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Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $
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 |
$ (-\Delta)^s $ |
$ s = m+\frac{\alpha}{2} $ |
$ \begin{equation*} (-\Delta)^{s} u(x) = f(u(x)), \qquad x\in\mathbb{R}^n_+, \end{equation*} $ |
$ m\in \mathbb{N}^* $ |
$ 0<\alpha<2 $ |
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781.![]() ![]() ![]() |
[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. |
[3] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, Cambridge University Press, Cambridge, 1996.
![]() ![]() |
[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.![]() ![]() ![]() |
[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. |
[6] |
H. Brézis and L. A. Peletier, Asymptotics for Elliptic Equations Involving Critical Growth, Report No.03, Mathematical Institute, Leiden University, 1988. |
[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. |
[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. |
[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. |
[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. |
[11] |
W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[22] |
N. S. Landkof, Foundations of modern potential theory, Springer–Verlag, Berlin, Heidelberg, New York, 1972. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781.![]() ![]() ![]() |
[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. |
[3] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathmatics, Cambridge University Press, Cambridge, 1996.
![]() ![]() |
[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.![]() ![]() ![]() |
[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. |
[6] |
H. Brézis and L. A. Peletier, Asymptotics for Elliptic Equations Involving Critical Growth, Report No.03, Mathematical Institute, Leiden University, 1988. |
[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. |
[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. |
[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. |
[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. |
[11] |
W. Chen, Y. Li and P. Ma, The fractional Laplacian, in press. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[22] |
N. S. Landkof, Foundations of modern potential theory, Springer–Verlag, Berlin, Heidelberg, New York, 1972. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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