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On the existence for an integral system including $$m$$ equations. (English) Zbl 1448.45006
Summary: 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 $$\alpha \in (0,n), p_i > 0 \; (i = 1,2,\ldots,m)$$, 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 $$\alpha \in (0,n), p_i < 0 (i = 1,2,\ldots,m)$$, the system has no positive solution for any double bounded $$K_i(x);  (i = 1,2,\ldots,m)$$. When $$\alpha > n, p_i < 0  (i = 1,2,\ldots,m)$$, and $$\max_i \{-p_i\} > \alpha/(\alpha-n)$$, then the system exists positive solutions increasing with the rate $$\alpha-n$$.
##### MSC:
 45G15 Systems of nonlinear integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 45M20 Positive solutions of integral equations
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