \alpha,\beta < 2, 1\leq p <\infty \,\,and\,\, n\geq 2. $ First we construct a decay result at infinity and a narrow region principle for related equations. Then we establish the radial symmetry of positive solutions for the above equation with the generalized direct method of moving planes.