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Symmetry of Positive Solutions to the Coupled Fractional System with Isolated Singularities

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Abstract

In this paper, we consider the following two-coupled fractional Laplacian system with two or more isolated singularities

$$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u}&{ = {\mu _1}{u^{2q + 1}} + \beta {u^{p1 - 1}}{v^{p2}},}&\; \\ {{{( - \Delta )}^s}u}&{ = {\mu _2}{v^{2q + 1}} + \beta {u^{p1}}{v^{p2 - 1}}}&{\text{in}\;{\mathbb{R}^n}\backslash \Lambda ,} \\ {u > 0,}&{v > 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&\; \end{array}} \right.$$

where s ∈ (0, 1), n > 2s and n ≥ 2. μ1, μ2 and β are all positive constants. p1, p2 > 1 and \({p_1} + {p_2} = 2q + 2 \in \left( {{{2n - 2s} \over {n - 2s}},\left. {{{2n} \over {n - 2s}}} \right]} \right.\). Λ ⊂ ℝn contains finitely many isolated points. By the method of moving plane, we obtain the symmetry results for positive solutions to above system.

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References

  1. Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bull. Braz. Math. Soc. (N.S.), 22, 1–37 (1991)

    Article  MathSciNet  Google Scholar 

  2. Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math., 42, 271–297 (1989)

    Article  MathSciNet  Google Scholar 

  3. Caffarelli, L., Jin, T. L., Sire, Y., et al.: Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities. Arch. Ration. Mech. Anal., 213, 245–268 (2014)

    Article  MathSciNet  Google Scholar 

  4. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32, 1245–1260 (2007)

    Article  MathSciNet  Google Scholar 

  5. Caju, R., doÓ, J. M., Silva Santos, A.: Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire, 36, 1575–1601 (2019)

    Article  MathSciNet  Google Scholar 

  6. Chen, C. C., Lin, C. S.: Local behavior of singular positive solutions of semilinear elliptic equations with sobolev exponent. Duke Math. J., 78, 315–334 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Chen, W. X., Li, C. M.: Classification of positive solutions for nonlinear differential and integral systems with critical exponents. Acta Math. Sci. Ser. B Engl. Ed., 29, 949–960 (2009)

    Article  MathSciNet  Google Scholar 

  8. Chen, W. X., Li, C. M., Li, Y.: A direct method of moving planes for the fractional Laplacian. Adv. Math., 308, 404–437 (2017)

    Article  MathSciNet  Google Scholar 

  9. Chen, W. X., Li, Y., Ma, P.: The Fractional Laplacian, World Scientific Publishing Company, New York, 2019

    Google Scholar 

  10. Chen, W. X., Qi, S. J.: Direct methods on fractional equations. Discrete Contin. Dyn. Syst., 39, 1269–1310 (2019)

    Article  MathSciNet  Google Scholar 

  11. Chen, Z. J., Lin, C. S.: Removable singularity of positive solutions for a critical elliptic system with isolated singularity. Math. Ann., 363, 501–523 (2015)

    Article  MathSciNet  Google Scholar 

  12. DelaTorre, A., González, M. del Mar: Isolated singularities for a semilinear equation for the fractional Laplacian arising in conformal geometry. Rev. Mat. Iberoamericana, 34, 1645–1678 (2018)

    Article  MathSciNet  Google Scholar 

  13. DelaTorre, A., del Pino, M., Gonzaález, M. del Mar, et al.: Delaunay-type singular solutions for the fractional Yamabe problem. Math. Ann., 369, 597–626 (2017)

    Article  MathSciNet  Google Scholar 

  14. Esposito, F., Farina, A., Sciunzi, B.: Qualitative properties of singular solutions to semilinear elliptic problems. J. Differential Equations, 265, 1962–1983 (2018)

    Article  MathSciNet  Google Scholar 

  15. Gidas, B., Ni, W. M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys., 68, 209–243 (1979)

    Article  MathSciNet  Google Scholar 

  16. Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math., 34, 525–598 (1981)

