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Ground states of nonlinear Schrödinger equations with fractional Laplacians
1. | Center for Applied Mathematics, Guangzhou University, Guangzhou 510405, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
$ \begin{equation*} \left\{ \begin{aligned} (-\Delta)^\alpha u+u& = f(u), x\in\mathbb{R}^N,\\ u(x)&\geq 0. \end{aligned} \right. \end{equation*} $ |
$ H^\alpha (\mathbb{R}^N) $ |
References:
[1] |
S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. Google Scholar |
[2] |
F. Almgren and E. Lieb,
Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc, 2 (1989), 683-773.
doi: 10.1090/S0894-0347-1989-1002633-4. |
[3] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
D. Applebaum, Lévy processes-from probability to finance and quantum groups, Not. Am. Math. Soc, 51 (2004), 1336-1347. |
[5] |
X. Chang and Z.-Q. Wang,
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.
doi: 10.1088/0951-7715/26/2/479. |
[6] |
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. |
[7] |
S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, LXVIII, 68 (2013), 201-216. |
[8] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrodinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[9] |
P. Felmer, A. Quaas, M. Tang and J. Yu,
Monotonicity properties for ground states of the scalar field equation, Ann.I.Poincare-AN, 25 (2008), 105-119.
doi: 10.1016/j.anihpc.2006.12.003. |
[10] |
M. Jara,
Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.
doi: 10.1002/cpa.20253. |
[11] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $\mathbb{R}^N$, Proc. Amer. Math. Soc, 131 (2002), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[12] |
L. Jeanjean and K. Tanaka,
A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N$ autonomous at infinity, ESAIM Control Optim. Calc. Var, 7 (2002), 597-614.
doi: 10.1051/cocv:2002068. |
[13] |
E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[14] |
S. Liu,
On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.
doi: 10.1007/s00526-011-0447-2. |
[15] |
J. Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math., 7 (2005), 463-481.
doi: 10.1142/S0219199705001817. |
[16] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[17] |
X. Tang,
Non-Nehari manifold method for asymptoticallyperiodic Schrödinger equations,, Sci. China Math., 58 (2015), 715-728.
doi: 10.1007/s11425-014-4957-1. |
[18] |
X. Tang,
Non-Nehari-manifold method for asymptoticallylinear Schrödinger equation,, J. Aust. Math. Soc., 98 (2015), 104-116.
doi: 10.1017/S144678871400041X. |
[19] |
X. Tang, X. Lin and J. Yu, Nontrivial solutions for Schrödinger equation with local supper-quadratic conditions, J. Dyn. Diff. Equat., accepted for publication.
doi: 10.1007/s10884-018-9662-2. |
[20] | L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, Order and Chaos, Patras University Press, 2008. Google Scholar |
[21] |
Y. Wei and X. Su,
Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124.
doi: 10.1007/s00526-013-0706-5. |
[22] |
H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281.
doi: 10.1016/S1007-5704(03)00049-2. |
[23] |
M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. Google Scholar |
[2] |
F. Almgren and E. Lieb,
Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc, 2 (1989), 683-773.
doi: 10.1090/S0894-0347-1989-1002633-4. |
[3] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
D. Applebaum, Lévy processes-from probability to finance and quantum groups, Not. Am. Math. Soc, 51 (2004), 1336-1347. |
[5] |
X. Chang and Z.-Q. Wang,
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.
doi: 10.1088/0951-7715/26/2/479. |
[6] |
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. |
[7] |
S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, LXVIII, 68 (2013), 201-216. |
[8] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrodinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[9] |
P. Felmer, A. Quaas, M. Tang and J. Yu,
Monotonicity properties for ground states of the scalar field equation, Ann.I.Poincare-AN, 25 (2008), 105-119.
doi: 10.1016/j.anihpc.2006.12.003. |
[10] |
M. Jara,
Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214.
doi: 10.1002/cpa.20253. |
[11] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $\mathbb{R}^N$, Proc. Amer. Math. Soc, 131 (2002), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[12] |
L. Jeanjean and K. Tanaka,
A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N$ autonomous at infinity, ESAIM Control Optim. Calc. Var, 7 (2002), 597-614.
doi: 10.1051/cocv:2002068. |
[13] |
E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[14] |
S. Liu,
On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.
doi: 10.1007/s00526-011-0447-2. |
[15] |
J. Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math., 7 (2005), 463-481.
doi: 10.1142/S0219199705001817. |
[16] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[17] |
X. Tang,
Non-Nehari manifold method for asymptoticallyperiodic Schrödinger equations,, Sci. China Math., 58 (2015), 715-728.
doi: 10.1007/s11425-014-4957-1. |
[18] |
X. Tang,
Non-Nehari-manifold method for asymptoticallylinear Schrödinger equation,, J. Aust. Math. Soc., 98 (2015), 104-116.
doi: 10.1017/S144678871400041X. |
[19] |
X. Tang, X. Lin and J. Yu, Nontrivial solutions for Schrödinger equation with local supper-quadratic conditions, J. Dyn. Diff. Equat., accepted for publication.
doi: 10.1007/s10884-018-9662-2. |
[20] | L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, Order and Chaos, Patras University Press, 2008. Google Scholar |
[21] |
Y. Wei and X. Su,
Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124.
doi: 10.1007/s00526-013-0706-5. |
[22] |
H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281.
doi: 10.1016/S1007-5704(03)00049-2. |
[23] |
M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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