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Entire solutions in nonlocal monostable equations: Asymmetric case
The properties of positive solutions to semilinear equations involving the fractional Laplacian
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China |
$Ω$ |
$\left\{ \begin{array}{*{35}{l}} \begin{align} & {{(-\Delta )}^{\frac{\alpha }{2}}}u=f(u),\ \ \text{in}\ \ \Omega , \\ & u=0, ~~~~~~~~~~~~~~~~~~~~ \text{in}\ \ {{\Omega }^{c}},\ \\ \end{align} & \ & {} \\\end{array} \right.~~~~(1)$ |
$α$ |
zhongwenzy$ |
$ |
$f$ |
$ \begin{align}u(x) \ = \ \int{{}}_{ Ω}G(x, y)f(u(y))dy, \end{align}~~~~(2) $ |
$G(x, y)$ |
$Ω$ |
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781. |
[2] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[3] |
K. Bogdan,
The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.
doi: 10.4064/sm-123-1-43-80. |
[4] |
J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990).
doi: 10.1016/0370-1573(90)90099-N. |
[5] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. in PDE, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[7] |
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. |
[8] |
W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv: 1309.7499v1. Google Scholar |
[9] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a
half space, Adv. in Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[10] |
W. Chen and C. Li,
Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8.
doi: 10.3934/cpaa.2005.4.1. |
[11] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst. vol.4 2010. |
[12] |
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. |
[13] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical
Foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math. 1–43, Springer,
Berlin, 2006.
doi: 10.1007/11545989_1. |
[14] |
P. Felmer and Y. Wang, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Comm. Cont. Math., 16 (2014), 1350023.
doi: 10.1142/S0219199713500235. |
[15] |
Q. Guan,
Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.
doi: 10.1007/s00220-006-0054-9. |
[16] |
T. Kulczycki,
Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364.
|
[17] |
Yan Li, A semilinear equation involving the fractional Laplacian in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 7 (2015), Google Scholar |
[18] |
E. Nezza, G. Palatucci and E. Valdinoci,
Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[19] |
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. |
[20] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.
doi: 10.1016/j.cnsns.2006.03.005. |
[21] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Radial symmetry of positive solutions to equations
involving the fractional Laplacian, Discrete Contin.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 ed, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781. |
[2] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[3] |
K. Bogdan,
The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.
doi: 10.4064/sm-123-1-43-80. |
[4] |
J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990).
doi: 10.1016/0370-1573(90)90099-N. |
[5] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. in PDE, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[6] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[7] |
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. |
[8] |
W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv: 1309.7499v1. Google Scholar |
[9] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a
half space, Adv. in Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[10] |
W. Chen and C. Li,
Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8.
doi: 10.3934/cpaa.2005.4.1. |
[11] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst. vol.4 2010. |
[12] |
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. |
[13] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical
Foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math. 1–43, Springer,
Berlin, 2006.
doi: 10.1007/11545989_1. |
[14] |
P. Felmer and Y. Wang, Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Comm. Cont. Math., 16 (2014), 1350023.
doi: 10.1142/S0219199713500235. |
[15] |
Q. Guan,
Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329.
doi: 10.1007/s00220-006-0054-9. |
[16] |
T. Kulczycki,
Properties of Green function of symmetric stable processes, Probability and Mathematical Statistics, 17 (1997), 339-364.
|
[17] |
Yan Li, A semilinear equation involving the fractional Laplacian in $\mathbb{R}^{n}$, J. Math. Anal. Appl., 7 (2015), Google Scholar |
[18] |
E. Nezza, G. Palatucci and E. Valdinoci,
Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[19] |
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. |
[20] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.
doi: 10.1016/j.cnsns.2006.03.005. |
[21] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Radial symmetry of positive solutions to equations
involving the fractional Laplacian, Discrete Contin.Dyn. Syst., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
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