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Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field

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  • In this paper we study the existence and orbital stability of ground states for logarithmic Schrödinger equation under a constant magnetic field. For this purpose we establish the well-posedness of the Cauchy Problem in a magnetic Sobolev space and an appropriate Orlicz space. Then we show the existence of ground state solutions via a constrained minimization on the Nehari manifold. We also show that the ground state is orbitally stable.

    Mathematics Subject Classification: Primary:35Q55, 35Q51, 35B35.


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  • [1] A. H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Diff. Eqns., 2016 (2016), 1-9. 
    [2] A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Analysis T.M.A., 155 (2017), 52-64.  doi: 10.1016/j.na.2017.01.006.
    [3] G. Arioli and A. Szulkin, A semilinear Schrödinger equations in the presence of a magnetic field, Arch. Ration. Mech. Anal. , 277-295. doi: 10.1007/s00205-003-0274-5.
    [4] I. Bialynicki-Birula and J. Mycielski, Gaussons: Solitons of the logarithmic Schrödinger equation, Physica Scripta, 20 (1978), 539-544.  doi: 10.1088/0031-8949/20/3-4/033.
    [5] Z. Binlin, M. Squassina and Z. Xia, Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscripta Math., to appear 26 pp.
    [6] P. H. BlanchardJ. Stubbe and L. Vázquez, On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri-Poncaré, Phys. Théor., 47 (1987), 309-336. 
    [7] P. Blanchard and J. Stubbe, Stability of ground states for nonlinear classical field theories vol. 347 of Lecture Notes in Physics, Springer Heidelberg, 1989, 19-35, doi: 10.1007/BFb0025759.
    [8] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.
    [9] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal., T.M.A., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.
    [10] T. Cazenave, Semilinear Schrödinger Equations Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.
    [11] T. Cazenave and M. Esteban, On the stability of stationary states for nonlinear Schrödinger equations with an external magnetic field, Mat. Apl. Comp., 7 (1988), 155-168. 
    [12] T. Cazenave and A. Haraux, Equations d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51. 
    [13] T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. 
    [14] P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation Commun. Contemp. Math. 16 (2014), 1350032, 15pp. doi: 10.1142/S0219199713500326.
    [15] P. d'Avenia and M. Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var. to appear, 22 pp.
    [16] P. d'AveniaM. Squassina and M. Zenari, Fractional logarithmic Schrödinger equations, Math. Meth. Appl. Sci., 38 (2015), 5207-5216.  doi: 10.1002/mma.3449.
    [17] M. J. Esteban and P.-L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Partial differential equations and the calculus of variations, (1989), 401-449. 
    [18] H. Hajaiej, Schrödinger systems arising in nonlinear optics and quantum mechanics, part Ⅰ, Math. Models Methods Appl, 22 (2012), 1250010.  doi: 10.1142/S0218202512500108.
    [19] A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981.
    [20] C. Ji and A. Szulkin, A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254.  doi: 10.1016/j.jmaa.2015.11.071.
    [21] S. Le Coz, Standing waves in nonlinear Schrödinger equations, In: Analytical and Numerical Aspects of Partial Differential Equations, Walter de Gruyter, Berlin, 151-192.
    [22] E. Lieb and M. Loss, Analysis vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.
    [23] H. Matsumoto and N. Ueki, Spectral analysis of Schrödinger operators with magnetic fields, J. Funct. Anal., 140 (1996), 218-225.  doi: 10.1006/jfan.1996.0106.
    [24] X. MingqiP. PucciM. Squassina and B. L. Zhang, Nonlocal Schrödinger-Kirchhoff equations with external magnetic field, Discrete Contin. Dyn. Syst. A, 37 (2017), 503-521.  doi: 10.3934/dcds.2017067.
    [25] J. G. Ribeiro, Finite time blow-up for some nonlinear Schrödinger equations with an external magnetic field, Nonlinear Analysis T.M.A., 16 (1991), 941-948.  doi: 10.1016/0362-546X(91)90098-L.
    [26] J. G. Ribeiro, Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. I.H.P. Sec. A, 4 (1991), 403-433. 
    [27] M. Squassina and A. Szulkin, Multiple solutions to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 54 (2015), 585-597.  doi: 10.1007/s00526-014-0796-8.
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