January  2021, 20(1): 449-465. doi: 10.3934/cpaa.2020276

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

* Corresponding author

Received  June 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: The first author is supported by CNPq/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.

Citation: Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
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.  Google Scholar

[2]

C. O. AlvesG. 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.  Google Scholar

[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. Google Scholar

[4]

C. O. AlvesR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. BonheureM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. CingolaniL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2$^{nd}$ edition, John Wiley & Sons, New York, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

G. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

M. Willem, Minimax Theorems, Birkhäuser Boston, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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.  Google Scholar

[2]

C. O. AlvesG. 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.  Google Scholar

[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. Google Scholar

[4]

C. O. AlvesR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

D. BonheureM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. CingolaniL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2$^{nd}$ edition, John Wiley & Sons, New York, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

G. LiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

M. Willem, Minimax Theorems, Birkhäuser Boston, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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