# American Institute of Mathematical Sciences

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, 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