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

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

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.

Citation: Gan Lu, Weiming Liu. Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1957-1975. doi: 10.3934/cpaa.2017096
References:
[1]

A. AmbrosettiE. 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.

[2]

T. BartschE. 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. 

[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.

[4]

S. CingolaniL. 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.

[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.

[6]

D. CaoE. 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.

[7]

D. CaoE. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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. 

[19]

G. LiS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

A. AmbrosettiE. 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.

[2]

T. BartschE. 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. 

[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.

[4]

S. CingolaniL. 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.

[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.

[6]

D. CaoE. 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.

[7]

D. CaoE. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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. 

[19]

G. LiS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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