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

Weiming  Liu; Lu  Gan

doi:10.3934/cpaa.2016.15.413

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

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

Weiming Liu ^1, and Lu Gan ^1,

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

Received March 2015 Revised December 2015 Published January 2016

Abstract
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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.

Keywords: multi-bump solutions., reduction method, Fractional nonlinear Schrödinger equation.

Mathematics Subject Classification: Primary: 35J10, 35B99; Secondary: 35J6.

Citation: Weiming Liu, Lu Gan. Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure and 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, Adv. Math., 224 (2010), 2052-2093.
[2]	W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.
[3]	X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2956-2992.
[4]	G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
[5]	J. Dávila, M. Del Pino and J. Wei, Concentration standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
[6]	S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216.
[7]	B. Feng, Ground states for the fractional nonlinear Schrödinger equation, J. Differential Equations, 127 (2013), 1-11.
[8]	Rupert Frank, Enno Lenzmann and Luis Silvestre, Uniqueness and nondegeneracy of ground states for $(-\Delta)^sQ + Q - Q^{\alpha+1} = 0$ in $\mathbb{R}^{N}$, Acta Math., 210 (2013), 261-318.
[9]	Rupert Frank and Enno Lenzmann, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., DOI: 10.1002/cpa.21591
[10]	P. Felmer, A. Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
[11]	N. Laskin, Fractional quantum mechanics and L'evy path integrals, Phys. Lett. A, 268 (2000), 29-305.
[12]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 31-35.
[13]	L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Phys. Rev. E, 58 (2009), 1659-1689.
[14]	W. Long, S. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations, Discret. Contin. Dynam. Syst., 36 (2016), 917-939.
[15]	E. S. Noussair and S. Yan, On positive multi-peak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.
[16]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
[17]	G. Palatucci and A. Pisante, Improved sobolev embeddings, profile decomposition and concentration-compactness for fractional sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
[18]	S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$, J. Math. Phys., 54 (2013), 031501, 17 pp.
[19]	X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ., 258 (2015), 1106-1128.
[20]	J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.
[21]	L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv:1403.0042.

show all references

References:

[1]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
[2]	W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.
[3]	X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2956-2992.
[4]	G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
[5]	J. Dávila, M. Del Pino and J. Wei, Concentration standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
[6]	S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216.
[7]	B. Feng, Ground states for the fractional nonlinear Schrödinger equation, J. Differential Equations, 127 (2013), 1-11.
[8]	Rupert Frank, Enno Lenzmann and Luis Silvestre, Uniqueness and nondegeneracy of ground states for $(-\Delta)^sQ + Q - Q^{\alpha+1} = 0$ in $\mathbb{R}^{N}$, Acta Math., 210 (2013), 261-318.
[9]	Rupert Frank and Enno Lenzmann, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., DOI: 10.1002/cpa.21591
[10]	P. Felmer, A. Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
[11]	N. Laskin, Fractional quantum mechanics and L'evy path integrals, Phys. Lett. A, 268 (2000), 29-305.
[12]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 31-35.
[13]	L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Phys. Rev. E, 58 (2009), 1659-1689.
[14]	W. Long, S. Peng and J. Yang, Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations, Discret. Contin. Dynam. Syst., 36 (2016), 917-939.
[15]	E. S. Noussair and S. Yan, On positive multi-peak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.
[16]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
[17]	G. Palatucci and A. Pisante, Improved sobolev embeddings, profile decomposition and concentration-compactness for fractional sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
[18]	S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$, J. Math. Phys., 54 (2013), 031501, 17 pp.
[19]	X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ., 258 (2015), 1106-1128.
[20]	J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.
[21]	L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv:1403.0042.

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