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\r\n\begin{equation}\label{eq_abstract}\r\n(- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P}\r\n\end{equation}\r\nwhere $N \geq 2$, $s\in (0,1)$, $\alpha \in (0,N)$, $\mu>0$ is fixed, $(-\Delta)^s$ denotes the fractional Laplacian and $I_{\alpha}$ is the Riesz potential. Here $F \in C^1(\mathbb{R})$ stands for a general nonlinearity of Berestycki-Lions type.\r\nWe obtain first some regularity result for the solutions of \eqref{eq_abstract}. Then, by assuming $F$ 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 \eqref{eq_abstract}.\r\nIn particular, we extend some results contained in \cite{DSS1}.\r\nSimilar qualitative properties of the ground states are obtained in the limiting case $s=1$, generalizing some results by Moroz and Van Schaftingen in \cite{MS2} when $F$ is odd.