Taiwanese Journal of Mathematics

On the Existence for an Integral System Including $m$ Equations

Xiaoqian Liu and Yutian Lei

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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 $\alpha-n$.

Article information

Taiwanese J. Math., Volume 24, Number 2 (2020), 421-437.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45G05: Singular nonlinear integral equations 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) [See also 47B35]

integral system positive solution Serrin-type condition radial solution asymptotic limit


Liu, Xiaoqian; Lei, Yutian. On the Existence for an Integral System Including $m$ Equations. Taiwanese J. Math. 24 (2020), no. 2, 421--437. doi:10.11650/tjm/190406. https://projecteuclid.org/euclid.twjm/1556784112

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