On some qualitative aspects for doubly nonlocal equations

In this paper we investigate some qualitative properties of the solutions to the following doubly nonlocal equation\begin{document}$\begin{equation} \label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \text{in } \mathbb{R}^N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(P)}\end{equation} $\end{document}where \begin{document}$ N \geq 2 $\end{document} , \begin{document}$ s\in (0, 1) $\end{document} , \begin{document}$ \alpha \in (0, N) $\end{document} , \begin{document}$ \mu>0 $\end{document} is fixed, \begin{document}$ (-\Delta)^s $\end{document} denotes the fractional Laplacian and \begin{document}$ I_{\alpha} $\end{document} is the Riesz potential. Here \begin{document}$ F \in C^1(\mathbb{R}) $\end{document} stands for a general nonlinearity of Berestycki-Lions type. We obtain first some regularity result for the solutions of (P). Then, by assuming \begin{document}$ F $\end{document} odd or even and positive on the half-line, we get constant sign and radial symmetry of the Pohozaev ground state solutions related to equation (P). In particular, we extend some results contained in [ 23 ]. Similar qualitative properties of the ground states are obtained in the limiting case \begin{document}$ s = 1 $\end{document} , generalizing some results by Moroz and Van Schaftingen in [ 52 ] when \begin{document}$ F $\end{document} is odd.