Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MULTIPLICITY RESULTS FOR NONLINEAR SCHRÖDINGER-POISSON SYSTEMS WITH SUBCRITICAL OR CRITICAL GROWTH

  • Guo, Shangjiang (College of Mathematics and Econometrics Hunan University) ;
  • Liu, Zhisu (School of Mathematics and Physics University of South China)
  • Received : 2014.09.27
  • Published : 2016.03.01
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Abstract

In this paper, we consider the following $Schr{\ddot{o}}dinger$-Poisson system: $$\{\begin{array}{lll}-{\Delta}u+u+{\lambda}{\phi}u={\mu}f(u)+{\mid}u{\mid}^{p-2}u,\;\text{ in }{\Omega},\\-{\Delta}{\phi}=u^2,\;\text{ in }{\Omega},\\{\phi}=u=0,\;\text{ on }{\partial}{\Omega},\end{array}$$ where ${\Omega}$ is a smooth and bounded domain in $\mathbb{R}^3$, $p{\in}(1,6]$, ${\lambda}$, ${\mu}$ are two parameters and $f:\mathbb{R}{\rightarrow}\mathbb{R}$ is a continuous function. Using some critical point theorems and truncation technique, we obtain three multiplicity results for such a problem with subcritical or critical growth.

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References

  1. A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrodinger-Poisson problem, Commun. Contemp. Math. 10 (2008), no. 3, 391-404. https://doi.org/10.1142/S021919970800282X
  2. G. Anello, A multiplicity theorem for critical points of functionals on reflexive Banach spaces, Arch. Math. 82 (2004), 172-179. https://doi.org/10.1007/s00013-003-0584-8
  3. J. Azorero and I. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), no. 2, 877-895. https://doi.org/10.1090/S0002-9947-1991-1083144-2
  4. A. Azzollini, P. d'Avenia, and V. Luisi, Generalized Schrodinger-Poisson type systems, Commun. Pure Appl. Anal. 12 (2013), no. 2, 867-879. https://doi.org/10.3934/cpaa.2013.12.867
  5. A. Azzollini and A. Pomponio, Ground state solutions for nonlinear Schrodinger-Maxwell equations, J. Math. Anal. Appl. 345 (2008), no. 1, 90-108. https://doi.org/10.1016/j.jmaa.2008.03.057
  6. T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schroodinger equation, SIAM J. Math. Anal. 37 (2005), 321-342. https://doi.org/10.1137/S0036141004442793
  7. P. D'Avenia, Non-radially symmetric solutions of nonlinear Schrodinger equation coupled with Maxwell equations, Adv. Nonlinear Stud. 2 (2002), no. 2, 177-192. https://doi.org/10.1515/ans-2002-0205
  8. P. D'Avenia, A. Pomponioa, and G. Vaira, Infinitely many positive solutions for a Schrodinger-Poisson system, Nonlinear Anal. 74 (2011), no. 16, 5705-5721. https://doi.org/10.1016/j.na.2011.05.057
  9. 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
  10. D. Bleecker, Gauge Theory and Variational Principles, Dover Publications, 2005.
  11. H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), no. 1, 55-64. https://doi.org/10.1016/0362-546X(86)90011-8
  12. G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrodinger-Poisson systems, J. Differential Equations 248 (2010), no. 3, 521-543. https://doi.org/10.1016/j.jde.2009.06.017
  13. G. Coclite, A multiplicity result for the nonlinear Schrodinger-Maxwell equations, Commun. Appl. Anal. 7 (2003), no. 2-3, 417-423.
  14. L. Evans, Partial Differential Equations, AMS, Providence, RI, 1998.
  15. I. Ianni and G. Vaira, On concentration of positive bound states for the Schrodinger-Poisson problem with potentials, Adv. Nonlinear Stud. 8 (2008), no. 3, 573-595. https://doi.org/10.1515/ans-2008-0305
  16. M. Kranolseskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964.
  17. P. Lions, The concentration compactness principle in the calculus of variations: The locally compact case. Parts 1, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), no. 2, 109-145 https://doi.org/10.1016/S0294-1449(16)30428-0
  18. P. Lions, The concentration compactness principle in the calculus of variations: Thelocally compact case. Parts 2, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), no. 4, 223-283. https://doi.org/10.1016/S0294-1449(16)30422-X
  19. Z. Liu and S. Guo, On ground state solutions for the Schrodinger-Poisson equations with critical growth, J. Math. Anal. Appl. 412 (2014), no. 1, 435-448. https://doi.org/10.1016/j.jmaa.2013.10.066
  20. Z. Liu, S. Guo, and Y. Fang, Multiple semiclassical states for coupled Schroinger-Poisson equations with critical exponential growth, J. Math. Phys. 56 (2015), no. 4, 041505, 22 pp.
  21. Z. Liu, S. Guo, and Z. Zhang, Existence and multiplicity of solutions for a class of sublinear Schrodinger-Maxwell equations, Taiwanese J. Math. 17 (2013), no. 3, 857-872. https://doi.org/10.11650/tjm.17.2013.2202
  22. L. Pisani and G. Siciliano, Note on a Schrodinger-Poisson system in a bounded domain, Appl. Math. Lett. 21 (2008), no. 5, 521-528. https://doi.org/10.1016/j.aml.2007.06.005
  23. P. Pucci and J. Serrin, A Mountain Pass theorem, J. Differential Equations 60 (1985), no. 1, 142-149. https://doi.org/10.1016/0022-0396(85)90125-1
  24. P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Reg. Conf. Ser. in Math. vol. 65, Amer. Math. Soc. Providence, RI, 1986.
  25. D. Ruiz, The Schrodinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), no. 2, 655-674. https://doi.org/10.1016/j.jfa.2006.04.005
  26. D. Ruiz and G. Siciliano, A note on the Schrodinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud. 8 (2008), no. 1, 179-190. https://doi.org/10.1515/ans-2008-0106
  27. A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems, in: Proceedings of the Second World Congress of Nonlinear Analysis, Part 8 (Athens, 1996), Nonlinear Anal. 30 (1997), no. 8, 4849-4857.
  28. M. Schechter and K. Tintarev, Spherical maxima in Hilbert space and semilinear elliptic eigenvalue problems, Differential Integral Equations 3 (1990), no. 5, 889-899.
  29. G. Siciliano, Multiple positive solutions for a Schrodinger-Poisson-Slater system, J. Math. Anal. Appl. 365 (2010), no. 1, 288-299. https://doi.org/10.1016/j.jmaa.2009.10.061
  30. J. Slater, A simplification of the Hartree-Fock method, Phys. Rev. 81 (1951), 385-390. https://doi.org/10.1103/PhysRev.81.385
  31. M. Struwe, Variational Methods, Application to Nonlinear Partial Differential Equation and Hamiltonian Syatem, Springer-Verlag, 2007.
  32. J. Sun, Infinitely many solutions for a class of sublinear Schrodinger-Maxwell equations, J. Math. Anal. Appl. 390 (2012), no. 2, 514-522. https://doi.org/10.1016/j.jmaa.2012.01.057
  33. J. Sun, H. Chen, and J. Nieto, On ground state solutions for some non-autonomous Schrodinger-Poisson systems, J. Differential Equations 252 (2012), no. 5, 3365-3380. https://doi.org/10.1016/j.jde.2011.12.007
  34. Z. Wang and H. Zhou, Positive solution for a nonlinear stationary Schrodinger-Poisson system in ${\mathbb{R}}^3$, Discrete Contin. Dyn. Syst. 18 (2007), no. 4, 809-816. https://doi.org/10.3934/dcds.2007.18.809
  35. L. Zhao and F. Zhao, On the existence of solutions for the Schroinger-Poisson equations, J. Math. Anal. Appl. 346 (2008), no. 1, 155-169. https://doi.org/10.1016/j.jmaa.2008.04.053

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