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Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities
Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field
Instituto Nacional de Matemática Pura e Aplicada -IMPA Estrada Dona Castorina 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil |
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.
References:
| [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. 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.
|
| [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'Avenia, M. 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, Ⅰ (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. 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. |
| [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. |
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.
|
| [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. 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.
|
| [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'Avenia, M. 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, Ⅰ (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. 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. |
| [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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