D’Ambrosio, Lorenzo; Mitidieri, Enzo Hardy-Littlewood-Sobolev systems and related Liouville theorems. (English) Zbl 1284.35116 Discrete Contin. Dyn. Syst., Ser. S 7, No. 4, 653-671 (2014). Summary: We prove some Liouville theorems for systems of integral equations and inequalities related to weighted Hardy-Littlewood-Sobolev inequality type on \(\mathbb{R}^N\). Some semilinear singular or degenerate higher order elliptic inequalities associated to polyharmonic operators are considered. Special cases include the Hénon-Lane-Emden system. Cited in 14 Documents MSC: 35C15 Integral representations of solutions to PDEs 35J48 Higher-order elliptic systems 45G15 Systems of nonlinear integral equations Keywords:representation formulae; Liouville theorems; Hardy-Littlewood-Sobolev inequality; Hardy-Hénon-emdem system PDFBibTeX XMLCite \textit{L. D'Ambrosio} and \textit{E. Mitidieri}, Discrete Contin. Dyn. Syst., Ser. S 7, No. 4, 653--671 (2014; Zbl 1284.35116) Full Text: DOI References: [1] H. Brezis, Sublinear elliptic equations in \(R^N\),, Manuscripta Math., 74, 87 (1992) · Zbl 0761.35027 [2] A. Björn, <em>Nonlinear Potential Theory on Metric Spaces</em>,, EMS Tracts in Mathematics (2011) · Zbl 1231.31001 [3] G. Caristi, Liouville Theorems for some nonlinear inequalities,, Proc. Steklov Inst. Math., 260, 90 (2008) · Zbl 1233.35207 [4] G. Caristi, Representation formulae for solutions to some classes of higher order systems and related liouville theorems,, Milan J. Math., 76, 27 (2008) · Zbl 1186.35026 [5] W. Chen, Weighted Hardy-Littlewood-Sobolev inequalities and Systems of integral equations,, Disc. and Cont. Dynamics Sys. Supplement, 164 (2005) · Zbl 1147.45301 [6] W. Chen, Classification of solutions for a system of integral equations,, Comm. in Partial Differential Equations, 30, 59 (2005) · Zbl 1073.45005 [7] W. Chen, Regularity of solutions for a system of integral equations,, Comm. Pure and Appl. Anal., 4, 1 (2005) [8] W. Chen, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59, 330 (2006) · Zbl 1093.45001 [9] C. Cowan, A Liouville theorem for a fourth order Hénon equation,, <a href= · Zbl 1301.35023 [10] L. D’Ambrosio, Representation formulae and inequalities for solutions of a class of second order partial differential equations,, Trans. Amer. Math. Soc., 358, 893 (2005) · Zbl 1081.35014 [11] L. Euler, Specimen transformationis singularis serierum,, Nova Acta Acad. Petropol., 7, 58 (1778) [12] M. Fazly, On the Hénon-Lane-Emden conjecture,, <a href= · Zbl 1285.35024 [13] J. Heinonen, <em>Nonlinear Potential Theory of Degenerate Elliptic Equations</em>,, Oxford University Press (1993) · Zbl 0780.31001 [14] W. K. Hayman, <em>Subharmonic functions</em>,, I (1976) · Zbl 0419.31001 [15] C. Jin, Qualitative Analysis of Some Systems of Integral Equations,, Cal. Var. PDEs, 26, 447 (2006) · Zbl 1113.45006 [16] C. Jin, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134, 1661 (2006) · Zbl 1156.45300 [17] Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres,, Journal of the European Mathematical Society, 6, 153 (2004) · Zbl 1075.45006 [18] J. Liu, Liouville-type theorem for polyharmonic systems in \(R^N\),, J. Differential Eq., 225, 685 (2006) · Zbl 1147.35316 [19] E. Mitidieri, Non existence of positive solutions of semilinear elliptic systems in \(R^N\),, Differential & Integral Eq., 9, 465 (1996) · Zbl 0848.35034 [20] E. Mitidieri, A priori estimates and nonexistence of solutions to nonlinear partial differential equations and inequalities,, Proc. Steklov Inst. Math., 234, 1 (2001) · Zbl 1074.35500 [21] E. M. Stein, Fractional Integrals in n-dimensional Euclidean space,, J. Math. Mech., 7 (1958) · Zbl 0082.27201 [22] X. Wei, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313, 207 (1999) · Zbl 0940.35082 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.