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The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field
1. | Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande – PB, 58109-970, Brazil |
2. | Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, São Carlos – SP, 13560-970, Brazil |
Nonlinear Schrödinger equations with an external magnetic field and a power nonlinearity with subcritical exponent $ p $ are considered. It is established a lower bound to the number of nontrivial solutions to these equations in terms of the topology of the domains in which the problem is given if $ p $ is suitably close to the critical exponent $ 2^* = 2N/(N-2) $, $ N \geq 3 $. To prove this lower bound, based on a proof of a result of Benci and Cerami, it is provided an abstract result that establishes Morse relations that are used to count solutions.
References:
[1] |
L. Abatangelo and S. Terracini,
Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent, J. Fixed Point Theory Appl., 10 (2011), 147-180.
doi: 10.1007/s11784-011-0053-0. |
[2] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
On the number of solutions of NLS equations with magnetics fields in expanding domains, J. Differ. Equ., 251 (2011), 2534-2548.
doi: 10.1016/j.jde.2011.03.003. |
[3] |
C. O. Alves, R. C. Nemer and S. H. M. Soares, The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger with magnetic fields, arXiv: 1408.3023. |
[4] |
C. O. Alves, R. C. Nemer and S. H. M. Soares,
Nontrivial solutions for a mixed boundary problem for Schrödinger equations with an external magnetic field, Topol. Methods Nonlinear Anal., 46 (2015), 329-362.
doi: 10.12775/TMNA.2015.050. |
[5] |
G. Arioli and A. Szulkin,
A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295.
doi: 10.1007/s00205-003-0274-5. |
[6] |
S. Barile,
A multiplicity result for singular NLS equations with magnetic potentials, Nonlinear Anal., 68 (2008), 3525-3540.
doi: 10.1016/j.na.2007.03.044. |
[7] |
V. Benci, Introduction to Morse theory: a new approach, in Topological Nonlinear Analysis, Birkhäuser Boston, (1995), 37–177.
doi: 10.1007/978-1-4612-2570-6_2. |
[8] |
V. Benci and G. Cerami,
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differ. Equ., 2 (1994), 29-48.
doi: 10.1007/BF01234314. |
[9] |
V. Benci and G. Cerami,
The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93.
doi: 10.1007/BF00375686. |
[10] |
D. Bonheure, M. Nys and J. V. Schaftingen,
Properties of ground states of nonlinear Schrödinger equations under a weak constant magnetic field, J. Math. Pures Appl., 124 (2019), 123-168.
doi: 10.1016/j.matpur.2018.05.007. |
[11] |
D. Cao and Z. Tang,
Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differ. Equ., 222 (2006), 381-424.
doi: 10.1016/j.jde.2005.06.027. |
[12] |
J. Chabrowski and A. Szulkin,
On the Schrödinger equation involving a critical Sobolev exponent and magnetic field, Topol. Methods Nonlinear Anal., 25 (2005), 3-21.
doi: 10.12775/TMNA.2005.001. |
[13] |
S. Cingolani,
On local Morse theory for p-area functionals, $p>2$, J. Fixed Point Theory Appl., 14 (2013), 355-373.
doi: 10.1007/s11784-014-0163-6. |
[14] |
S. Cingolani,
Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differ. Equ., 188 (2003), 52-79.
doi: 10.1016/S0022-0396(02)00058-X. |
[15] |
S. Cingolani and M. Clapp,
Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation, Nonlinearity, 22 (2009), 2309-2331.
doi: 10.1088/0951-7715/22/9/013. |
[16] |
S. Cingolani, L. Jeanjean and S. Secchi,
Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions, ESAIM: COCV, 15 (2009), 653-675.
doi: 10.1051/cocv:2008055. |
[17] |
S. Cingolani and S. Secchi,
Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields, J. Math. Anal. Appl., 275 (2002), 108-130.
doi: 10.1016/S0022-247X(02)00278-0. |
[18] |
S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths, J. Math. Phys., 46 (2005), 053503, 19 pp.
