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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 and 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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

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