December  2016, 21(10): 3407-3428. doi: 10.3934/dcdsb.2016104

Infinitely many solutions of the nonlinear fractional Schrödinger equations

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China, China

Received  January 2015 Revised  March 2016 Published  November 2016

In this paper, we study the fractional Schrödinger equation \begin{equation*} (-\Delta)^{s} u+V(x)u=f(x,u), \quad x\in\mathbb{R}^{N}, \end{equation*} where $0< s <1$, $(-\Delta)^{s}$ denotes the fractional Laplacian of order $s$ and the nonlinearity $f$ is sublinear or superlinear at infinity. Under certain assumptions on $V$ and $f$, we prove that this equation has infinitely many solutions via variational methods, which unifies and sharply improves the recent results of Teng (2015) [33]. Moreover, we also consider the above equation with concave and critical nonlinearities, and obtain the existence of infinitely many solutions.
Citation: 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
References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion,, Physica A, 356 (2005), 403.  doi: 10.1016/j.physa.2005.03.035.  Google Scholar

[2]

B. Barrios, E. Colorado, A. de Pablo and U. Sáchez, On some critical problems for the fractional Laplacian operator,, J. Differential Equations, 252 (2012), 6133.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

[3]

T. Bartsch, A. Pankov and Z. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549.  doi: 10.1142/S0219199701000494.  Google Scholar

[4]

T. Bartsch, Z. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations,, in: Handbook of Differential Equations-Stationary Partial Differential Equations, 2 (2005), 1.  doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[6]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Mathematics, (1996).   Google Scholar

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Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions,, Nonlinearity, 28 (2015), 2247.  doi: 10.1088/0951-7715/28/7/2247.  Google Scholar

[8]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financ. Math. Ser., (2004).   Google Scholar

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S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schröinger type problem involving the fractional laplacian,, Matematiche (Catania), 68 (2013), 201.   Google Scholar

[14]

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

[15]

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

[16]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Appl. Math., 62 (2009), 198.  doi: 10.1002/cpa.20253.  Google Scholar

[17]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer type problem set on $\mathbbR^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787.  doi: 10.1017/S0308210500013147.  Google Scholar

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N. Laskin, Fractional quantum mechanics and lévy path integrals,, Phys. Lett. A, 268 (2000), 298.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[19]

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

[20]

G. Molica Bisci, D. Repovš and R. Servadei, Nontrivial solutions for superlinear nonlocal problems,, preprint., ().   Google Scholar

[21]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

R. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Reg. Conf. Ser. in Math., (1986).  doi: 10.1090/cbms/065.  Google Scholar

[23]

P. H. Rabinowitz, On a class of nonlinear Schröinger equations,, Z. Angew. Math. Phys., 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[24]

S. Secchi, Ground state solutions for nonlinear fractional Schröinger equations in $\mathbbR^N$,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4793990.  Google Scholar

[25]

R. Servadei, A critical fractional Laplace equation in the resonant case,, Topol. Methods Nonlinear Anal., 43 (2014), 251.  doi: 10.12775/TMNA.2014.015.  Google Scholar

[26]

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

[27]

R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, Commun. Pure Appl. Anal., 12 (2013), 2445.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[28]

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

[29]

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

[30]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67.  doi: 10.1002/cpa.20153.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[32]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21.  doi: 10.1007/s00526-010-0378-3.  Google Scholar

[33]

K. M. Teng, Multiple solutions for a class of fractional Schröngdinger equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 71 (2015), 4927.  doi: 10.1016/j.nonrwa.2014.06.008.  Google Scholar

[34]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial,, Order and Chaos, (2008).   Google Scholar

[35]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics,, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273.  doi: 10.1016/S1007-5704(03)00049-2.  Google Scholar

[36]

Willem, Minimax Theorems,, Birkhäser, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[37]

W. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343.  doi: 10.1007/s002290170032.  Google Scholar

show all references

References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion,, Physica A, 356 (2005), 403.  doi: 10.1016/j.physa.2005.03.035.  Google Scholar

[2]

B. Barrios, E. Colorado, A. de Pablo and U. Sáchez, On some critical problems for the fractional Laplacian operator,, J. Differential Equations, 252 (2012), 6133.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

[3]

T. Bartsch, A. Pankov and Z. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549.  doi: 10.1142/S0219199701000494.  Google Scholar

[4]

T. Bartsch, Z. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations,, in: Handbook of Differential Equations-Stationary Partial Differential Equations, 2 (2005), 1.  doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[6]

J. Bertoin, Lévy Processes,, Cambridge Tracts in Mathematics, (1996).   Google Scholar

[7]

Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions,, Nonlinearity, 28 (2015), 2247.  doi: 10.1088/0951-7715/28/7/2247.  Google Scholar

[8]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financ. Math. Ser., (2004).   Google Scholar

[13]

S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schröinger type problem involving the fractional laplacian,, Matematiche (Catania), 68 (2013), 201.   Google Scholar

[14]

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

[15]

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

[16]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Appl. Math., 62 (2009), 198.  doi: 10.1002/cpa.20253.  Google Scholar

[17]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer type problem set on $\mathbbR^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787.  doi: 10.1017/S0308210500013147.  Google Scholar

[18]

N. Laskin, Fractional quantum mechanics and lévy path integrals,, Phys. Lett. A, 268 (2000), 298.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[19]

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

[20]

G. Molica Bisci, D. Repovš and R. Servadei, Nontrivial solutions for superlinear nonlocal problems,, preprint., ().   Google Scholar

[21]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

R. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Reg. Conf. Ser. in Math., (1986).  doi: 10.1090/cbms/065.  Google Scholar

[23]

P. H. Rabinowitz, On a class of nonlinear Schröinger equations,, Z. Angew. Math. Phys., 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[24]

S. Secchi, Ground state solutions for nonlinear fractional Schröinger equations in $\mathbbR^N$,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4793990.  Google Scholar

[25]

R. Servadei, A critical fractional Laplace equation in the resonant case,, Topol. Methods Nonlinear Anal., 43 (2014), 251.  doi: 10.12775/TMNA.2014.015.  Google Scholar

[26]

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

[27]

R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, Commun. Pure Appl. Anal., 12 (2013), 2445.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[28]

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

[29]

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

[30]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67.  doi: 10.1002/cpa.20153.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[32]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21.  doi: 10.1007/s00526-010-0378-3.  Google Scholar

[33]

K. M. Teng, Multiple solutions for a class of fractional Schröngdinger equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 71 (2015), 4927.  doi: 10.1016/j.nonrwa.2014.06.008.  Google Scholar

[34]

L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial,, Order and Chaos, (2008).   Google Scholar

[35]

H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations, Chaotic transport and complexity in classical and quantum dynamics,, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273.  doi: 10.1016/S1007-5704(03)00049-2.  Google Scholar

[36]

Willem, Minimax Theorems,, Birkhäser, (1996).  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[37]

W. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343.  doi: 10.1007/s002290170032.  Google Scholar

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