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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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##### References:
  G. Caristi, L. D’Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math. 76 (2008), 27-67. · Zbl 1186.35026  S.-Y. A. Chang and P. C. Yang, On uniqueness of solutions of $$n$$ th order differential equations in conformal geometry, Math. Res. Lett. 4 (1997), no. 1, 91-102. · Zbl 0903.53027  W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst. 24 (2009), no. 4, 1167-1184. · Zbl 1176.35067  —-, Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci. Ser. B (Engl. Ed.) 29 (2009), no. 4, 949-960. · Zbl 1212.35103  —-, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2497-2514. · Zbl 1270.35224  W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations 30 (2005), no. 1-3, 59-65. · Zbl 1073.45005  —-, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330-343. · Zbl 1093.45001  J. Dou and M. Zhu, Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN 2015 (2015), no. 19, 9696-9726. · Zbl 1329.26033  F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett. 14 (2007), no. 3, 373-383. · Zbl 1144.26031  C. Jin and C. Li, Quantitative analysis of some system of integral equations, Calc. Var. Partial Differential Equations 26 (2006), no. 4, 447-457. · Zbl 1113.45006  Y. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst. 35 (2015), no. 3, 1039-1057. · Zbl 1304.45006  Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst. 36 (2016), no. 6, 3277-3315. · Zbl 1336.35092  Y. Lei and C. Ma, Radial symmetry and decay rates of positive solutions of a Wolff type integral system, Proc. Amer. Math. Soc. 140 (2012), no. 2, 541-551. · Zbl 1241.45005  Y.-Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2, 153-180. · Zbl 1075.45006  E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349-374. · Zbl 0527.42011  C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $$\mathbb{R}^n$$, Comment. Math. Helv. 73 (1998), no. 2, 206-231. · Zbl 0933.35057  C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Commun. Pure Appl. Anal. 8 (2009), no. 6, 1925-1932. · Zbl 1185.45011  J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $$\mathbb{R}^n$$, J. Differential Equations 225 (2006), no. 2, 685-709. · Zbl 1147.35316  Y. Lü and C. Zhou, Symmetry for an integral system with general nonlinearity, Discrete Contin. Dyn. Syst. 39 (2019), no. 3, 1533-1543. · Zbl 1417.45003  È. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math. 234 (2001), no. 3, 1-362. · Zbl 0987.35002  Q. A. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality on $$\mathbb{R}^n$$, Israel J. Math. 220 (2017), no. 1, 189-223. · Zbl 1379.26027  S. Sun and Y. Lei, Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials, J. Funct. Anal. 263 (2012), no. 12, 3857-3882. · Zbl 1260.45004  S. D. Taliaferro, Local behavior and global existence of positive solutions of $$au^{\lambda} \leq -\Delta u \leq u^{\lambda}$$, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 6, 889-901. · Zbl 1039.35145  J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999), no. 2, 207-228. · Zbl 0940.35082  X. Xu, Exact solutions of nonlinear conformally invariant integral equations in $$\mathbb{R}^3$$, Adv. Math. 194 (2005), no. 2, 485-503. · Zbl 1073.45003  —-, Uniqueness theorem for integral equations and its application, J. Funct. Anal. 247 (2007), no. 1, 95-109. · Zbl 1153.45005  W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics 120, Springer-Verlag, New York, 1989. · Zbl 0692.46022
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