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July  2020, 19(7): 3597-3612. doi: 10.3934/cpaa.2020157

Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $

Mei Yu 1,2, , Xia Zhang 3, and  Binlin Zhang 4,,

1. 

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, 710129, China
2. 

Department of Mathematics, Yeshiva University, New York, 10033, USA
3. 

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
4. 

College of Mathematics and Systems Sciences, Shandong University of Science and Technology, Qingdao, 266590, China

*Corresponding author

Received  June 2019 Revised  January 2020 Published  April 2020

Fund Project: M. Yu was supported by National Natural Science Foundation of China (Grant No. 11801446, Grant No. 11971385), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JQ1037, Grant No. 2019JM-275); X. Zhang was supported by National Natural Science Foundation of China (Grant No. 11671111, 11871177); B. Zhang was supported by National Natural Science Foundation of China (Grant No. 11871199), Heilongjiang Province Postdoctoral Startup Foundation (LBH-Q18109), and the Introduction and Cultivation Project of Young and Innovative Talents in Universities of Shandong Province

In this paper, we consider the following equation with the higher-order fractional Laplacian
$ (-\Delta)^s $
 for
$ s = m+\frac{\alpha}{2} $
:
$ \begin{equation*} (-\Delta)^{s} u(x) = f(u(x)), \qquad x\in\mathbb{R}^n_+, \end{equation*} $
where
$ m\in \mathbb{N}^* $
,
$ 0<\alpha<2 $
. By developing a narrow region principle in unbounded domain and establishing a equivalence of differential equation and integral equation, together with the method of moving planes, we deduce the monotonicity property of positive solutions and the Liouville theorem of nonnegative solutions.
Keywords: Higher-order fractional Laplacian, Liouville theorem, method of moving planes, monotonicity.
Mathematics Subject Classification: Primary: 35S15, 35B06; Secondary: 35J61.
Citation: Mei Yu, Xia Zhang, Binlin Zhang. Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3597-3612. doi: 10.3934/cpaa.2020157
References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511809781.  Google Scholar
[2]

F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differ. Equ., 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2.  Google Scholar

[3] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, Cambridge University Press, Cambridge, 1996.   Google Scholar
[4] G. M. Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
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J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

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H. Brézis and L. A. Peletier, Asymptotics for Elliptic Equations Involving Critical Growth, Report No.03, Mathematical Institute, Leiden University, 1988. Google Scholar

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

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

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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

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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

[11]

W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press. Google Scholar

[12]

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

[13]

C. V. Coffman, Uniqueness of the ground state solution for $ $\bigtriangleup$ u-u+u^3$ and a variational characterization of other solutions, Arch. Ration. Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684.  Google Scholar

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

[15]

X. Cui and M. Yu, Non-existence of positive solutions for a higher order fractional equation, Discrete Contin. Dyn. Syst., 39 (2019), 1379-1387.  doi: 10.3934/dcds.2019059.  Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

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P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1-24.  doi: 10.1142/S0219199713500235.  Google Scholar

[18]

D. G. Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.   Google Scholar

[19]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[20]

H. G. Kaper and M. K. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, in Nonlinear Diffusion Equations and Their Equilibrium States II, vol. 13, (1988), 1–17. doi: 10.1007/978-1-4613-9608-6_1.  Google Scholar

[21]

M. K. Kwong, Uniqueness of positive solutions of $ $\bigtriangleup$ u-u+u^p = 0$ in $\mathbb{R}^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[22]

N. S. Landkof, Foundations of modern potential theory, Springer–Verlag, Berlin, Heidelberg, New York, 1972.  Google Scholar

[23]

K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $ $\bigtriangleup$ u+f(u)=0$ in $\mathbb{R}^n$, Arch. Ration. Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874.  Google Scholar

[24]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^n$, J. Differ. Equ., 61 (1986), 380-397.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar

[25]

A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differ. Equ., 52 (2014), 641-659.  doi: 10.1007/s00526-014-0727-8.  Google Scholar

[26]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range inthraction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-889.  doi: 10.1016/j.cnsns.2006.03.005.  Google Scholar

[27]

M. Xiang, B. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional $p$–Laplacian equations, J. Differ. Equ., 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028.  Google Scholar

[28]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differ. Equ., 46 (2013), 75-95.  doi: 10.1007/s00526-011-0474-z.  Google Scholar

[29]

L. Zhan and M. Yu, A Liouville theorem for a class of fractional systems in $\mathbb{R}^n_+$, J. Differ. Equ., 263 (2017), 6025-6065.  doi: 10.1016/j.jde.2017.07.009.  Google Scholar

[30]

L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$, Discrete Contin. Dyn. Syst., 36 (2016), 1721-1736.  doi: 10.3934/dcds.2016.36.1721.  Google Scholar

show all references

References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511809781.  Google Scholar
[2]

F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differ. Equ., 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2.  Google Scholar

[3] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, Cambridge University Press, Cambridge, 1996.   Google Scholar
[4] G. M. Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.  Google Scholar
[5]

J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[6]

H. Brézis and L. A. Peletier, Asymptotics for Elliptic Equations Involving Critical Growth, Report No.03, Mathematical Institute, Leiden University, 1988. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press. Google Scholar

[12]

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

[13]

C. V. Coffman, Uniqueness of the ground state solution for $ $\bigtriangleup$ u-u+u^3$ and a variational characterization of other solutions, Arch. Ration. Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684.  Google Scholar

[14]

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

[15]

X. Cui and M. Yu, Non-existence of positive solutions for a higher order fractional equation, Discrete Contin. Dyn. Syst., 39 (2019), 1379-1387.  doi: 10.3934/dcds.2019059.  Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[17]

P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1-24.  doi: 10.1142/S0219199713500235.  Google Scholar

[18]

D. G. Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.   Google Scholar

[19]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ., 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[20]

H. G. Kaper and M. K. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, in Nonlinear Diffusion Equations and Their Equilibrium States II, vol. 13, (1988), 1–17. doi: 10.1007/978-1-4613-9608-6_1.  Google Scholar

[21]

M. K. Kwong, Uniqueness of positive solutions of $ $\bigtriangleup$ u-u+u^p = 0$ in $\mathbb{R}^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[22]

N. S. Landkof, Foundations of modern potential theory, Springer–Verlag, Berlin, Heidelberg, New York, 1972.  Google Scholar

[23]

K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $ $\bigtriangleup$ u+f(u)=0$ in $\mathbb{R}^n$, Arch. Ration. Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874.  Google Scholar

[24]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^n$, J. Differ. Equ., 61 (1986), 380-397.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar

[25]

A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space, Calc. Var. Partial Differ. Equ., 52 (2014), 641-659.  doi: 10.1007/s00526-014-0727-8.  Google Scholar

[26]

V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range inthraction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-889.  doi: 10.1016/j.cnsns.2006.03.005.  Google Scholar

[27]

M. Xiang, B. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional $p$–Laplacian equations, J. Differ. Equ., 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028.  Google Scholar

[28]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differ. Equ., 46 (2013), 75-95.  doi: 10.1007/s00526-011-0474-z.  Google Scholar

[29]

L. Zhan and M. Yu, A Liouville theorem for a class of fractional systems in $\mathbb{R}^n_+$, J. Differ. Equ., 263 (2017), 6025-6065.  doi: 10.1016/j.jde.2017.07.009.  Google Scholar

[30]

L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$, Discrete Contin. Dyn. Syst., 36 (2016), 1721-1736.  doi: 10.3934/dcds.2016.36.1721.  Google Scholar

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