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March  2019, 39(3): 1379-1387. doi: 10.3934/dcds.2019059

Non-existence of positive solutions for a higher order fractional equation

Xuewei Cui and  Mei Yu ,

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China

* Corresponding author: Mei Yu

Received  December 2017 Revised  April 2018 Published  December 2018

Fund Project: This work was supported by the National Nature Science Foundation of China (Grant No.11271299 and No.11801446) and the Natural Science Basic Research Plan in Shaanxi Province of China (Program No.2016JM1023 and No.2018JQ1037.).

In this paper, we consider a nonlinear equation involving fractional Laplacian of higher order on the whole space. We establish the equivalence between the pseudo-differential equation and an integral equation by applying the maximum principle and the Liouville theorem. For positive solutions to the equation, we obtained non-existence by applying the method of moving planes.

Keywords: Fractional Laplacian of higher order, non-existence of solutions, maximum principle, Liouville theorem, the method of moving planes in integral form.
Mathematics Subject Classification: Primary: 35A01, 35B09; Secondary: 35C15.
Citation: Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. 

[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), 39-71.  doi: 10.1017/S0308210511000175.

[4]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann Inst H Poincaré Anal NonLinéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[5]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. in Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.

[6]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[8]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[9]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Syst., 38 (2018), 4603-4615.  doi: 10.3934/dcds.2018201.

[10]

G. Caristi, L. D Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.

[11]

W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[12]

W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[14]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[15]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 2758-2785.  doi: 10.1016/j.jde.2015.11.029.

[16]

T. Cheng, Monotonicity and symmetry of solutions to fractional Laplacian equation, Disc. Cont. Dyn. Syst., 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.

[17]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math, (2006), Springer, Berlin, 1–43. doi: 10.1007/11545989_1.

[18]

L. D Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems, Discrete Contin. Dyn. Syst., 7 (2014), 653-671.  doi: 10.3934/dcdss.2014.7.653.

[19]

L. Dupaigne and Y. Sire, A Liouville theorem for nonlocal elliptic equations, in: Symmetry for Elliptic PDEs, in: Contemp. Math., vol. 528, Amer. Math. Soc. Providence, RI, 2010,105–114. doi: 10.1090/conm/528/10417.

[20]

P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh, 142A (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[21]

P. Felmer and W. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1350023, 24 pp.  doi: 10.1142/S0219199713500235.

[22]

B. Gidas, W. Ni and L. Nirenberg, Symmetry and the related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[23]

N. Guillen and R. W. Schwab, Aleksandrov-Bakelman-Pucci Type Estimates for Integro-Differential Equations, Arch. Rat. Mech. Anal., 206 (2012), 111-157.  doi: 10.1007/s00205-012-0529-0.

[24]

T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. 

[25]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag Berlin Heidelberg, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.

[26]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[27]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. 

[28]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rational Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.

[29]

R. Metzler and J. Klafter, The restaurant at the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.

[30]

A. Quaas and X. Aliang, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var., 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[32]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[33]

J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Disc. Cont. Dyn. Syst., 31 (2011), 975-983.  doi: 10.3934/dcds.2011.31.975.

[34]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-898.  doi: 10.1016/j.cnsns.2006.03.005.

[35]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

show all references

References:
[1]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[2]

K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. 

[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), 39-71.  doi: 10.1017/S0308210511000175.

[4]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann Inst H Poincaré Anal NonLinéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[5]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. in Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.

[6]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[8]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.  doi: 10.4007/annals.2010.171.1903.

[9]

M. Cai and L. Ma, Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Syst., 38 (2018), 4603-4615.  doi: 10.3934/dcds.2018201.

[10]

G. Caristi, L. D Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.

[11]

W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013.

[12]

W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038.

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[14]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[15]

W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 2758-2785.  doi: 10.1016/j.jde.2015.11.029.

[16]

T. Cheng, Monotonicity and symmetry of solutions to fractional Laplacian equation, Disc. Cont. Dyn. Syst., 37 (2017), 3587-3599.  doi: 10.3934/dcds.2017154.

[17]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math, (2006), Springer, Berlin, 1–43. doi: 10.1007/11545989_1.

[18]

L. D Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems, Discrete Contin. Dyn. Syst., 7 (2014), 653-671.  doi: 10.3934/dcdss.2014.7.653.

[19]

L. Dupaigne and Y. Sire, A Liouville theorem for nonlocal elliptic equations, in: Symmetry for Elliptic PDEs, in: Contemp. Math., vol. 528, Amer. Math. Soc. Providence, RI, 2010,105–114. doi: 10.1090/conm/528/10417.

[20]

P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh, 142A (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[21]

P. Felmer and W. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1350023, 24 pp.  doi: 10.1142/S0219199713500235.

[22]

B. Gidas, W. Ni and L. Nirenberg, Symmetry and the related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[23]

N. Guillen and R. W. Schwab, Aleksandrov-Bakelman-Pucci Type Estimates for Integro-Differential Equations, Arch. Rat. Mech. Anal., 206 (2012), 111-157.  doi: 10.1007/s00205-012-0529-0.

[24]

T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. 

[25]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag Berlin Heidelberg, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.

[26]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[27]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. 

[28]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rational Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.

[29]

R. Metzler and J. Klafter, The restaurant at the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.

[30]

A. Quaas and X. Aliang, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var., 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8.

[31]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[32]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.

[33]

J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Disc. Cont. Dyn. Syst., 31 (2011), 975-983.  doi: 10.3934/dcds.2011.31.975.

[34]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-898.  doi: 10.1016/j.cnsns.2006.03.005.

[35]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Syst., 36 (2016), 1125-1141.  doi: 10.3934/dcds.2016.36.1125.

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