Existence and regularity for an entire Grushin-Choquard equation

TL;DR

This paper studies an entire Grushin-Choquard equation, proving existence of solutions within certain parameter ranges, establishing their regularity, and deriving nonexistence results via a Pohozaev identity.

Contribution

It introduces new existence and regularity results for solutions to a Grushin-Choquard equation and provides a Pohozaev identity for nonexistence proofs.

Findings

Existence of mountain pass solutions for specific parameter ranges.

Solutions belong to L^q spaces for all q in [2, ∞] and are locally Hölder continuous.

A Pohozaev identity is established, leading to nonexistence results.

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 $C^{0,\alpha}_{\textrm{loc}}(\mathbb{R}^N)$ 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.