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Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations
A direct method of moving planes for fractional Laplacian equations in the unit ball
1. | Department of Mathematics, Henan Normal University, Xinxiang, 453007, China |
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, 2st edition, Cambridge Studies in Advanced Mathematics, 116. Cambridge: Cambridge University, 2009.
doi: 10.1017/CBO9780511809781. |
[2] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge: Cambridge University Press, 1996. |
[3] |
J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications. Physics reports, 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[4] |
C. Brändle, E. Colorado, A. De Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc Royal Soc Edinburgh, A143 (2013), 39-71. |
[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 and L. Silvestre, An extension problem related to the fractional Laplacian, Comm Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and A. 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. |
[8] |
W. X. Chen, C. C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, preprint, arXiv:1411.1697. |
[9] |
W. X. Chen, C. 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. |
[10] |
W. X. Chen, C. C. Li and B. Ou, Qualitative properities of solutions for an integral equation, Disc Cont Dyn Sys, 12 (2005), 347-354. |
[11] |
W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, preprint, arXiv:1404.1640. |
[12] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Mathematics, 1871 (2006), 1-43.
doi: 10.1007/11545989_1. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm Math Phys, 68 (1979), 209-243. |
[14] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, \textbfA (1981), 369-402. |
[15] |
C. C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM Journal on Mathematical Analysis, 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[16] |
L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, Journal of Differential Equations, 245 (2008), 2551-2565.
doi: 10.1016/j.jde.2008.04.008. |
[17] |
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. |
[18] |
V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of systems with long-range interaction, Comm Non1 Sci Numer Simul, 11 (2006), 885-898.
doi: 10.1016/j.cnsns.2006.03.005. |
show all references
References:
[1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, 2st edition, Cambridge Studies in Advanced Mathematics, 116. Cambridge: Cambridge University, 2009.
doi: 10.1017/CBO9780511809781. |
[2] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge: Cambridge University Press, 1996. |
[3] |
J. P. Bouchard and A. Georges, Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications. Physics reports, 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[4] |
C. Brändle, E. Colorado, A. De Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc Royal Soc Edinburgh, A143 (2013), 39-71. |
[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 and L. Silvestre, An extension problem related to the fractional Laplacian, Comm Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and A. 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. |
[8] |
W. X. Chen, C. C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, preprint, arXiv:1411.1697. |
[9] |
W. X. Chen, C. 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. |
[10] |
W. X. Chen, C. C. Li and B. Ou, Qualitative properities of solutions for an integral equation, Disc Cont Dyn Sys, 12 (2005), 347-354. |
[11] |
W. X. Chen and J. Y. Zhu, Indefinite fractional elliptic problem and Liouville theorems, preprint, arXiv:1404.1640. |
[12] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Mathematics, 1871 (2006), 1-43.
doi: 10.1007/11545989_1. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm Math Phys, 68 (1979), 209-243. |
[14] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, \textbfA (1981), 369-402. |
[15] |
C. C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM Journal on Mathematical Analysis, 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[16] |
L. Ma and L. Zhao, Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application, Journal of Differential Equations, 245 (2008), 2551-2565.
doi: 10.1016/j.jde.2008.04.008. |
[17] |
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. |
[18] |
V. E. Tarasov and G. M. Zaslavsky, Fractional dynamics of systems with long-range interaction, Comm Non1 Sci Numer Simul, 11 (2006), 885-898.
doi: 10.1016/j.cnsns.2006.03.005. |
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