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 and 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. 

[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 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.

[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. 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.

[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.

[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. VlahosH. IslikerY. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, Order and Chaos, Patras University Press, 2008. 
[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. 

[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 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.

[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. 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.

[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.

[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. VlahosH. IslikerY. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, Order and Chaos, Patras University Press, 2008. 
[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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