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

Xuewei Cui; Mei Yu

doi:10.3934/dcds.2019059

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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 & Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059

References:

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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	D. Applebaum, Lévy Processes and Stochastic Calculus, 2^nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	T. Kulczycki, Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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

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