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
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show all references

References:
[1]

Physica A, 356 (2005), 403-407. doi: 10.1016/j.physa.2005.03.035.  Google Scholar

[2]

J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.  Google Scholar

[3]

Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.  Google Scholar

[4]

in: Handbook of Differential Equations-Stationary Partial Differential Equations, 2 (2005), 1-55. doi: 10.1016/S1874-5733(05)80009-9.  Google Scholar

[5]

Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.  Google Scholar

[6]

Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.  Google Scholar

[7]

Nonlinearity, 28 (2015), 2247-2264. doi: 10.1088/0951-7715/28/7/2247.  Google Scholar

[8]

Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[9]

Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

[10]

Commun. Pure Appl. Anal., 13 (2014), 2359-2376. doi: 10.3934/cpaa.2014.13.2359.  Google Scholar

[11]

J. Math. Phys., 53 (2012), 043507, 7pp. doi: 10.1063/1.3701574.  Google Scholar

[12]

Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC press, Boca Raton, FL, 2004.  Google Scholar

[13]

Matematiche (Catania), 68 (2013), 201-216.  Google Scholar

[14]

Comm. Math. Phys., 329 (2014), 383-404. doi: 10.1007/s00220-014-1919-y.  Google Scholar

[15]

Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746.  Google Scholar

[16]

Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.  Google Scholar

[17]

Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar

[18]

Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[19]

Phys. Rev. E, 66 (2002), 056108, 7pp. 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]

Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[22]

CBMS Reg. Conf. Ser. in Math., vol 65, Amer. math. Soc., Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[23]

Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[24]

J. Math. Phys., 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990.  Google Scholar

[25]

Topol. Methods Nonlinear Anal., 43 (2014), 251-267. doi: 10.12775/TMNA.2014.015.  Google Scholar

[26]

Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  Google Scholar

[27]

Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[28]

Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[29]

Nonlinearity, 27 (2014), 187-207. doi: 10.1088/0951-7715/27/2/187.  Google Scholar

[30]

Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.  Google Scholar

[31]

Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar

[32]

Calc. Var. Partial Differential Equations, 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.  Google Scholar

[33]

Nonlinear Anal. Real World Appl., 71 (2015), 4927-4934. doi: 10.1016/j.nonrwa.2014.06.008.  Google Scholar

[34]

Order and Chaos, Patras University Press, 2008. Google Scholar

[35]

Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273-281. doi: 10.1016/S1007-5704(03)00049-2.  Google Scholar

[36]

Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[37]

Manuscripta Math., 104 (2001), 343-358. doi: 10.1007/s002290170032.  Google Scholar

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