American Institute of Mathematical Sciences

Advanced
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

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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

B. Feng, Ground states for the fractional nonlinear Schrödinger equation, J. Differential Equations, 127 (2013), 1-11. 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 $R$, Acta Math., 210 (2013), 261-318. 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, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. Google Scholar

[11]

N. Laskin, Fractional quantum mechanics and L'evy path integrals, Phys. Lett. A, 268 (2000), 29-305. Google Scholar

[12]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 31-35. Google Scholar

[13]

L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Phys. Rev. E, 58 (2009), 1659-1689. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. Google Scholar

[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. Google Scholar

[18]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$, J. Math. Phys., 54 (2013), 031501, 17 pp. Google Scholar

[19]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ., 258 (2015), 1106-1128. Google Scholar

[20]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41. 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, Adv. Math., 224 (2010), 2052-2093. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

B. Feng, Ground states for the fractional nonlinear Schrödinger equation, J. Differential Equations, 127 (2013), 1-11. 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 $R$, Acta Math., 210 (2013), 261-318. 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, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. Google Scholar

[11]

N. Laskin, Fractional quantum mechanics and L'evy path integrals, Phys. Lett. A, 268 (2000), 29-305. Google Scholar

[12]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 31-35. Google Scholar

[13]

L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Phys. Rev. E, 58 (2009), 1659-1689. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. Google Scholar

[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. Google Scholar

[18]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $R^N$, J. Math. Phys., 54 (2013), 031501, 17 pp. Google Scholar

[19]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ., 258 (2015), 1106-1128. Google Scholar

[20]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41. Google Scholar

[21]

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

[1]

Yuxia Guo, Zhongwei Tang. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3393-3415. doi: 10.3934/dcds.2015.35.3393
[2]

Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058
[3]

Claudianor O. Alves, Olímpio H. Miyagaki, Sérgio H. M. Soares. Multi-bump solutions for a class of quasilinear equations on $R$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 829-844. doi: 10.3934/cpaa.2012.11.829
[4]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[5]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[6]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[7]

Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3723-3744. doi: 10.3934/cpaa.2021128
[8]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[9]

Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137
[10]

Masoumeh Hosseininia, Mohammad Hossein Heydari, Carlo Cattani. A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2273-2295. doi: 10.3934/dcdss.2020295
[11]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587
[12]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
[13]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085
[14]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[15]

Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
[16]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[17]

Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188
[18]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3589-3610. doi: 10.3934/dcdss.2021021
[19]

Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104
[20]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (134)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy