Existence and regularity for an entire Grushin-Choquard equation

Abstract

We consider the following Choquard equation $$ -Δ_γu + u = \left(d(z)^{-μ} \ast |u|^p\right)|u|^{p-2}u, \text{ in } \mathbb{R}^N, $$ where $Δ_γ$ is the Grushin operator. For a suitable range of the parameter $p$ we prove the existence of a mountain pass solution of the equation. We also establish that the solutions belong to $L^q(\mathbb{R}^N)$ for all $q\in [2,\infty]$ and to $C^{0,α}_\mathrm{loc}(\mathbb{R}^N)$ for some $α\in (0,1)$.

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: AH23
• Lemma 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Lemma 3.1
• proof