## Liouville type theorems for general integral system with negative exponents.(English)Zbl 1401.45007

Summary: In this paper, we establish a Liouville type theorem for the following integral system with negative exponents $\begin{cases} u(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} f(u,v)(y)dy,\quad & x \in \mathbb{R}^n, \\ v(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} g(u,v)(y) dy, & x \in \mathbb{R}^n, \end{cases}$ where $$n \geq 1$$, $$\nu > 0$$, and $$f$$, $$g$$ are continuous functions defined on $$\mathbb{R}_{+} \times \mathbb{R}_{+}$$. Under nature structure conditions on $$f$$ and $$g$$, we classify each pair of positive solutions for above integral system by using the method of moving sphere in integral forms. Moreover, some other Liouville theorems are established for similar integral systems.

### MSC:

 45G15 Systems of nonlinear integral equations 45M20 Positive solutions of integral equations

### Keywords:

positive solution; method of moving sphere
Full Text:

### References:

 [1] 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 [2] 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 [3] ——–, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), no. 3, 330–343. · Zbl 1093.45001 [4] L. Chen, Z. Liu, G. Lu and C. Tao, Reverse Stein-Weiss inequalities and existence of their extremal functions, to appear in Trans. Amer. Math. Soc. · Zbl 1418.42041 [5] Y. S. Choi and X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations 246 (2009), no. 1, 216–234. · Zbl 1165.35014 [6] J. Dou, Q. Guo and M. Zhu, Subcritical approach to sharp Hardy-Littlewood-Sobolev type inequalities on the upper half space, Adv. Math. 312 (2017), 1–45. · Zbl 1423.46048 [7] J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math. 54 (2011), no. 4, 753–768. · Zbl 1222.45003 [8] J. Dou, F. Ren and J. Villavert, Classification of positive solutions to a Lane-Emden type integral system with negative exponents, Discrete Contin. Dyn. Syst. 36 (2016), no. 12, 6767–6780. · Zbl 1354.45008 [9] J. Dou and M. Zhu, Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN 2015 (2015), no. 19, 9696–9726. · Zbl 1329.26033 [10] Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst. 34 (2014), no. 6, 2561–2580. · Zbl 1286.35090 [11] Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications, J. Differential Equations 260 (2016), no. 1, 1–25. · Zbl 1331.35014 [12] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett. 14 (2007), no. 3, 373–383. · Zbl 1144.26031 [13] F. Hang, X. Wang and X. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math. 61 (2008), no. 1, 54–95. · Zbl 1173.26321 [14] Y. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst. 35 (2015), no. 3, 1039–1057. · Zbl 1304.45006 [15] 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 [16] Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math. 90 (2003), 27–87. · Zbl 1173.35477 [17] Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J. 80 (1995), no. 2, 383–417. · Zbl 0846.35050 [18] L. Ma and J. C. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal. 254 (2008), no. 4, 1058–1087. · Zbl 1136.35036 [19] 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 [20] 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 [21] ——–, Uniqueness theorem for integral equations and its application, J. Funct. Anal. 247 (2007), no. 1, 95–109. · Zbl 1153.45005 [22] X. Yu Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differential Equations 46 (2013), no. 1-2, 75–95. · Zbl 1262.45004
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