Chen, Wenxiong; Li, Congming; Li, Guanfeng Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions. (English) Zbl 1368.35110 Calc. Var. Partial Differ. Equ. 56, No. 2, Paper No. 29, 18 p. (2017). The authors consider equations involving fully nonlinear non-local operators involving fractional derivatives of order \(\alpha\). Such kind of operators were introduced by Caffarelli and Silvestre. A maximum principle is proved. The method of moving planes plays a crucial role to establish radial symmetry and monotonicity for positive solutions to Dirichlet problems associated to such fully nonlinear fractional order equations in the unit ball and in the whole space, as well as the non-existence of solutions on the half space. The limit of this operator when \(\alpha\) coverges to 2 is also studied. Reviewer: Vincenzo Vespri (Firenze) Cited in 1 ReviewCited in 53 Documents MSC: 35J60 Nonlinear elliptic equations 35R11 Fractional partial differential equations Keywords:nonlocal operators; fully non linear elliptic equations; fractional derivatives PDFBibTeX XMLCite \textit{W. Chen} et al., Calc. Var. Partial Differ. Equ. 56, No. 2, Paper No. 29, 18 p. (2017; Zbl 1368.35110) Full Text: DOI arXiv References: [1] Brandle, C., Colorado, E., de Pablo, A., Sanchez, U.: A concave-convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. 143, 39-71 (2013) · Zbl 1290.35304 [2] Chen, W., Fang, Y., Yang, R.: Liouville theorems involving the fractional Laplacian on a half space. Adv. Math. (2015) 274(2015), 167-198 · Zbl 1372.35332 [3] Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615-622 (1991) · Zbl 0768.35025 [4] Chen, W., Li, C.: Methods on Nonlinear Elliptic Equations. AIMS Book Series 4. 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