    Article  MathSciNet  Google Scholar 

  17. González, M. del Mar, Mazzeo, R., Sire, Y.: Singular solutions of fractional order conformal Laplacians. J. Geom. Anal., 22, 845–863 (2012)

    Article  MathSciNet  Google Scholar 

  18. Guo, Y. X., Liu, J. Q.: Liouville type theorems for positive solutions of elliptic system in ℝN. Comm. Partial Differential Equations, 33, 263–284 (2008)

    Article  MathSciNet  Google Scholar 

  19. Guo, Y. X., Li, B., Wei, J. C.: Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in ℝ3. J. Differential Equations, 256, 3463–3495 (2014)

    Article  MathSciNet  Google Scholar 

  20. Jin, T. L., de Queiroz, O. S., Sire, Y., et al.: On local behavior of singular positive solutions to nonlocal elliptic equations. Calc. Var. Partial Differential Equations, 56, Paper No. 9, 25 pp. (2017)

  21. Joseph, D. D., Lundgren, T. S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal., 49, 241–269 (1973)

    Article  MathSciNet  Google Scholar 

  22. Li, C. M.: Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent. Math., 123, 221–231 (1996)

    Article  MathSciNet  Google Scholar 

  23. Li, C. M., Wu, Z. G., Xu, H.: Maximum principles and Bôcher type theorems. Proc. Natl. Acad. Sci. USA, 115, 6976–6979 (2018)

    Article  MathSciNet  Google Scholar 

  24. Li, Y. M., Bao, J. G.: Liouville theorem and isolated singularity of fractional Laplacian system with critical exponents. Nonlinear Anal., 191, 111636, 24 pp. (2020)

    Article  MathSciNet  Google Scholar 

  25. Li, Y., Ni, W. M.: Radial symmetry of positive solutions of nonlinear elliptic equations in ℝn. Comm. Partial Differential Equations, 18, 1043–1054 (1993)

    Article  MathSciNet  Google Scholar 

  26. Lin, C. S.: Estimates of the scalar curvature equation via the method of moving planes III. Comm. Pure Appl. Math., 53, 611–646 (2000)

    Article  MathSciNet  Google Scholar 

  27. Lin, C. S., Prajapat, J. V.: Asymptotic symmetry of singular solutions of semilinear elliptic equations. J. Differential Equations, 245, 2534–2550 (2008)

    Article  MathSciNet  Google Scholar 

  28. Lions, P. L.: Isolated singularities in semilinear problems. J. Differential Equations, 38, 441–450 (1980)

    Article  MathSciNet  Google Scholar 

  29. Ros-Oton, X.: Regularity for the fractional Gelfand problem up to dimension 77. J. Math. Anal. Appl., 419, 10–19 (2014)

    Article  MathSciNet  Google Scholar 

  30. Serrin, J., Zou, H. H.: Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math., 189, 79–142 (2002)

    Article  MathSciNet  Google Scholar 

  31. Yang, H., Zou, W. M.: Symmetry ofcomponents and Liouville-type theorems for semilinear elliptic systems involving the fractional Laplacian. Nonlinear Anal., 180, 208–224 (2019)

    Article  MathSciNet  Google Scholar 

  32. Yang, H., Zou, W. M.: Qualitative analysis for an elliptic system in the punctured space. Z. Angew. Math. Phys., 71, Paper No. 47, 21 pp. (2020)

  33. Zhuo, R., Chen, W. X., Cui, X. W., et al.: Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete Contin. Dyn. Syst., 36, 1125–1141 (2016)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China

    Meng Hui Li, Jin Chun He, Hao Yuan Xu & Mei Hua Yang

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  1. Meng Hui Li
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  2. Jin Chun He
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  3. Hao Yuan Xu
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Correspondence to Hao Yuan Xu.

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Supported by NSFC (Grant Nos. 11971184, 61873320 and 51836003)

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Li, M.H., He, J.C., Xu, H.Y. et al. Symmetry of Positive Solutions to the Coupled Fractional System with Isolated Singularities. Acta. Math. Sin.-English Ser. 37, 1437–1452 (2021). https://doi.org/10.1007/s10114-021-0259-z

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