Existence and regularity for an entire Grushin-Choquard equation
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910061119406080 |
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| author | Bernini, Federico Malanchini, Paolo |
| author_facet | Bernini, Federico Malanchini, Paolo |
| contents | 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 and we establish that the solution belongs to $L^q(\mathbb{R}^N)$ for all $q\in [2,\infty]$ and to $C^{0,α}_{\textrm{loc}}(\mathbb{R}^N)$ for some $α\in (0,1)$. Additionally, we provide a Poho\v zaev type identity, which allows us to derive a nonexistence result for smooth solutions to our equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05389 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and regularity for an entire Grushin-Choquard equation Bernini, Federico Malanchini, Paolo Analysis of PDEs 35J70, 35J20, 35B65 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 and we establish that the solution belongs to $L^q(\mathbb{R}^N)$ for all $q\in [2,\infty]$ and to $C^{0,α}_{\textrm{loc}}(\mathbb{R}^N)$ for some $α\in (0,1)$. Additionally, we provide a Poho\v zaev type identity, which allows us to derive a nonexistence result for smooth solutions to our equation. |
| title | Existence and regularity for an entire Grushin-Choquard equation |
| topic | Analysis of PDEs 35J70, 35J20, 35B65 |
| url | https://arxiv.org/abs/2603.05389 |