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November  2019, 12(7): 2115-2125. doi: 10.3934/dcdss.2019136

Ground states of nonlinear Schrödinger equations with fractional Laplacians

Zupei Shen 1, , Zhiqing Han 2, and  Qinqin Zhang 1,,

1. 

Center for Applied Mathematics, Guangzhou University, Guangzhou 510405, China
2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Qinqin Zhang

Received  December 2017 Revised  April 2018 Published  December 2018

Fund Project: This work was supported by National Natural Science Foundation (11701114, 11471085) and the program for Changjiang scholars and Innovative Recearch Team in univesity (Grant No.IRT1226).

Inspired by Schaftingen [15], we develop a symmetric variational principle for the field equation involving a fractional Laplacians
$ \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*} $
As an application, we prove the existence of symmetric ground states in the fractional Sobolev space
$ H^\alpha (\mathbb{R}^N) $
. These results improve some known ones in the literature. An example is also given to illustrate our results.
Keywords: Schrödinger equation, fractional Laplacian, ground state, rearrangement.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136
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.  Google Scholar

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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.  Google Scholar

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D. Applebaum, Lévy processes-from probability to finance and quantum groups, Not. Am. Math. Soc, 51 (2004), 1336-1347.  Google Scholar

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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.  Google Scholar

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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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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

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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.  Google Scholar

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J. Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math., 7 (2005), 463-481. doi: 10.1142/S0219199705001817.  Google Scholar

[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.  Google Scholar

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X. Tang, Non-Nehari manifold method for asymptoticallyperiodic Schrödinger equations,, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.  Google Scholar

[18]

X. Tang, Non-Nehari-manifold method for asymptoticallylinear Schrödinger equation,, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[4]

D. Applebaum, Lévy processes-from probability to finance and quantum groups, Not. Am. Math. Soc, 51 (2004), 1336-1347.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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.  Google Scholar

[15]

J. Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math., 7 (2005), 463-481. doi: 10.1142/S0219199705001817.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

X. Tang, Non-Nehari-manifold method for asymptoticallylinear Schrödinger equation,, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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