doi: 10.1063/1.1874333. |
[19] |
M. Clapp,
On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem, Nonlinear Anal., 42 (2000), 405-422.
doi: 10.1016/S0362-546X(98)00354-X. |
[20] |
M. J. Esteban and P. L. Lions, Partial differential equations and the calculus of variations, Birkhäuser Boston, 1989.
doi: 10.1007/978-1-4615-9828-2_18. |
[21] |
G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2$^{nd}$ edition, John Wiley & Sons, New York, 1999. |
[22] |
M. F. Furtado,
A relation between the domain topology and the number of minimal nodal solutions for a quasilinear elliptic problem, Nonlinear Anal., 62 (2005), 615-628.
doi: 10.1016/j.na.2005.03.073. |
[23] |
C. Ji and V. D. Radulescu, Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in $\mathbb{R}^2$, Manuscripta Math., (2020).
doi: 10.1007/s00229-020-01195-1. |
[24] |
K. Kurata,
Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal., 41 (2000), 763-778.
doi: 10.1016/S0362-546X(98)00308-3. |
[25] |
G. Li, S. Peng and C. Wang,
Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields, J. Differ. Equ., 251 (2011), 3500-3521.
doi: 10.1016/j.jde.2011.08.038. |
[26] |
S. Liang and J. Zhang,
Solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity, Proc. Edinb. Math. Soc., 54 (2011), 131-147.
doi: 10.1017/S0013091509000492. |
[27] |
M. Squassina,
Soliton dynamics for the nonlinear Schrödinger equation with magnetic field, Manuscripta Math., 130 (2009), 461-494.
doi: 10.1007/s00229-009-0307-y. |
[28] |
Z. Tang,
Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency, J. Differ. Equ., 245 (2008), 2723-2748.
doi: 10.1016/j.jde.2008.07.035. |
[29] |
Z. Tang,
On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl., 54 (2007), 627-637.
doi: 10.1016/j.camwa.2006.12.031. |
[30] |
Z. Tang,
Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields, Z. Angew. Math. Phys., 59 (2008), 810-833.
doi: 10.1007/s00033-007-7032-8. |
[31] |
M. Willem, Minimax Theorems, Birkhäuser Boston, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
L. Abatangelo and S. Terracini,
Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent, J. Fixed Point Theory Appl., 10 (2011), 147-180.
doi: 10.1007/s11784-011-0053-0. |
[2] |
C. O. Alves, G. M. Figueiredo and M. F. Furtado,
On the number of solutions of NLS equations with magnetics fields in expanding domains, J. Differ. Equ., 251 (2011), 2534-2548.
doi: 10.1016/j.jde.2011.03.003. |
[3] |
C. O. Alves, R. C. Nemer and S. H. M. Soares, The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger with magnetic fields, arXiv: 1408.3023. |
[4] |
C. O. Alves, R. C. Nemer and S. H. M. Soares,
Nontrivial solutions for a mixed boundary problem for Schrödinger equations with an external magnetic field, Topol. Methods Nonlinear Anal., 46 (2015), 329-362.
doi: 10.12775/TMNA.2015.050. |
[5] |
G. Arioli and A. Szulkin,
A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295.
doi: 10.1007/s00205-003-0274-5. |
[6] |
S. Barile,
A multiplicity result for singular NLS equations with magnetic potentials, Nonlinear Anal., 68 (2008), 3525-3540.
doi: 10.1016/j.na.2007.03.044. |
[7] |
V. Benci, Introduction to Morse theory: a new approach, in Topological Nonlinear Analysis, Birkhäuser Boston, (1995), 37–177.
doi: 10.1007/978-1-4612-2570-6_2. |
[8] |
V. Benci and G. Cerami,
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differ. Equ., 2 (1994), 29-48.
doi: 10.1007/BF01234314. |
[9] |
V. Benci and G. Cerami,
The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93.
