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http://dx.doi.org/10.4134/JKMS.j190156

THE FRACTIONAL SCHRÖDINGER-POISSON SYSTEMS WITH INFINITELY MANY SOLUTIONS  

Jin, Tiankun (College of Teacher Education Daqing Normal University)
Yang, Zhipeng (Department of Mathematics Yunnan Normal University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.2, 2020 , pp. 489-506 More about this Journal
Abstract
In this paper, we study the existence of infinitely many large energy solutions for the supercubic fractional Schrödinger-Poisson systems. We consider different superlinear growth assumptions on the non-linearity, starting from the well-know Ambrosetti-Rabinowitz type condition. We obtain three different existence results in this setting by using the Fountain Theorem, all these results extend some results for semelinear Schrödinger-Poisson systems to the nonlocal fractional setting.
Keywords
Fractional $Schr{\ddot{o}}dinger$-Poisson system; variational method; fountain theorem;
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1 A. Azzollini, Concentration and compactness in nonlinear Schrodinger-Poisson system with a general nonlinearity, J. Differential Equations 249 (2010), no. 7, 1746-1763. https://doi.org/10.1016/j.jde.2010.07.007   DOI
2 P. Bartolo, V. Benci, and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity, Nonlinear Anal. 7 (1983), no. 9, 981-1012. https://doi.org/10.1016/0362-546X(83)90115-3   DOI
3 T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), no. 10, 1205-1216. https://doi.org/10.1016/0362-546X(93)90151-H   DOI
4 V. Benci and D. Fortunato, An eigenvalue problem for the Schrodinger-Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 283-293. https://doi.org/10.12775/TMNA.1998.019   DOI
5 Z. Binlin, G. Molica Bisci, and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity 28 (2015), no. 7, 2247-2264. https://doi.org/10.1088/0951-7715/28/7/2247   DOI
6 S.-J. Chen and C.-L. Tang, High energy solutions for the superlinear Schrodinger-Maxwell equations, Nonlinear Anal. 71 (2009), no. 10, 4927-4934. https://doi.org/10.1016/j.na.2009.03.050   DOI
7 E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521-573. https://doi.org/10.1016/j.bulsci.2011.12.004   DOI
8 L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbf{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 4, 787-809. https://doi.org/10.1017/S0308210500013147   DOI
9 N. S. Landkof, Foundations of Modern Potential Theory, translated from the Russian by A. P. Doohovskoy, Springer-Verlag, New York, 1972.
10 N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298-305. https://doi.org/10.1016/S0375-9601(00)00201-2   DOI
11 N. Laskin, Fractional Schrodinger equation, Phys. Rev. E (3) 66 (2002), no. 5, 056108, 7 pp. https://doi.org/10.1103/PhysRevE.66.056108   DOI
12 L. Li and S.-J. Chen, Infinitely many large energy solutions of superlinear Schrodinger-Maxwell equations, Electron. J. Differential Equations 2012 (2012), no. 224, 9 pp.
13 E. G. Murcia and G. Siciliano, Positive semiclassical states for a fractional Schrodinger-Poisson system, Differential Integral Equations 30 (2017), no. 3-4, 231-258. http://projecteuclid.org/euclid.die/1487386824
14 S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal. 73 (2010), no. 3, 788-795. https://doi.org/10.1016/j.na.2010.04.016   DOI
15 W. Liu, Existence of multi-bump solutions for the fractional Schrodinger-Poisson system, J. Math. Phys. 57 (2016), no. 9, 091502, 17 pp. https://doi.org/10.1063/1.4963172
16 G. Molica Bisci, D. Repovs, and R. Servadei, Nontrivial solutions of superlinear nonlocal problems, Forum Math. 28 (2016), no. 6, 1095-1110. https://doi.org/10.1515/forum-2015-0204   DOI
17 S. Secchi, Ground state solutions for nonlinear fractional Schrodinger equations in $\mathbb{R}^N$, J. Math. Phys. 54 (2013), no. 3, 031501, 17 pp. https://doi.org/10.1063/1.4793990   DOI
18 M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990. https://doi.org/10.1007/978-3-662-02624-3
19 K. Teng, Multiple solutions for a class of fractional Schrodinger equations in $\mathbb{R}^N$, Nonlinear Anal. Real World Appl. 21 (2015), 76-86. https://doi.org/10.1016/j.nonrwa.2014.06.008   DOI
20 K. Teng, Existence of ground state solutions for the nonlinear fractional Schrodinger-Poisson system with critical Sobolev exponent, J. Differential Equations 261 (2016), no. 6, 3061-3106. https://doi.org/10.1016/j.jde.2016.05.022   DOI
21 Y. Yu, F. Zhao and L. Zhao, The concentration behavior of ground state solutions for a fractional Schrodinger-Poisson system, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Art. 116, 25 pp. https://doi.org/10.1007/s00526-017-1199-4   DOI
22 M.-H. Yang and Z.-Q. Han, Existence and multiplicity results for the nonlinear Schrodinger-Poisson systems, Nonlinear Anal. Real World Appl. 13 (2012), no. 3, 1093-1101. https://doi.org/10.1016/j.nonrwa.2011.07.008   DOI
23 Z. Yang, Y. Yu, and F. Zhao, The concentration behavior of ground state solutions for a critical fractional Schrodinger-Poisson system, Math. Nachr. 292 (2019), 1837-1868. https://doi.org/10.1002/mana.201700398   DOI
24 Z. Yang, Y. Yu, and F. Zhao, Concentration behavior of ground state solutions for a fractional Schrodinger-Poisson system involving critical exponent, Commun. Contemp. Math. (2019), 1850027. https://doi.org/10.1142/S021919971850027X
25 J. Zhang, J. M. do O, and M. Squassina, Fractional Schrodinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud. 16 (2016), no. 1, 15-30. https://doi.org/10.1515/ans-2015-5024   DOI
26 L. Zhao, H. Liu, and F. Zhao, Existence and concentration of solutions for the Schrodinger-Poisson equations with steep well potential, J. Differential Equations 255 (2013), no. 1, 1-23. https://doi.org/10.1016/j.jde.2013.03.005   DOI
27 L. Zhao and F. Zhao, On the existence of solutions for the Schrodinger-Poisson equations, J. Math. Anal. Appl. 346 (2008), no. 1, 155-169. https://doi.org/10.1016/j.jmaa.2008.04.053   DOI
28 G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), no. 2, 332-336 (1979).
29 A. Ambrosetti, On Schrodinger-Poisson systems, Milan J. Math. 76 (2008), 257-274. https://doi.org/10.1007/s00032-008-0094-z   DOI
30 A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7   DOI
31 G. Cerami, On the existence of eigenvalues for a nonlinear boundary value problem, Ann. Mat. Pura Appl. (4) 124 (1980), 161-179. https://doi.org/10.1007/BF01795391   DOI
32 X. Chang and Z.-Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations 256 (2014), no. 8, 2965-2992. https://doi.org/10.1016/j.jde.2014.01.027   DOI
33 M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhauser Boston, Inc., Boston, MA, 1996. https://doi.org/10.1007/978-1-4612-4146-1
34 P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33-97. http://projecteuclid.org/euclid.cmp/1104116712   DOI
35 S. Liu, On ground states of superlinear p-Laplacian equations in $\mathbf{R}^N$, J. Math. Anal. Appl. 361 (2010), no. 1, 48-58. https://doi.org/10.1016/j.jmaa.2009.09.016   DOI