Taiwanese Journal of Mathematics

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

Xiaoqian Liu and Yutian Lei

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Abstract

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

Source
Taiwanese J. Math., Advance publication (2019), 17 pages.

Dates
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1556784112

Digital Object Identifier
doi:10.11650/tjm/190406

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

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

Citation

Liu, Xiaoqian; Lei, Yutian. On the Existence for an Integral System Including $m$ Equations. Taiwanese J. Math., advance publication, 2 May 2019. doi:10.11650/tjm/190406. https://projecteuclid.org/euclid.twjm/1556784112


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