doi: 10.1007/BF00375686. |
[10] |
D. Bonheure, M. Nys and J. V. Schaftingen,
Properties of ground states of nonlinear Schrödinger equations under a weak constant magnetic field, J. Math. Pures Appl., 124 (2019), 123-168.
doi: 10.1016/j.matpur.2018.05.007. |
[11] |
D. Cao and Z. Tang,
Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differ. Equ., 222 (2006), 381-424.
doi: 10.1016/j.jde.2005.06.027. |
[12] |
J. Chabrowski and A. Szulkin,
On the Schrödinger equation involving a critical Sobolev exponent and magnetic field, Topol. Methods Nonlinear Anal., 25 (2005), 3-21.
doi: 10.12775/TMNA.2005.001. |
[13] |
S. Cingolani,
On local Morse theory for p-area functionals, $p>2$, J. Fixed Point Theory Appl., 14 (2013), 355-373.
doi: 10.1007/s11784-014-0163-6. |
[14] |
S. Cingolani,
Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differ. Equ., 188 (2003), 52-79.
doi: 10.1016/S0022-0396(02)00058-X. |
[15] |
S. Cingolani and M. Clapp,
Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation, Nonlinearity, 22 (2009), 2309-2331.
doi: 10.1088/0951-7715/22/9/013. |
[16] |
S. Cingolani, L. Jeanjean and S. Secchi,
Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions, ESAIM: COCV, 15 (2009), 653-675.
doi: 10.1051/cocv:2008055. |
[17] |
S. Cingolani and S. Secchi,
Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields, J. Math. Anal. Appl., 275 (2002), 108-130.
doi: 10.1016/S0022-247X(02)00278-0. |
[18] |
S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths, J. Math. Phys., 46 (2005), 053503, 19 pp.
doi: 10.1063/1.1874333. |
[19] |
M. Clapp,
On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem, Nonlinear Anal., 42 (2000), 405-422.
doi: 10.1016/S0362-546X(98)00354-X. |
[20] |
M. J. Esteban and P. L. Lions, Partial differential equations and the calculus of variations, Birkhäuser Boston, 1989.
doi: 10.1007/978-1-4615-9828-2_18. |
[21] |
G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2$^{nd}$ edition, John Wiley & Sons, New York, 1999. |
[22] |
M. F. Furtado,
A relation between the domain topology and the number of minimal nodal solutions for a quasilinear elliptic problem, Nonlinear Anal., 62 (2005), 615-628.
doi: 10.1016/j.na.2005.03.073. |
[23] |
C. Ji and V. D. Radulescu, Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in $\mathbb{R}^2$, Manuscripta Math., (2020).
doi: 10.1007/s00229-020-01195-1. |
[24] |
K. Kurata,
Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal., 41 (2000), 763-778.
doi: 10.1016/S0362-546X(98)00308-3. |
[25] |
G. Li, S. Peng and C. Wang,
Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields, J. Differ. Equ., 251 (2011), 3500-3521.
doi: 10.1016/j.jde.2011.08.038. |
[26] |
S. Liang and J. Zhang,
Solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity, Proc. Edinb. Math. Soc., 54 (2011), 131-147.
doi: 10.1017/S0013091509000492. |
[27] |
M. Squassina,
Soliton dynamics for the nonlinear Schrödinger equation with magnetic field, Manuscripta Math., 130 (2009), 461-494.
doi: 10.1007/s00229-009-0307-y. |
[28] |
Z. Tang,
Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency, J. Differ. Equ., 245 (2008), 2723-2748.
doi: 10.1016/j.jde.2008.07.035. |
[29] |
Z. Tang,
On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl., 54 (2007), 627-637.
doi: 10.1016/j.camwa.2006.12.031. |
[30] |
Z. Tang,
Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields, Z. Angew. Math. Phys., 59 (2008), 810-833.
doi: 10.1007/s00033-007-7032-8. |
[31] |
M. Willem, Minimax Theorems, Birkhäuser Boston, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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