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January  2018, 17(1): 163-175. doi: 10.3934/cpaa.2018010

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

Alex H. Ardila

Instituto Nacional de Matemática Pura e Aplicada -IMPA Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil

Received  February 2017 Revised  June 2017 Published  September 2017

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.

Keywords: Logarithmic Schrödinger equation, magnetic field, orbital stability.
Mathematics Subject Classification: Primary:35Q55, 35Q51, 35B35.
Citation: Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010
References:
[1]

A. H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Diff. Eqns., 2016 (2016), 1-9.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[6]

P. H. Blanchard, J. Stubbe and L. Vázquez, On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri-Poncaré, Phys. Théor., 47 (1987), 309-336.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

T. Cazenave and A. Haraux, Equations d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.   Google Scholar

[13]

T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.   Google Scholar

[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.  Google Scholar

[15]

P. d'Avenia and M. Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var. to appear, 22 pp. Google Scholar

[16]

P. d'Avenia, M. Squassina and M. Zenari, Fractional logarithmic Schrödinger equations, Math. Meth. Appl. Sci., 38 (2015), 5207-5216.  doi: 10.1002/mma.3449.  Google Scholar

[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, Ⅰ (1989), 401-449.   Google Scholar

[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.  Google Scholar

[19]

A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

X. Mingqi, P. Pucci, M. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Diff. Eqns., 2016 (2016), 1-9.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[6]

P. H. Blanchard, J. Stubbe and L. Vázquez, On the stability of solitary waves for classical scalar fields, Ann. Inst. Henri-Poncaré, Phys. Théor., 47 (1987), 309-336.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

T. Cazenave and A. Haraux, Equations d'évolution avec non-linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21-51.   Google Scholar

[13]

T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.   Google Scholar

[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.  Google Scholar

[15]

P. d'Avenia and M. Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var. to appear, 22 pp. Google Scholar

[16]

P. d'Avenia, M. Squassina and M. Zenari, Fractional logarithmic Schrödinger equations, Math. Meth. Appl. Sci., 38 (2015), 5207-5216.  doi: 10.1002/mma.3449.  Google Scholar

[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, Ⅰ (1989), 401-449.   Google Scholar

[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.  Google Scholar

[19]

A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

X. Mingqi, P. Pucci, M. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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