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November  2017, 16(6): 2105-2123. doi: 10.3934/cpaa.2017104

Multiple solutions for a fractional nonlinear Schrödinger equation with local potential

Wulong Liu 1,, and  Guowei Dai 2,

1. 

School of Science, Jiangxi University of Science and Technology, Ganzhou, Jiangxi 341000, China
2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

* Corresponding author

Received  December 2016 Revised  April 2017 Published  July 2017

Fund Project: The first author is partially supported by the Youth Science Foundation of Jiangxi Provincial Department of Education (GJJ14460), the NSFC Grant(61364015), the Foundation of the Jiangxi University of Science and Technology (NSFJ2015-G25). The second author is supported by NNSF of China (No. 11261052,11401477) and the Fundamental Research Funds for the Central Universities (No. DUT15RC(3)018, DUT17LK05).

Using penalization techniques and the Ljusternik-Schnirelmann theory, we establish the multiplicity and concentration of solutions for the following fractional Schrödinger equation
$\left\{ \begin{align} &{{\varepsilon }^{2\alpha }}{{\left( -\Delta \right)}^{a}}u+V\left( x \right)u=f\left( u \right),\ \ x\in {{\mathbb{R}}^{N}}, \\ &u\in {{H}^{a}}\left( {{\mathbb{R}}^{N}} \right),u>0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ \end{align} \right.$
where
$0<α<1$
,
$N>2α$
,
$\varepsilon>0$
is a small parameter,
$V$
satisfies the local condition, and
$f$
is superlinear and subcritical nonlinearity. We show that this equation has at least
$\text{cat}_{M_{δ}}(M)$
single spike solutions.
Keywords: Penalization techniques, Ljusternik-Schnirelmann theory, fractional Schrödinger equation, multiple solutions, single spike solutions.
Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35S15.
Citation: 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
References:
[1]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.  doi: 10.1080/03605302.2011.593013.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618260.  Google Scholar

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B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

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C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^n$ with critical growth and convex nonlinearities, preprint, arXiv: 1609.01911. Google Scholar

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J. Byeon, O. Kwon and J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.   Google Scholar

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X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

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X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 927-949.  doi: 10.1088/0951-7715/28/4/927.  Google Scholar

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G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.  doi: 10.3934/cpaa.2014.13.2359.  Google Scholar

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M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential J. Math. Phys. 53 (2012), 043507. doi: 10.1063/1.3701574.  Google Scholar

[14]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.  doi: 10.1006/jdeq.1999.3662.  Google Scholar

[15]

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

[16]

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

[17]

M. Del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[18]

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

[19]

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. (2016), doi: 10.1007/s00229-016-0878-3.  Google Scholar

[20]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^n$ in Appunti. Edizioni della Normale Scuola Normale di Pisa (2017). arXiv: 1506.01748. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[21]

M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $(Δ)^s u+u=u^p$ in $\mathbb{R}^N$ when $s$ is close to $1$, Comm. Math. Phys., 329 (2014), 383-404.  doi: 10.1007/s00220-014-1919-y.  Google Scholar

[22]

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

[23]

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

[24]

G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrdinger equation in $\mathbb{R}^N$ Nonlinear Differ. Equ. Appl. 23 (2016), 12. doi: 10.1007/s00030-016-0355-4.  Google Scholar

[25]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-53.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[26]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[27]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[28]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.   Google Scholar

[29]

W. Liu, Multiple solutions for a fractional nonlinear Schrödinger equation with a general nonlinearity, prepared. Google Scholar

[30]

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

[31]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.  doi: 10.4171/RMI/750.  Google Scholar

[32]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[33]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[34]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[35]

X. Shang, J. Zhang and Y. Yang, On fractional Schödinger equation in $\mathbb{R}^N$ with critical growth J. Math. Phys. 54 (2013), 121502. doi: 10.1063/1.4835355.  Google Scholar

[36]

X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.  doi: 10.1088/0951-7715/27/2/187.  Google Scholar

[37]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128.  doi: 10.1016/j.jde.2014.10.012.  Google Scholar

show all references

References:
[1]

C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.  doi: 10.1080/03605302.2011.593013.  Google Scholar

[2]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618260.  Google Scholar

[3]

B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

[4]

V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational. Mech. Anal., 114 (1991), 79-93.  doi: 10.1007/BF00375686.  Google Scholar

[5]

C. Bucur and M. Medina, A fractional elliptic problem in $\mathbb{R}^n$ with critical growth and convex nonlinearities, preprint, arXiv: 1609.01911. Google Scholar

[6]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[7]

J. Byeon, O. Kwon and J. Seok, Nonlinear scalar field equations involving the fractional Laplacian, Nonlinearity, 30 (2017), 1659-1681.   Google Scholar

[8]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.  Google Scholar

[9]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar

[10]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[11]

G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations, Nonlinearity, 28 (2015), 927-949.  doi: 10.1088/0951-7715/28/4/927.  Google Scholar

[12]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.  doi: 10.3934/cpaa.2014.13.2359.  Google Scholar

[13]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential J. Math. Phys. 53 (2012), 043507. doi: 10.1063/1.3701574.  Google Scholar

[14]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138.  doi: 10.1006/jdeq.1999.3662.  Google Scholar

[15]

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

[16]

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

[17]

M. Del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[18]

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

[19]

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. (2016), doi: 10.1007/s00229-016-0878-3.  Google Scholar

[20]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^n$ in Appunti. Edizioni della Normale Scuola Normale di Pisa (2017). arXiv: 1506.01748. doi: 10.1007/978-88-7642-601-8.  Google Scholar

[21]

M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $(Δ)^s u+u=u^p$ in $\mathbb{R}^N$ when $s$ is close to $1$, Comm. Math. Phys., 329 (2014), 383-404.  doi: 10.1007/s00220-014-1919-y.  Google Scholar

[22]

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

[23]

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

[24]

G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrdinger equation in $\mathbb{R}^N$ Nonlinear Differ. Equ. Appl. 23 (2016), 12. doi: 10.1007/s00030-016-0355-4.  Google Scholar

[25]

A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-53.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[26]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[27]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[28]

N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 31-35.   Google Scholar

[29]

W. Liu, Multiple solutions for a fractional nonlinear Schrödinger equation with a general nonlinearity, prepared. Google Scholar

[30]

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

[31]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.  doi: 10.4171/RMI/750.  Google Scholar

[32]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[33]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[34]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[35]

X. Shang, J. Zhang and Y. Yang, On fractional Schödinger equation in $\mathbb{R}^N$ with critical growth J. Math. Phys. 54 (2013), 121502. doi: 10.1063/1.4835355.  Google Scholar

[36]

X. Shang and J. Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.  doi: 10.1088/0951-7715/27/2/187.  Google Scholar

[37]

X. Shang and J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differential Equations, 258 (2015), 1106-1128.  doi: 10.1016/j.jde.2014.10.012.  Google Scholar

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