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On a class of rotationally symmetric $p$-harmonic maps
Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials
School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, China |
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.
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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.
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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.
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D. Cao, E. S. Noussair and S. Yan,
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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. |
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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.
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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. |
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S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magenetic potentials having polynomial growths J. Math. Phys. , 46 (2005), 053503, 19 pp.
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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. |
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Y. Ding and X. Liu,
Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82.
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Y. Ding and Z. Wang,
Bound states of nonlinear Schrödinger equations with magnetic fields, Annali di Matematica, 190 (2011), 427-451.
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C. Gui,
Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820.
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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.
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On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928.
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Multiple positive solutions for a type of nonlinear Schrödinger equations, Annali della Scuola Normale Superiore di Pisa Classe di Scienze, 16 (2016), 603-623.
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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. |
[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. 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. |
[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.
|
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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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