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Symmetry and monotonicity of positive solutions for Choquard equations involving a generalized tempered fractional p-Laplacian in \({\mathbb {R}}^{n}\)

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Abstract

In this paper, we study a nonlinear system involving a generalized tempered fractional p-Laplacian in \({\mathbb {R}}^{n}\):

$$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta -\lambda _{f})_{p}^{s}u(x)+\omega u(x)=C_{n,t}(|x|^{2t-n}*u^{q})u^{q-1}, &{}x\in {\mathbb {R}}^{n},\\ u(x)>0,&{}x\in {\mathbb {R}}^{n}, \end{array} \right. \end{aligned}$$

where \(0<s,\ t<1\), \(p>2,\ p-1<q<\infty ,\ n\ge 2,\ \omega >0\). By using the direct method of moving planes, we prove that the positive solutions of system above must be radially symmetric and monotone decreasing about some point in the whole space. In particular, decay at infinity and narrow region principle play an important role in getting the main results.

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References

  1. Benci, V., Micheletti, A.M., Visetti, D.: An eigenvalue problem for a quasilinear elliptic field equation. J. Differential Equations 184(2), 299–320 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Br\(\ddot{a}\)ndle, C., Colorado, E., de Pablo, A., S\(\acute{a}\)nchez, U.: A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 143(1), 39–71 (2013)

  3. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32(7–9), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, L.F., Fan, L.L.: Symmetry and monotonicity of positive solutions for a system involving fractional \(p\) &\(q\)-Laplacian in a ball. Complex Var. Elliptic Equ. 68(4), 667–679 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, L.F., Fan, L.L.: Symmetry and monotonicity of positive solutions for a system involving fractional \(p\) &\(q\)-Laplacian in \({\mathbb{R}}^{n}\). Anal. Math. Phys. 12(2), Paper No. 42 (2022)

  6. Cao, L.F., Wang, X.S.: Radial symmetry of positive solutions to a class of fractional Laplacian with a singular nonlinearity. J. Korean Math. Soc. 58(6), 1449–1460 (2021)

    MathSciNet  MATH  Google Scholar 

  7. Cao, L.F., Wang, X.S., Dai, Z.H.: Radial symmetry and monotonicity of solutions to a system involving fractional \(p\)-Laplacian in a ball. Adv. Math. Phys. 1565731 (2018)

  8. Chen, W.X., Fang, Y.Q., Yang, R.: Liouville theorems involving the fractional laplacian on a half space. Adv. Math. 274, 167–198 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, W.X., Li, C.M.: Maximum principles for the fractional \(p\)-Laplacian and symmetry of solutions. Adv. Math. 335, 735–758 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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  MATH  Google Scholar 

  11. Chen, W.X., Li, C.M., Ou, B.: Qualitative properties of solutions for an integral equation. Discrete Contin. Dyn. Syst. 12(2), 347–354 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, W.X., Li, C.M., Ou, B.: Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59(3), 330–343 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, W.X., Li, C.M., Zhu, J.Y.: Fractional equations with indefinite nonlinearities. Discrete Contin. Dyn. Syst. 39(3), 1257–1268 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, W.X., Wu, L.Y.: A maximum principle on unbounded domains and a liouville theorem for fractional \(p\)-harmonic functions. preprint ar**v: 1905.09986 (2019)

  15. Chen, W.X., Zhu, J.Y.: Indefinite fractional elliptic problem and liouville theorems. J. Differential Equations 260(5), 4758–4785 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, Y.G., Liu, B.Y.: Symmetry and non-existence of positive solutions for fractional \(p\)-Laplacian systems. Nonlinear Anal. 183, 303–322 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. d’Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations. Math. Models Methods Appl. Sci. 25(8), 1447–1476 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deng, W.H., Li, B.Y., Tian, W.Y., Zhang, P.W.: Boundary problems for the fractional and tempered fractional operators. Multiscale Model. Simul. 16(1), 125–149 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems, vol. 28. Springer-Verlag, Berlin-New York (1979)

    MATH  Google Scholar 

  20. Jarohs, S., Weth, T.: Symmetry via antisymmetric maximum principles in nonlocal problems of variable order. Ann. Mat. Pura Appl. 195(1), 273–291 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Le, P.: Symmetry of singular solutions for a weighted Choquard equation involving the fractional \(p\)-Laplacian. Commun. Pure Appl. Anal. 19(1), 527–539 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Le, P.: Symmetry of solutions for a fractional \(p\)-Laplacian equation of Choquard type. Internat. J. Math. 31(4), 2050026 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Le, P.: Symmetry of positive solutions to Choquard type equations involving the fractional \(p\)-Laplacian. Acta Appl. Math. 170, 387–398 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ma, L.W., Zhang, Z.Q.: Symmetry of positive solutions for Choquard equations with fractional \(p\)-Laplacian. Nonlinear Anal. 182, 248–262 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Marano, S.A., Papageorgiou, N.S.: Constant-sign and nodal solutions of coercive (\(p, q\))-Laplacian problems. Nonlinear Anal. 77, 118–129 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Medeiros, E., Perera, K.: Multiplicity of solutions for a quasilinear elliptic problem via the cohomological index. Nonlinear Anal. 71(9), 3654–3660 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shiri, B., Wu, G.C., Baleanu, D.: Collocation methods for terminal value problems of tempered fractional differential equations. Appl. Numer. Math. 156, 385–395 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, P.Y., Chen, L., Niu, P.C.: Symmetric properties for Choquard equations involving fully nonlinear nonlocal operators. Bull. Braz. Math. Soc. (N.S.) 52(4), 841–862 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, P.Y., Chen, W.X.: Hopf’s lemmas for parabolic fractional laplacians and parabolic fractional \(p\)-Laplacians. Commun. Pure Appl. Anal. 21(9), 3055–3069 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, P.Y., Dai, Z.H., Cao, L.F.: Radial symmetry and monotonicity for fractional H\(\acute{e}\)non equation in \({\mathbb{R} }^{n}\). Complex Var. Elliptic Equ. 60(12), 1685–1695 (2015)

    Article  MathSciNet  Google Scholar 

  31. Wang, X.S., Yang, Z.D.: Symmetry and monotonicity of positive solutions for a Choquard equation with the fractional laplacian. Complex Var. Elliptic Equ. 67(5), 1211–1228 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, L.H., Hou, W.W., Ahmad, B., Wang, G.T.: Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional \(p\)-Laplacian. Discrete Contin. Dyn. Syst. Ser. S 14(10), 3851–3863 (2021)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first and the second authors are supported by Key Scientific Research Project for Colleges and Universities in Henan Province (No. 23A110007). The third author is supported by the Natural Science Foundation of China (No. 12271254; 12141104). We thank the anonymous referees for the received constructive comments after a very careful reading of the previous version of this manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nan**g University of Science and Technology, Nan**g, 210094, Jiangsu, People’s Republic of China

    Linlin Fan & Peibiao Zhao

  2. College of Mathematics and Information Science, Henan Normal University, **nxiang, 453007, Henan, People’s Republic of China

    Linfen Cao

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  1. Linlin Fan
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Correspondence to Peibiao Zhao.

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Fan, L., Cao, L. & Zhao, P. Symmetry and monotonicity of positive solutions for Choquard equations involving a generalized tempered fractional p-Laplacian in \({\mathbb {R}}^{n}\). Fract Calc Appl Anal 26, 2757–2773 (2023). https://doi.org/10.1007/s13540-023-00207-7

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