×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] 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
[2] 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
[3] 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
[4] —-, 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
[5] —-, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal. 12 (2013), no. 6, 2497-2514. · Zbl 1270.35224
[6] 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
[7] —-, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330-343. · Zbl 1093.45001
[8] J. Dou and M. Zhu, Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN 2015 (2015), no. 19, 9696-9726. · Zbl 1329.26033
[9] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett. 14 (2007), no. 3, 373-383. · Zbl 1144.26031
[10] 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
[11] Y. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst. 35 (2015), no. 3, 1039-1057. · Zbl 1304.45006
[12] 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
[13] 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
[14] 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
[15] 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
[16] 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
[17] 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
[18] 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
[19] 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
[20] 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
[21] 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
[22] 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
[23] 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
[24] 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
[25] 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
[26] —-, Uniqueness theorem for integral equations and its application, J. Funct. Anal. 247 (2007), no. 1, 95-109. · Zbl 1153.45005
[27] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics 120, Springer-Verlag, New York, 1989. · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.