Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PROPERTIES OF POSITIVE SOLUTIONS FOR THE FRACTIONAL LAPLACIAN SYSTEMS WITH POSITIVE-NEGATIVE MIXED POWERS

  • Zhongxue Lu (School of Mathematics and Statistics Jiangsu Normal University) ;
  • Mengjia Niu (School of Mathematics and Statistics Jiangsu Normal University) ;
  • Yuanyuan Shen (School of Mathematics and Statistics Jiangsu Normal University) ;
  • Anjie Yuan (School of Mathematics and Statistics Jiangsu Normal University)
  • Received : 2023.05.10
  • Accepted : 2023.12.07
  • Published : 2024.05.01
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Abstract

In this paper, by establishing the direct method of moving planes for the fractional Laplacian system with positive-negative mixed powers, we obtain the radial symmetry and monotonicity of the positive solutions for the fractional Laplacian systems with positive-negative mixed powers in the whole space. We also give two special cases.

Keywords

Acknowledgement

The authors would like to thank the anonymous referee for her/his useful comments and valuable suggestions which improved and clarified the paper.

References

  1. D. Applebaum, Levy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), no. 11, 1336-1347. 
  2. B. Barrios, I. De Bonis, M. Medina, and I. Peral, Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math. 13 (2015), no. 1, 390-407. https://doi.org/10.1515/math-2015-0038 
  3. C. Brandle, E. Colorado, A. de Pablo, and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 1, 39-71. https://doi.org/10.1017/S0308210511000175 
  4. L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. https://doi.org/10.1080/03605300600987306 
  5. M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Discrete Contin. Dyn. Syst. 38 (2018), no. 9, 4603-4615. https://doi.org/10.3934/dcds.2018201 
  6. L. Cao and X. Wang, Radial symmetry of positive solutions to a class of fractional Laplacian with a singular nonlinearity, J. Korean Math. Soc. 58 (2021), no. 6, 1449-1460. https://doi.org/10.4134/JKMS.j210091 
  7. W. Chen, Y. Fang, and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math. 274 (2015), 167-198. https://doi.org/10.1016/j.aim.2014.12.013 
  8. W. Chen and C. Li, Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018), 735-758. https://doi.org/10.1016/j.aim.2018.07.016 
  9. W. Chen, C. Li, and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations 56 (2017), no. 2, Paper No. 29, 18 pp. https://doi.org/10.1007/s00526-017-1110-3 
  10. W. Chen, C. Li, and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math. 308 (2017), 404-437. https://doi.org/10.1016/j.aim.2016.11.038 
  11. W. Chen, Y. Li, and P. Ma, The Fractional Laplacian, World Sci. Publ., Hackensack, NJ, 2020. https://doi.org/10.1142/10550 
  12. W. Chen, Y. Li, and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal. 272 (2017), no. 10, 4131-4157. https://doi.org/10.1016/j.jfa.2017.02.022 
  13. C. Cheng, Z. Lu, and Y. Lu, A direct method of moving planes for the system of the fractional Laplacian, Pacific J. Math. 290 (2017), no. 2, 301-320. https://doi.org/10.2140/pjm.2017.290.301 
  14. W. Dai, G. Qin, and D. Wu, Direct methods for pseudo-relativistic Schrodinger operators, J. Geom. Anal. 31 (2021), no. 6, 5555-5618. https://doi.org/10.1007/s12220-020-00492-1 
  15. R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in R, Acta Math. 210 (2013), no. 2, 261-318. https://doi.org/10.1007/s11511-013-0095-9 
  16. R. L. Frank, E. Lenzmann, and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671-1726. https://doi.org/10.1002/cpa.21591 
