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April, 2020 On the Existence for an Integral System Including $m$ Equations
Xiaoqian Liu, Yutian Lei
Taiwanese J. Math. 24(2): 421-437 (April, 2020). DOI: 10.11650/tjm/190406


In this paper, we study an integral system \[ \begin{cases} u_{i}(x) = K_{i}(x) (|x|^{\alpha-n} \ast u^{p_{i+1}}_{i+1})(x), &i = 1,2,\ldots,m-1, \\ u_{m}(x) = K_{m}(x) (|x|^{\alpha-n} \ast u^{p_{1}}_{1})(x). \end{cases} \] When $\alpha \in (0,n)$, $p_{i} \gt 0$ ($i = 1,2,\ldots,m$), the Serrin-type condition is critical for existence of positive solutions for some double bounded functions $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \in (0,n)$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), the system has no positive solution for any double bounded $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \gt n$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), and $\max_{i} \{-p_{i}\} \gt \alpha/(\alpha-n)$, then the system exists positive solutions increasing with the rate $\alpha-n$.


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Xiaoqian Liu. Yutian Lei. "On the Existence for an Integral System Including $m$ Equations." Taiwanese J. Math. 24 (2) 421 - 437, April, 2020.


Received: 1 December 2018; Revised: 18 April 2019; Accepted: 23 April 2019; Published: April, 2020
First available in Project Euclid: 2 May 2019

zbMATH: 07192942
MathSciNet: MR4078205
Digital Object Identifier: 10.11650/tjm/190406

Primary: 45E10 , 45G05

Keywords: asymptotic limit , integral system , positive solution , radial solution , Serrin-type condition

Rights: Copyright © 2020 The Mathematical Society of the Republic of China


Vol.24 • No. 2 • April, 2020
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