Chen, Wenxiong; Li, Congming; Li, Yan A direct blowing-up and rescaling argument on nonlocal elliptic equations. (English) Zbl 1348.35080 Int. J. Math. 27, No. 8, Article ID 1650064, 20 p. (2016). In this paper, the authors develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. The novelty of this paper is that the authors work directly on the nonlocal operator. Instead, in their seminal papers, Caffarelli and Silvestre used the conventional extension method to localise the problem. More precisely, the authors carry on a blowing-up and rescaling argument in order to obtain a priori estimates on the positive solutions. Based on this estimate and the Leray-Schauder degree theory, they prove the existence of positive solutions. Reviewer: Vincenzo Vespri (Firenze) Cited in 1 ReviewCited in 22 Documents MSC: 35J61 Semilinear elliptic equations 35B09 Positive solutions to PDEs 35B44 Blow-up in context of PDEs 35B45 A priori estimates in context of PDEs Keywords:nonlocal elliptic operators; fractional Laplacian; blowing-up; rescaling; a priori estimates; existence of solutions PDFBibTeX XMLCite \textit{W. Chen} et al., Int. J. Math. 27, No. 8, Article ID 1650064, 20 p. (2016; Zbl 1348.35080) Full Text: DOI arXiv References: [1] 1. B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations252 (2012) 6133-6162. genRefLink(16, ’S0129167X16500646BIB001’, ’10.1016 [2] 2. B. Barrios, L. Del Pezzo, J. Garcia-Melian and A. Quaas, A priori bounds and existence of solutions for some nonlocal elliptic problems, preprint (2015), arXiv:1506.04289. [3] 3. C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh143 (2013) 39-71. genRefLink(16, ’S0129167X16500646BIB003’, ’10.1017 · Zbl 1290.35304 [4] 4. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations32 (2007) 1245-1260. genRefLink(16, ’S0129167X16500646BIB004’, ’10.1080 [5] 5. L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math.62 (2009) 597-638. genRefLink(16, ’S0129167X16500646BIB005’, ’10.1002 [6] 6. K. C. Chang, Methods of Nonlinear Analysis, Monographs in Mathematics (Springer-Verlag, 2005). [7] 7. W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math.274 (2014) 167-198. genRefLink(16, ’S0129167X16500646BIB007’, ’10.1016 [8] 8. W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal.4 (2005) 1-8. genRefLink(128, ’S0129167X16500646BIB008’, ’000227935900001’); · Zbl 1073.45004 [9] 9. W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, preprint (2014), arXiv:1411.1697. · Zbl 1362.35320 [10] 10. W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst.12 (2005) 347-354. genRefLink(128, ’S0129167X16500646BIB010’, ’000226741600011’); · Zbl 1081.45003 [11] 11. W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations30 (2005) 59-65. genRefLink(16, ’S0129167X16500646BIB011’, ’10.1081 [12] 12. W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math.59 (2006) 330-343. genRefLink(16, ’S0129167X16500646BIB012’, ’10.1002 [13] 13. W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations260 (2016) 2758-2785. [14] 14. H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations, Discrete Contin. Dyn. Syst.33 (2013) 2319-2347. genRefLink(16, ’S0129167X16500646BIB014’, ’10.3934 [15] 15. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics (Springer, Berlin, 2001). Reprint of the 1998 edition. · Zbl 1042.35002 [16] 16. L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math.60 (2007) 67-112. genRefLink(16, ’S0129167X16500646BIB016’, ’10.1002 [17] 17. R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Discrete Contin. Dyn. Syst.36 (2016) 1125-1141. · Zbl 1322.31007 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.