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November  2017, 16(6): 1957-1975. doi: 10.3934/cpaa.2017096

Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials

Gan Lu and  Weiming Liu ,

School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China

* Corresponding author

Received  September 2016 Revised  April 2017 Published  July 2017

Fund Project: The authors would like to thank the referees for their suggestions of this work. The paper is supported by the fund from NSFC (No.11601139)

This paper is concerned with the following nonlinear Schrödinger equations with magnetic potentials

$\begin{equation}\label{0} \Bigl(\frac{\nabla}{i}-α A(|x|)\Bigl)^{2}u+(1+α V(|x|))u=|u|^{p-2}u,\,\,u∈ H^{1}(\mathbb{R}^{N},\mathbb{C}),\ \ \ \ \ \ \ \ \ \ \left( 0.1 \right) \end{equation}$

where $2<p<\frac{2N}{N-2}$ if $N≥q 3$ and $2<p<+∞$ if $N=2$. $α$ can be regarded as a parameter. $A(|x|)=(A_{1}(|x|),A_{2}(|x|),···,A_{N}(|x|))$ is a magnetic field satisfying that $A_{j}(|x|)>0(j=1,...,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V(|x|)>0$ is a real continuous electric potential. Under some decaying conditions of both electric and magnetic potentials which are given in section 1, we prove that the equation has multiple complex-valued solutions by applying the finite reduction method.

Keywords: Lyapunov-Schmidt reduction, magnetic potentials, nonlinear Schrödinger equations, multiple complex-valued solutions.
Mathematics Subject Classification: Primary: 35J10, 35B99; Secondary: 35J60.
Citation: Gan Lu, Weiming Liu. Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1957-1975. doi: 10.3934/cpaa.2017096
References:
[1]

A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112. doi: 10.1007/s00526-006-0079-0. Google Scholar

[2]

T. Bartsch, E. N. Dancer and S. Peng, On multi-bump semiclassical bound states of nonlinear Schrödinger euqations with electromagnetic fields, Adv. Differential Equations, 7 (2006), 781-812. Google Scholar

[3]

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

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S. Cingolani, L. Jeanjean and S. Secchi, Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions, ESAIM Control Optim. Calc. Var., 15 (2009), 653-675. doi: 10.1051/cocv:2008055. Google Scholar

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D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^{N}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 567-588. doi: 10.1016/S0294-1449(16)30115-9. Google Scholar

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D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 73-111. doi: 10.1016/S0294-1449(99)80021-3. Google Scholar

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D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh. Sect. A, (1999), 235-264. doi: 10.1017/S030821050002134X. Google Scholar

[8]

J. Cosmo and J. Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field, J. Differential Equations, 259 (2015), 596-627. doi: 10.1016/j.jde.2015.02.016. Google Scholar

[9]

S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equations with electromagenetic fields, J. Math. Anal. Appl., 275 (2002), 108-130. doi: 10.1016/S0022-247X(02)00278-0. Google Scholar

[10]

S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magenetic potentials having polynomial growths J. Math. Phys. , 46 (2005), 053503, 19 pp. doi: 10.1063/1.1874333. Google Scholar

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D. Cao and Z. Tang, Existence and Uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 222 (2006), 381-424. doi: 10.1016/j.jde.2005.06.027. Google Scholar

[12]

Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82. doi: 10.1007/s00229-011-0530-1. Google Scholar

[13]

Y. Ding and Z. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Annali di Matematica, 190 (2011), 427-451. doi: 10.1007/s10231-010-0157-y. Google Scholar

[14]

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. doi: 10.1080/03605309608821208. Google Scholar

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K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagenetic fields, Nonlinear Anal., 41 (2000), 763-778. doi: 10.1016/S0362-546X(98)00308-3. Google Scholar

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M. K. Kwong, Uniqueness of the positive solution of $Δ u-u+u^{p}=0$ in $\mathbb{R}^{n}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar

[17]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928. Google Scholar

[18]

W. Long and S. Peng, Multiple positive solutions for a type of nonlinear Schrödinger equations, Annali della Scuola Normale Superiore di Pisa Classe di Scienze, 16 (2016), 603-623. Google Scholar

[19]

G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 251 (2011), 3500-3521. doi: 10.1016/j.jde.2011.08.038. Google Scholar

[20]

W. Liu, Infinitely many solutions for nonlinear Schrödinger systems with magnetic potentials in $\mathbb{R}^{3}$, Math. Meth. Appl. Sci., 39 (2016), 1452-1479. doi: 10.1002/mma.3581. Google Scholar

[21]

W. Liu and C. Wang, Infinitely many solutions for the nonlinear Schrödinger equations with magnetic potentials in $\mathbb{R}^{N}$ J. Math. Phys. , 54 (2013), 121508, 23pp. doi: 10.1063/1.4851756. Google Scholar

[22]

W. Liu and C. Wang, Multi-peak solutions of a nonlinear Schrödinger equation with magnetic fields, Adv. Nonlinear Stud., 14 (2014), 951-975. doi: 10.1515/ans-2014-0408. Google Scholar

[23]

W. Liu and C. Wang, Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields, Discrete Contin. Dyn. Syst. A, 36 (2016), 7081-7115. doi: 10.3934/dcds.2016109. Google Scholar

[24]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227. doi: 10.1112/S002461070000898X. Google Scholar

[25]

E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc., 76 (1998), 427-452. doi: 10.1112/S0024611598000148. Google Scholar

[26]

M. Del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar

[27]

S. Shirai, Multi-bump solutions to a nonlinear Schrödinger equation with steep magnetic wells J. Math. Phys. , 56(2015), 091510, 19pp. doi: 10.1063/1.4930247. Google Scholar

