Existence and regularity for an entire Grushin-Choquard equation

Abstract

We consider the following Choquard equation $$-{\Delta }_{\gamma }u+u=\left(d(z{)}^{-\mu }\ast |u{|}^p\right)|u{|}^{p-2}u,\text{ in }{\mathbb{R}}^N,$$ where ${\Delta }_{\gamma }$ 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 $q\in [2,\infty ]$ 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.