Infinitely many solutions of the nonlinear fractional Schrödinger equations

Miao  Du; Lixin  Tian

doi:10.3934/dcdsb.2016104

Article Contents

2016, Volume 21, Issue 10: 3407-3428. Doi: 10.3934/dcdsb.2016104

This issue Previous Article Computational methods for asynchronous basins Next Article Long-time behavior of an SIR model with perturbed disease transmission coefficient

Infinitely many solutions of the nonlinear fractional Schrödinger equations

Miao Du^1, and
Lixin Tian^1,

1.
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023

Received: January 2015

Revised: March 2016

Published: December 2016

Abstract / Introduction Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35R11, 35A15; Secondary: 35S05.

Citation:

Related Papers

Cited by

References

[1]	S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407.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), 6133-6162.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), 549-569.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), 1-55.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), 313-345.doi: 10.1007/BF00250555.
[6]	J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.
[7]	Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.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), 2052-2093.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), 1245-1260.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), 2359-2376.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), 043507, 7pp.doi: 10.1063/1.3701574.
[12]	R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC press, Boca Raton, FL, 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), 201-216.
[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), 383-404.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), 1237-1262.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), 198-214.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), 787-809.doi: 10.1017/S0308210500013147.
[18]	N. Laskin, Fractional quantum mechanics and lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.doi: 10.1016/S0375-9601(00)00201-2.
[19]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp.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), 521-573.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., vol 65, Amer. math. Soc., Providence, RI, 1986.doi: 10.1090/cbms/065.
[23]	P. H. Rabinowitz, On a class of nonlinear Schröinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.doi: 10.1007/BF00946631.
[24]	S. Secchi, Ground state solutions for nonlinear fractional Schröinger equations in $\mathbbR^N$, J. Math. Phys., 54 (2013), 031501, 17pp.doi: 10.1063/1.4793990.
[25]	R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., 43 (2014), 251-267.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), 2105-2137.
[27]	R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.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), 67-102.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), 187-207.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), 67-112.doi: 10.1002/cpa.20153.
[31]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.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), 21-41.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), 4927-4934.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, Patras University Press, 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), 273-281.doi: 10.1016/S1007-5704(03)00049-2.
[36]	Willem, Minimax Theorems, Birkhäser, Boston, 1996.doi: 10.1007/978-1-4612-4146-1.
[37]	W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.doi: 10.1007/s002290170032.