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Regularity and classification of solutions to static Hartree equations involving fractional Laplacians
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A Liouville theorem for the subcritical Lane-Emden system
Non-existence of positive solutions for a higher order fractional equation
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China |
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.
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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