Home Positive solutions to nonlinear systems involving fully nonlinear fractional operators
Article
Licensed
Unlicensed Requires Authentication

Positive solutions to nonlinear systems involving fully nonlinear fractional operators

  • Pengcheng Niu , Leyun Wu EMAIL logo and Xiaoxue Ji
Published/Copyright: June 9, 2018
Become an author with De Gruyter Brill
Author Information Explore this Subject
Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 2

Abstract

In this paper we consider the following fractional system

F(x,u(x),v(x),Fα(u(x)))=0,G(x,v(x),u(x),Gβ(v(x)))=0,

where 0 < α, β < 2, 𝓕α and 𝓖β are the fully nonlinear fractional operators:

Fα(u(x))=Cn,αPV∫Rnf(u(x)−u(y))x−yn+αdy,Gβ(v(x))=Cn,βPV∫Rng(v(x)−v(y))x−yn+βdy.

A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system.

MSC 2010: 35B09; 35A01; 35B53; 35J47
Key Words and Phrases: nonlinear fractional system; fully nonlinear fractional operator; decay at infinity; narrow region principle; direct method of moving planes; radial symmetry; nonexistence

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11771354).

References

[1] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. 22, No 1 (1991), 1–37; http://dx.doi.org/10.1007/BF01244896.10.1007/BF01244896Search in Google Scholar

[2] J.P. Bouchard, A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications. Phys. Rep. 195 (1990), 127–293; http://dx.doi.org/10.1016/0370-1573(90)90099-N.10.1016/0370-1573(90)90099-NSearch in Google Scholar

[3] C. Brandle, E. Colorado, A. de Pablo, U. Sanchez, A concave convex elliptic problem involving the fractional Laplacian. Proc. Royal Soc. Edinburgh143 (2013), 39–71; http://dx.doi.org/10.1017/S0308210511000175.10.1017/S0308210511000175Search in Google Scholar

[4] J. Busca, B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space. J. Differential Equations163 (2000), 41–56; http://dx.doi.org/10.1006/jdeq.1999.3701.10.1006/jdeq.1999.3701Search in Google Scholar

[5] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations32 (2007), 1245–1260; http://dx.doi.org/10.1080/03605300600987306.10.1080/03605300600987306Search in Google Scholar

[6] L. Caffarelli, L. Silvestre, Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62, No 5 (2009), 597–638; http://dx.doi.org/10.1002/cpa.20274.10.1002/cpa.20274Search in Google Scholar

[7] L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation. Ann. Math. 171 (2010), 1903–1930; http://dx.doi.org/10.4007/annals.2010.171.1903.10.4007/annals.2010.171.1903Search in Google Scholar

[8] W.X. Chen, Y.Q. Fang, R. Yang, Liouville theorems involving the fractional Laplacian on a half space. Advances in Math. 274 (2015), 167–198; http://dx.doi.org/10.1016/j.aim.2014.12.013.10.1016/j.aim.2014.12.013Search in Google Scholar

[9] W.X. Chen, C.M. Li, A priori estimates for prescribing scalar curvature equations. Ann. of Math. 145, No 3 (1997), 547–564; http://dx.doi.org/10.2307/2951844.10.2307/2951844Search in Google Scholar

[10] W.X. Chen, C.M. Li, Methods on Nonlinear Elliptic Equations. In: AIMS Book Series on Diff. Equa. Dyn. Sys., 2010.Search in Google Scholar

[11] W.X. Chen, C.M. Li, Radial symmetry of solutions for some integral systems of Wolff type. Disc. Cont. Dyn. Sys. 30 (2011), 1083–1093; http://dx.doi.org/10.3934/dcds.2011.30.1083.10.3934/dcds.2011.30.1083Search in Google Scholar

[12] W.X. Chen, C.M. Li, Y. Li, A direct method of moving planes for the fractional Laplacian. Adv. Math. 308 (2017), 404–437; http://dx.doi.org/10.1016/j.aim.2016.11.03.10.1016/j.aim.2016.11.038Search in Google Scholar

[13] W.X. Chen, C.M. Li, B. Ou, Classification of solutions for a system of integral equations. Comm. Partial Differential Equations30, No 1-3 (2005), 59–65; http://dx.doi.org/10.1081/PDE-200044445.10.1081/PDE-200044445Search in Google Scholar

[14] W.X. Chen, J.Y. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems. J. Math. Anal. Appl. 2 (2011), 744–753; http://dx.doi.org/10.1016/j.jmaa.2010.11.035.10.1016/j.jmaa.2010.11.035Search in Google Scholar

[15] P. Constantin, Euler Equations, Navier-Stokes Equations and Turbulence, Mathematical Foundation of Turbulent Viscous Flows. In: Lecture Notes in Math., Vol. 1871, Springer-verlag, New York, 2006, 1–43.Search in Google Scholar

[16] G. Di Blasio, B. Volzone, Comparison and regularity results for the fractional Laplacian via symmetrization methods. J. Differential Equations253, No 9 (2012), 2593–2615; http://dx.doi.org/10.1016/j.jde.2012.07.004.10.1016/j.jde.2012.07.004Search in Google Scholar

