Electron. J. Differential Equations, Vol. 2017 (2017), No. 284, pp. 1-8.

Regularity lifting result for an integral system involving Riesz potentials

Yayun Li, Deyun Xu

Abstract:
In this article, we study the integral system involving the Riesz potentials
$$\displaylines{ u(x)=\sqrt{p} \int_{\mathbb{R}^n}\frac{u^{p-1}(y)v(y)dy}{|x-y|^{n-\alpha}}, \quad u>0 \text{ in } \mathbb{R}^n,\cr v(x)=\sqrt{p} \int_{\mathbb{R}^n}\frac{u^p(y)dy}{|x-y|^{n-\alpha}} \quad v>0 \text{ in } \mathbb{R}^n, }$$
where $n \geq 1$, $0<\alpha<n$ and $p>1$. Such a system is related to the study of a static Hartree equation and the Hardy-Littlewood-Sobolev inequality. We investigate the regularity of positive solutions and prove that some integrable solutions belong to $C^1(\mathbb{R}^n)$. An essential regularity lifting lemma comes into play, which was established by Chen, Li and Ma [20].

Submitted May 16, 2017. Published November 14, 2017.
Math Subject Classifications: 35J10, 35Q55, 45E10, 45G05.
Key Words: Riesz potential; integral system; regularity lifting lemma; Hartree equation; Hardy-Littlewood-Sobolev inequality.

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Yayun Li
Institute of Mathematics
School of Mathematical Sciences
Nanjing Normal University
Nanjing, 210023, China
email: liyayun.njnu@qq.com
  Deyun Xu
Institute of Mathematics
School of Mathematical Sciences
Nanjing Normal University
Nanjing, 210023, China
email: 954816700@qq.com

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