  17. J. Giacomoni, T. Mukherjee, and K. Sreenadh, Positive solutions of fractional elliptic equation with critical and singular nonlinearity, Adv. Nonlinear Anal. 6 (2017), no. 3, 327-354. https://doi.org/10.1515/anona-2016-0113 
  18. S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl. (4) 195 (2016), no. 1, 273-291. https://doi.org/10.1007/s10231-014-0462-y 
  19. T. Jin, Y. Y. Li, and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 6, 1111-1171. https://doi.org/10.4171/JEMS/456 
  20. T. Jin and J. Xiong, A fractional Yamabe flow and some applications, J. Reine Angew. Math. 696 (2014), 187-223. https://doi.org/10.1515/crelle-2012-0110 
  21. P. Le, Symmetry of positive solutions to quasilinear fractional systems, Taiwanese J. Math. 25 (2021), no. 3, 517-534. https://doi.org/10.11650/tjm/201203 
  22. C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Comm. Partial Differential Equations 16 (1991), no. 2-3, 491-526. https://doi.org/10.1080/03605309108820766 
  23. B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal. 146 (2016), 120-135. https://doi.org/10.1016/j.na.2016.08.022 
  24. Y. Lu and Z. Lu, Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system, Discrete Contin. Dyn. Syst. 36 (2016), no. 7, 3791-3810. https://doi.org/10.3934/dcds.2016.36.3791 
  25. T. Mukherjee and K. Sreenadh, Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differential Equations 2016 (2016), Paper No. 54, 23 pp.
  26. Y. Niu, Monotonicity of solutions for a class of nonlocal Monge-Amp'ere problem, Commun. Pure Appl. Anal. 19 (2020), no. 11, 5269-5283. https://doi.org/10.3934/cpaa.2020237 
  27. M. Qie, Z. Lu, and X. Zhang, A direct method of moving planes for the fractional p-Laplacian system with negative powers, Indian J. Pure Appl. Math. 54 (2023), no. 2, 344-358. https://doi.org/10.1007/s13226-022-00257-2 
  28. L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67-112. https://doi.org/10.1002/cpa.20153 
  29. P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure Appl. Anal. 16 (2017), no. 5, 1707-1718. https://doi.org/10.3934/cpaa.2017082 
  30. G. Wang and X. Ren, Radial symmetry of standing waves for nonlinear fractional Laplacian Hardy-Schrodinger systems, Appl. Math. Lett. 110 (2020), 106560, 8 pp. https://doi.org/10.1016/j.aml.2020.106560 
  31. P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl. 450 (2017), no. 2, 982-995. https://doi.org/10.1016/j.jmaa.2017.01.070 
  32. X. Wang and L. Zhang, Existence and multiplicity of weak positive solutions to a class of fractional Laplacian with a singular nonlinearity, Results Math. 74 (2019), no. 2, Paper No. 81, 18 pp. https://doi.org/10.1007/s00025-019-1004-0 
  33. L. Wu and W. Chen, The sliding methods for the fractional p-Laplacian, Adv. Math. 361 (2020), 106933, 26 pp. https://doi.org/10.1016/j.aim.2019.106933 
  34. L. Wu and P. Niu, Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations, Discrete Contin. Dyn. Syst. 39 (2019), no. 3, 1573-1583. https://doi.org/10.3934/dcds.2019069 
  35. Z. Wu and H. Xu, Symmetry properties in systems of fractional Laplacian equations, Discrete Contin. Dyn. Syst. 39 (2019), no. 3, 1559-1571. https://doi.org/10.3934/dcds.2019068 
  36. L. Zhang, B. Ahmad, G. Wang, and X. Ren, Radial symmetry of solution for fractional p-Laplacian system, Nonlinear Anal. 196 (2020), 111801, 16 pp. https://doi.org/10.1016/j.na.2020.111801 
  37. B. Zhang and Z. Lu, Symmetry and nonexistence of solutions for a fully nonlinear nonlocal system, Pacific J. Math. 299 (2019), no. 1, 237-255. https://doi.org/10. 2140/pjm.2019.299.237  https://doi.org/10.2140/pjm.2019.299.237
  38. R. Zhuo, W. Chen, X. Cui, and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst. 36 (2016), no. 2, 1125-1141. https://doi.org/10.3934/dcds.2016.36.1125