[17] J. Fröhlich, B.L.G. Jonsson, E. Lenzmann, Effective dynamics for boson stars. Nonlinearity20, No 5 (2007), 1031–1075; http://dx.doi.org/10.1088/0951-7715/20/5/001.10.1088/0951-7715/20/5/001Search in Google Scholar

[18] B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, No 3 (1979), 209–243; http://projecteuclid.org/euclid.cmp/1103905359.10.1007/BF01221125Search in Google Scholar

[19] C. Jin, C.M. Li, Symmetry of solutions to some systems of integral equations. Proc. Amer. Math. Soc. 134, No 6 (2006), 1661–1670; http://dx.doi.org/10.1090/S0002-9939-05-08411-X.10.1090/S0002-9939-05-08411-XSearch in Google Scholar

[20] C.M. Li, Monotonicity and symmetry of solutions of fully non-linear elliptic equations on unbounded domains. Comm. Partial Differential Equations16, No 4-5 (1991), 585–615; http://dx.doi.org/10.1080/03605309108820770.10.1080/03605309108820770Search in Google Scholar

[21] C.M. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains. Comm. Partial Differential Equations16, No 2-3 (1991), 491–526; http://dx.doi.org/10.1080/03605309108820766.10.1080/03605309108820766Search in Google Scholar

[22] L. Li, J.J. Sun, S. Tersian, Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian. Fract. Calc. Appl. Anal. 20, No 5 (2017), 1146–11164; https://doi.org/10.1515/fca-2017-0061; https://www.degruyter.com/view/j/fca.2017.20.issue-5/issue-files/fca.2017.20.issue-5.xml.10.1515/fca-2017-0061Search in Google Scholar

[23] Y. Li, W.M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in Rn. Comm. Partial Differential Equations18, No 5-6 (1993), 1043–1054; http://dx.doi.org/10.1080/03605309308820960.10.1080/03605309308820960Search in Google Scholar

[24] Y.Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. 6, No 2 (2004), 153–180; http://dx.doi.org/10.4171/JEMS/6.10.4171/JEMS/6Search in Google Scholar

[25] B.Y. Liu, L. Ma, Radial symmetry results for fractional Laplacian systems. Nonlinear Anal. 146 (2016), 120–135; http://dx.doi.org/10.1016/j.na.2016.08.022.10.1016/j.na.2016.08.022Search in Google Scholar

[26] L. Ma, B.Y. Liu, Symmetry results for decay solutions of elliptic systems in the whole space. Adv. Math. 225, No 6 (2010), 3052–3063; http://dx.doi.org/10.1016/j.aim.2010.05.022.10.1016/j.aim.2010.05.022Search in Google Scholar

[27] A.F. Nowakowski, F.C.G.A. Nicolleau, M. Rahman, The fractional Laplacian as a limiting case of a self-similar spring model and applications to n-dimensional anomalous diffusion. Adv. Math. 16, No 4 (2013), 827–859; 10.2478/s13540-013-0052-5.Search in Google Scholar

[28] V.E. Tarasov, G.M. Zaslavsky, Fractional dynamics of systems with long-range interaction. Commun. Nonlinear Sci. Numer. Simul. 11, No 8 (2006), 885–898; http://dx.doi.org/10.1016/j.cnsns.2006.03.005.10.1016/j.cnsns.2006.03.005Search in Google Scholar

[29] J.L. Vázquez, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. J. Eur. Math. Soc. 16, No 4 (2014), 769–803; http://dx.doi.org/10.4171/JEMS/446.10.4171/JEMS/446Search in Google Scholar

[30] J.L. Vázquez, B. Volzone, Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type. J. Math. Pures Appl. 101 (2014), 553–582; http://dx.doi.org/10.4171/JEMS/446.10.1016/j.matpur.2013.07.001Search in Google Scholar

[31] J.L. Vázquez, B. Volzone, Optimal estimates for fractional fast diffusion equations. J. Math. Pures Appl. 9, No 103 (2015), 535–556; http://dx.doi.org/10.1016/j.matpur.2014.07.002.10.1016/j.matpur.2014.07.002Search in Google Scholar

[32] P.Y. Wang, M. Yu, Solutions of fully nonlinear nonlocal systems. J. Math. Anal. Appl. 450, No 2 (2017), 982–995; http://doi.org/10.1016/j.jmaa.2017.01.070.10.1016/j.jmaa.2017.01.070Search in Google Scholar

[33] Q.M. Zhou, K.Q. Wang, Existence and multiplicity of solutions for non-linear elliptic problems with the fractional Laplacian. Fract. Calc. Appl. Anal. 18, No 1 (2015), 133–145; https://doi.org/10.1515/fca-2015-0009; https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.10.1515/fca-2015-0009Search in Google Scholar

Received: 2017-4-26
Published Online: 2018-6-9
Published in Print: 2018-4-25

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
https://doi.org/10.1515/fca-2018-0030
Keywords for this article
nonlinear fractional system; fully nonlinear fractional operator; decay at infinity; narrow region principle; direct method of moving planes; radial symmetry; nonexistence

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 29.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2018-0030/html
Scroll to top button