American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities
  • CPAA Home
  • This Issue
  • Next Article
    A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications
November  2018, 17(6): 2239-2259. doi: 10.3934/cpaa.2018107

Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

Xudong Shang 1,, and  Jihui Zhang 2,

1. 

School of Mathematics, Nanjing Normal University Taizhou College, Taizhou 225300, China
2. 

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

* Corresponding author

Received  April 2017 Revised  September 2017 Published  June 2018

In this paper we consider the multiplicity and concentration behavior of positive solutions for the following fractional nonlinear Schrödinger equation
$\left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{s}}\left( {{\mathbb{R}}^{N}} \right)\ \ \ \ \ \ \ \ u\left( x \right)>0, \\ \end{align} \right.$
where
$\varepsilon$
 is a positive parameter,
$(-Δ)^{s}$
 is the fractional Laplacian,
$s ∈ (0,1)$
 and
$N> 2s$
. Suppose that the potential
$V(x) ∈\mathcal{C}(\mathbb{R}^{N})$
 satisfies
$\text{inf}_{\mathbb{R}^{N}} V(x)>0$
, and there exist
$k$
 points
$x^{j} ∈ \mathbb{R}^{N}$
 such that for each
$j = 1,···,k$
,
$V(x^{j})$
 are strict global minimum. When
$f$
 is subcritical, we prove that the problem has at least
$k$
 positive solutions for
$\varepsilon>0$
 small. Moreover, we establish the concentration property of the solutions as
$\varepsilon$
 tends to zero.
Keywords: Fractional Schrödinger equations, Multiplicity of solutions, Ekeland's variational principle.
Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35B38.
Citation: 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
References:
[1]

C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. Google Scholar

[2]

K-C Chang, Methods in nonlinear analysis, Springer-verlag Berlin Heidelberg, 2005. Google Scholar

[3]

D. Cao and E. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $ \mathbb{R}^{N} $, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.  Google Scholar

[4]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.  Google Scholar

[5]

S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $ \mathbb{R}^{N} $, Manuscripta Math., 153 (2017), 183-230.  doi: 10.1007/s00229-016-0878-3.  Google Scholar

[6]

J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for nonlocal equtions with Dirichlet datum, Anal. PDE., 8 (2015), 1165-1235.  doi: 10.2140/apde.2015.8.1165.  Google Scholar

[7]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $ \mathbb{R}^{N}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. Google Scholar

[8]

E. Di Nezza, G. 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.  Google Scholar

[9]

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.  doi: 10.4418/2013.68.1.15.  Google Scholar

[10]

J. Dávila, M. Del Pino and J. C. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

[11]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.  Google Scholar

[12]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect A., 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[13]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, (2016), 55-91.  doi: 10.1007/s00526-016-1045-0.  Google Scholar

[14]

N. Laskin, Fractional Schrödinger equation, Phy. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact case, Part Ⅱ, Ann. Inst. H. Poincaré Analy. Non linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[16]

R. Pabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[17]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R}^{N} $, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990.  Google Scholar

[18]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[19]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. Google Scholar

show all references

References:
[1]

C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. Google Scholar

[2]

K-C Chang, Methods in nonlinear analysis, Springer-verlag Berlin Heidelberg, 2005. Google Scholar

[3]

D. Cao and E. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $ \mathbb{R}^{N} $, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.  Google Scholar

[4]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.  Google Scholar

[5]

S. Dipierro, M. Medina, I. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $ \mathbb{R}^{N} $, Manuscripta Math., 153 (2017), 183-230.  doi: 10.1007/s00229-016-0878-3.  Google Scholar

[6]

J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for nonlocal equtions with Dirichlet datum, Anal. PDE., 8 (2015), 1165-1235.  doi: 10.2140/apde.2015.8.1165.  Google Scholar

[7]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $ \mathbb{R}^{N}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. Google Scholar

[8]

E. Di Nezza, G. 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.  Google Scholar

[9]

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.  doi: 10.4418/2013.68.1.15.  Google Scholar

[10]

J. Dávila, M. Del Pino and J. C. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar

[11]

M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.  doi: 10.1088/0951-7715/28/6/1937.  Google Scholar

[12]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect A., 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[13]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, (2016), 55-91.  doi: 10.1007/s00526-016-1045-0.  Google Scholar

[14]

N. Laskin, Fractional Schrödinger equation, Phy. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact case, Part Ⅱ, Ann. Inst. H. Poincaré Analy. Non linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[16]

R. Pabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[17]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R}^{N} $, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990.  Google Scholar

[18]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[19]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. Google Scholar

[1]

Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911
[2]

Peng Gao, Yong Li. Averaging principle for the Schrödinger equations†. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089
[3]

Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254
[4]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[5]

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
[6]

Gerd Grubb. Limited regularity of solutions to fractional heat and Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3609-3634. doi: 10.3934/dcds.2019148
[7]

Ricardo Almeida, Agnieszka B. Malinowska. Fractional variational principle of Herglotz. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2367-2381. doi: 10.3934/dcdsb.2014.19.2367
[8]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[9]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[10]

Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1267-1282. doi: 10.3934/cpaa.2014.13.1267
[11]

Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359
[12]

Weiwei Ao, Juncheng Wei, Wen Yang. Infinitely many positive solutions of fractional nonlinear Schrödinger equations with non-symmetric potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5561-5601. doi: 10.3934/dcds.2017242
[13]

Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131
[14]

Song Peng, Aliang Xia. Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1201-1217. doi: 10.3934/cpaa.2018058
[15]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83
[16]

Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735
[17]

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
[18]

Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
[19]

Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2795-2808. doi: 10.3934/dcds.2018118
[20]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (152)
  • HTML views (153)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences