Advanced Search
Article Contents
Article Contents

Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system

Abstract Related Papers Cited by
  • In this paper, we study the following system

    $-\epsilon^2\Delta v+V(x)v+\psi(x)v=v^p, \quad x\in R^3,$

    $-\Delta\psi=\frac{1}{\epsilon}v^2,\quad \lim_{|x|\rightarrow\infty}\psi(x)=0,\quad x\in R^3,$

    where $\epsilon>0$, $p\in (3,5)$, $V$ is positive potential. We relate the number of solutions with topology of the set where $V$ attain their minimum value. By applying Ljusternik-Schnirelmann theory, we prove the multiplicity of solutions.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J10, 35B35.


    \begin{equation} \\ \end{equation}
  • [1]

    C. O. Alves and S. H. M. Soares, Existence and concentration of positive solutions for a class of gradient systems, Nonlinear Differ. Equ. Appl., 12 (2005), 437-457.doi: 10.1007/s00030-005-0021-8.


    C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method, Comm. Pure. Appl. Anal., 8 (2009), 1745-1758.doi: 10.3934/cpaa.2009.8.1745.


    C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in $R^N$, J. Differential Equations, 246 (2009), 1288-1311.doi: 10.1016/j.jde.2008.08.004.


    A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271.doi: 10.1007/s002050100152.


    A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.doi: 10.1016/j.jmaa.2008.03.057.


    A. Azzollini and A. Pomponio, A note on the ground state solutions for the nonlinear Schrödinger-Maxwell equations, Boll. Unione Mat. Ital., 9 (2009), 93-104.


    V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Cal. Var. Partial Differential Equations, 2 (1994), 29-48.doi: 10.1007/BF01234314.


    R. Benguria and H. Brezis, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.


    J. Byeon and Z. Q. Wang, Standing waves for nonlinear Schrödinger equations with singular potentials, Ann. I. H. Poincaré-AN, 26 (2009), 943-958.doi: 10.1016/j.anihpc.2008.03.009.


    S. Cingolani and M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.


    G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.


    I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.doi: 10.1080/03605309208820878.


    T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.doi: 10.1017/S030821050000353X.


    I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Advanced Nonlinear Studies, 8 (2008), 573-595.


    I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, I. Necessary conditions, Math. Models Methods Appl. Sci., 19 (2009), 707-720.doi: 10.1142/S0218202509003589.


    I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, II. Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.doi: 10.1142/S0218202509003656.


    P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. II, Ann. Inst. H. Poincare Anal. Non Linéaire, 1 (1984), 223-283.


    M. Del Pino and P. Felmer, Semiclassical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.


    D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, Journ. Func. Anal., 237 (2006), 655-674.doi: 10.1016/j.jfa.2006.04.005.


    D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.


    O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), 179-204.doi: 10.1023/B:JOSS.0000003109.97208.53.


    M. Yang, Z. Shen and Y. Ding, Multiple semiclassical solutions for the nonlinear Maxwell-Schrödinger system, Nonlinear Analysis, 71 (2009), 730-739.doi: 10.1016/j.na.2008.10.105.


    H. Yin and P. Chang, Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity, J. Differential Equations, 247 (2009), 618-647.doi: 10.1016/j.jde.2009.03.002.


    L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Analysis, 70 (2009), 2150-2164.doi: 10.1016/j.na.2008.02.116.

  • 加载中

Article Metrics

HTML views() PDF downloads(65) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint