March  2016, 15(2): 413-428. doi: 10.3934/cpaa.2016.15.413

Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$

1. 

School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China, China

Received  March 2015 Revised  December 2015 Published  January 2016

In this paper, we study a fractional nonlinear Schrödinger equation. Applying the finite reduction method, we prove that the equation has multi-bump positive solutions under some suitable conditions which are given in section 1.
Citation: Weiming Liu, Lu Gan. Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2016, 15 (2) : 413-428. doi: 10.3934/cpaa.2016.15.413
References:
[1]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. Google Scholar

[2]

W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian,, \emph{J. Funct. Anal.}, 266 (2014), 6531. Google Scholar

[3]

X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian,, \emph{J. Differential Equations}, 256 (2014), 2956. Google Scholar

[4]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 2359. Google Scholar

[5]

J. Dávila, M. Del Pino and J. Wei, Concentration standing waves for the fractional nonlinear Schrödinger equation,, \emph{J. Differential Equations}, 256 (2014), 858. Google Scholar

[6]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian,, \emph{Matematiche}, 68 (2013), 201. Google Scholar

[7]

B. Feng, Ground states for the fractional nonlinear Schrödinger equation,, \emph{J. Differential Equations}, 127 (2013), 1. Google Scholar

[8]

Rupert Frank, Enno Lenzmann and Luis Silvestre, Uniqueness and nondegeneracy of ground states for $(-\Delta)^sQ + Q - Q^{\alpha+1} = 0$ in $R$,, \emph{Acta Math.}, 210 (2013), 261. Google Scholar

[9]

Rupert Frank and Enno Lenzmann, Uniqueness of radial solutions for the fractional Laplacian,, \emph{Commun. Pure Appl. Math.}, (). Google Scholar

[10]

P. Felmer, A. Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 142 (2012), 1237. Google Scholar

[11]

N. Laskin, Fractional quantum mechanics and L'evy path integrals,, \emph{Phys. Lett. A}, 268 (2000), 29. Google Scholar

[12]

N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002), 31. Google Scholar

[13]

L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation,, \emph{Phys. Rev. E}, 58 (2009), 1659. Google Scholar

[14]

W. Long, S. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations,, \emph{Discret. Contin. Dynam. Syst.}, 36 (2016), 917. Google Scholar

[15]

E. S. Noussair and S. Yan, On positive multi-peak solutions of a nonlinear elliptic problem,, \emph{J. Lond. Math. Soc.}, 62 (2000), 213. Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521. Google Scholar

[17]

G. Palatucci and A. Pisante, Improved sobolev embeddings, profile decomposition and concentration-compactness for fractional sobolev spaces,, \emph{Calc. Var. Partial Differential Equations}, 50 (2014), 799. Google Scholar

[18]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$,, \emph{J. Math. Phys.}, 54 (2013). Google Scholar

[19]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials,, \emph{J. Differ. Equ.}, 258 (2015), 1106. Google Scholar

[20]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 42 (2011), 21. Google Scholar

[21]

L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation,, arXiv:1403.0042., (). Google Scholar

show all references

References:
[1]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. Google Scholar

[2]

W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian,, \emph{J. Funct. Anal.}, 266 (2014), 6531. Google Scholar

[3]

X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian,, \emph{J. Differential Equations}, 256 (2014), 2956. Google Scholar

[4]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, \emph{Commun. Pure Appl. Anal.}, 13 (2014), 2359. Google Scholar

[5]

J. Dávila, M. Del Pino and J. Wei, Concentration standing waves for the fractional nonlinear Schrödinger equation,, \emph{J. Differential Equations}, 256 (2014), 858. Google Scholar

[6]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian,, \emph{Matematiche}, 68 (2013), 201. Google Scholar

[7]

B. Feng, Ground states for the fractional nonlinear Schrödinger equation,, \emph{J. Differential Equations}, 127 (2013), 1. Google Scholar

[8]

Rupert Frank, Enno Lenzmann and Luis Silvestre, Uniqueness and nondegeneracy of ground states for $(-\Delta)^sQ + Q - Q^{\alpha+1} = 0$ in $R$,, \emph{Acta Math.}, 210 (2013), 261. Google Scholar

[9]

Rupert Frank and Enno Lenzmann, Uniqueness of radial solutions for the fractional Laplacian,, \emph{Commun. Pure Appl. Math.}, (). Google Scholar

[10]

P. Felmer, A. Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 142 (2012), 1237. Google Scholar

[11]

N. Laskin, Fractional quantum mechanics and L'evy path integrals,, \emph{Phys. Lett. A}, 268 (2000), 29. Google Scholar

[12]

N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002), 31. Google Scholar

[13]

L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation,, \emph{Phys. Rev. E}, 58 (2009), 1659. Google Scholar

[14]

W. Long, S. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations,, \emph{Discret. Contin. Dynam. Syst.}, 36 (2016), 917. Google Scholar

[15]

E. S. Noussair and S. Yan, On positive multi-peak solutions of a nonlinear elliptic problem,, \emph{J. Lond. Math. Soc.}, 62 (2000), 213. Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521. Google Scholar

[17]

G. Palatucci and A. Pisante, Improved sobolev embeddings, profile decomposition and concentration-compactness for fractional sobolev spaces,, \emph{Calc. Var. Partial Differential Equations}, 50 (2014), 799. Google Scholar

[18]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$,, \emph{J. Math. Phys.}, 54 (2013). Google Scholar

[19]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials,, \emph{J. Differ. Equ.}, 258 (2015), 1106. Google Scholar

[20]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 42 (2011), 21. Google Scholar

[21]

L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation,, arXiv:1403.0042., (). Google Scholar

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