Existence and regularity for an entire Grushin-Choquard equation

We consider the following Choquard equation $$-{\mathrm{\Delta }}_{\gamma }u+u=(d(z{)}^{-\mu }\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,\text{ in }{\mathbb{R}}^N,$$ where ${\mathrm{\Delta }}_{\gamma }$ is the Grushin operator. For a suitable range of the parameter $p$ we prove the existence of a mountain pass solution of the equation and we establish that the solution belongs to $L^q({\mathbb{R}}^N)$ for all $q\in [2,\mathrm{\infty }]$ and to $C_{\text{loc}}^{0,\alpha }({\mathbb{R}}^N)$ for some $\alpha \in (0,1)$. Additionally, we provide a Poho\v zaev type identity, which allows us to derive a nonexistence result for smooth solutions to our equation.