Existence of ground state solutions for the general Choquard equation with Riesz fractional derivative operator

This work examines the existence of a ground state solution for the following general nonlinear Choquard equation:

$$-\mathrm{\Delta }u+u=\left({(-\mathrm{\Delta })}_E^{\frac{-\alpha }{2}}\ast \mathcal{Q}\left(u\right)\right)\mathcal{q}\left(u\right),in{\mathbb{R}}^N,$$

where $N\ge 3,$ $\mathcal{Q}$ is the primitive function of $\mathcal{q},$ $\mathcal{Q}\in C^1\left(\mathbb{R};\mathbb{R}\right)$ fulfils the general Berestycki–Lions conditions, and ${(-\mathrm{\Delta })}_E^{\frac{-\alpha }{2}}$ is the equivalent Riesz fractional operator of order $\alpha \in \left(0,2\right)$. In this case, the Riesz potential has not previously been investigated. The existence of a solution is established through the application of variational techniques. This modification not only expands the theoretical understanding of such equations but also opens up new avenues for practical applications, particularly in fields such as quantum mechanics and astrophysics.