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On Liouville theorems of a Hartree–Poisson system

Part of: Partial differential equations Elliptic equations and systems Nonlinear integral equations Singular integral equations

Published online by Cambridge University Press:  26 October 2023

Ling Li  and
Yutian Lei Open the ORCID record for Yutian Lei [Opens in a new window]
Ling Li
Affiliation:
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, China (liling.njnu@qq.com; leiyutian@njnu.edu.cn)
Yutian Lei
Affiliation:
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, China (liling.njnu@qq.com; leiyutian@njnu.edu.cn)
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Abstract

In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system:

\begin{equation*}\left\{\begin{aligned}&-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\&-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n},\end{aligned}\right.\end{equation*}
where $n \geq3$ and $\min\{p,q\} \gt 1$. We prove that the system has no positive solution under a Serrin-type condition. In addition, the system has no positive radial classical solution in a Sobolev-type subcritical case. In addition, the system has no positive solution with some integrability in this Sobolev-type subcritical case. Finally, the relation between a Liouville theorem and the estimate of boundary blowing-up rates is given.

Keywords

Hartree–Poisson systemLiouville theoremradial solutionsmethod of moving planes

MSC classification

Primary: 35J60: Nonlinear elliptic equations 35B33: Critical exponents 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 45G15: Systems of nonlinear integral equations

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 1154 - 1178
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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