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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[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.

[23]

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

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[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.

[23]

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

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

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