[28]

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

[29]

D. Salazar, Vortex-type solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 66 (2015), 663-675. doi: 10.1007/s00033-014-0412-y. Google Scholar

[30]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation Self-Focusing and Wave Collapse, Applied Mathematical Sciences 139. Springer-Verlag, New York, Berlin, Heidelberg, 1999. Google Scholar

[31]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1. Google Scholar

[32]

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb{S}^N$, J. Funct. Anal., 258 (2010), 3048-3081. doi: 10.1016/j.jfa.2009.12.008. Google Scholar

[33]

M. Yang and Y. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities, J. Math. Anal. Appl., 403 (2013), 680-694. doi: 10.1016/j.jmaa.2013.02.062. Google Scholar

show all references

References:
[1]

A. Ambrosetti, E. Colorado and D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 30 (2007), 85-112. doi: 10.1007/s00526-006-0079-0. Google Scholar

[2]

T. Bartsch, E. N. Dancer and S. Peng, On multi-bump semiclassical bound states of nonlinear Schrödinger euqations with electromagnetic fields, Adv. Differential Equations, 7 (2006), 781-812. Google Scholar

[3]

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

[4]

S. Cingolani, L. Jeanjean and S. Secchi, Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions, ESAIM Control Optim. Calc. Var., 15 (2009), 653-675. doi: 10.1051/cocv:2008055. Google Scholar

[5]

D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^{N}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 567-588. doi: 10.1016/S0294-1449(16)30115-9. Google Scholar

[6]

D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 73-111. doi: 10.1016/S0294-1449(99)80021-3. Google Scholar

[7]

D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh. Sect. A, (1999), 235-264. doi: 10.1017/S030821050002134X. Google Scholar

[8]

J. Cosmo and J. Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field, J. Differential Equations, 259 (2015), 596-627. doi: 10.1016/j.jde.2015.02.016. Google Scholar

[9]

S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equations with electromagenetic fields, J. Math. Anal. Appl., 275 (2002), 108-130. doi: 10.1016/S0022-247X(02)00278-0. Google Scholar

[10]

S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magenetic potentials having polynomial growths J. Math. Phys. , 46 (2005), 053503, 19 pp. doi: 10.1063/1.1874333. Google Scholar

[11]

D. Cao and Z. Tang, Existence and Uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 222 (2006), 381-424. doi: 10.1016/j.jde.2005.06.027. Google Scholar

[12]

Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82. doi: 10.1007/s00229-011-0530-1. Google Scholar

[13]

Y. Ding and Z. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Annali di Matematica, 190 (2011), 427-451. doi: 10.1007/s10231-010-0157-y. Google Scholar

[14]

C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820. doi: 10.1080/03605309608821208. Google Scholar

[15]

K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagenetic fields, Nonlinear Anal., 41 (2000), 763-778. doi: 10.1016/S0362-546X(98)00308-3. Google Scholar

[16]

M. K. Kwong, Uniqueness of the positive solution of $Δ u-u+u^{p}=0$ in $\mathbb{R}^{n}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar

[17]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928. Google Scholar

[18]

W. Long and S. Peng, Multiple positive solutions for a type of nonlinear Schrödinger equations, Annali della Scuola Normale Superiore di Pisa Classe di Scienze, 16 (2016), 603-623. Google Scholar

[19]

G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields, J. Differential Equations, 251 (2011), 3500-3521. doi: 10.1016/j.jde.2011.08.038. Google Scholar

[20]

W. Liu, Infinitely many solutions for nonlinear Schrödinger systems with magnetic potentials in $\mathbb{R}^{3}$, Math. Meth. Appl. Sci., 39 (2016), 1452-1479. doi: 10.1002/mma.3581. Google Scholar

[21]

W. Liu and C. Wang, Infinitely many solutions for the nonlinear Schrödinger equations with magnetic potentials in $\mathbb{R}^{N}$ J. Math. Phys. , 54 (2013), 121508, 23pp. doi: 10.1063/1.4851756. Google Scholar

[22]

W. Liu and C. Wang, Multi-peak solutions of a nonlinear Schrödinger equation with magnetic fields, Adv. Nonlinear Stud., 14 (2014), 951-975. doi: 10.1515/ans-2014-0408. Google Scholar

[23]

W. Liu and C. Wang, Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields, Discrete Contin. Dyn. Syst. A, 36 (2016), 7081-7115. doi: 10.3934/dcds.2016109. Google Scholar

[24]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227. doi: 10.1112/S002461070000898X. Google Scholar

[25]

E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc., 76 (1998), 427-452. doi: 10.1112/S0024611598000148. Google Scholar

[26]

M. Del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar

[27]

S. Shirai, Multi-bump solutions to a nonlinear Schrödinger equation with steep magnetic wells J. Math. Phys. , 56(2015), 091510, 19pp. doi: 10.1063/1.4930247. Google Scholar

[28]

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

[29]

D. Salazar, Vortex-type solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 66 (2015), 663-675. doi: 10.1007/s00033-014-0412-y. Google Scholar

[30]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation Self-Focusing and Wave Collapse, Applied Mathematical Sciences 139. Springer-Verlag, New York, Berlin, Heidelberg, 1999. Google Scholar

[31]

J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439. doi: 10.1007/s00526-009-0270-1. Google Scholar

[32]

J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb{S}^N$, J. Funct. Anal., 258 (2010), 3048-3081. doi: 10.1016/j.jfa.2009.12.008. Google Scholar

[33]

M. Yang and Y. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities, J. Math. Anal. Appl., 403 (2013), 680-694. doi: 10.1016/j.jmaa.2013.02.062. Google Scholar

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