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

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

* 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.
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:
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S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407.   Google Scholar

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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 NezzaG. 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. FelmerA. 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. FelmerA. QuaasM. 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

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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

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

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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. VlahosH. IslikerY. 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 NezzaG. 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. FelmerA. 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. FelmerA. QuaasM. 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. VlahosH. IslikerY. 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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