December  2016, 36(12): 7081-7115. doi: 10.3934/dcds.2016109

Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields

1. 

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

2. 

School of Mathematics and Statistics and Hubei Key Laboratory Mathematical Sciences, Central China Normal University, Wuhan 430079, China

Received  December 2015 Revised  July 2016 Published  October 2016

In this paper, we study the nonlinear Schrödinger equation with non-symmetric electromagnetic fields $$ \Big(\frac{\nabla}{i}-A_{\epsilon}(x)\Big)^{2}u+V_{\epsilon}(x)u=f(u),~~~~~~u\in H^{1}(\mathbb{R}^{N},\mathbb{C}), $$ where $A_{\epsilon}(x)=(A_{\epsilon,1}(x),A_{\epsilon,2}(x),\cdots,A_{\epsilon,N}(x))$ is a magnetic field satisfying that $A_{\epsilon,j}(x)(j=1,\ldots,N)$ is a real $C^{1}$ bounded function on $\mathbb{R}^{N}$ and $V_{\epsilon}(x)$ is an electric potential. Both of them satisfy some decay conditions but without any symmetric conditions and $f(u)$ is a superlinear nonlinearity satisfying some non-degeneracy condition. Applying two times finite reduction methods and localized energy method, we prove that there exists some $\epsilon_{0 }> 0$ such that for $0 < \epsilon < \epsilon_{0 }$, the above problem has infinitely many complex-valued solutions.
Citation: Weiming Liu, Chunhua Wang. Infinitely many solutions for a nonlinear Schrödinger equation with non-symmetric electromagnetic fields. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109
References:
[1]

W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differential Equ., 51 (2014), 761-798. doi: 10.1007/s00526-013-0694-5.

[2]

R. Brummelhuis, Expotential decay in the semi-classical limit for eigenfunctions of Schrödinger operators with magnetic fields and potentials which degenerate at infinity, Comm. Partial Differential Equ., 16 (1991), 1489-1502. doi: 10.1080/03605309108820807.

[3]

T. Bartsch, E. N. Dancer and S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields, Adv. Differential Equ., 11 (2006), 781-812.

[4]

A. Bahri and Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbbR^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15. doi: 10.4171/RMI/92.

[5]

A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincare, 14 (1997), 365-413. doi: 10.1016/S0294-1449(97)80142-4.

[6]

S. Barile, S. Cingolani and S. Secchi, Single-peaks for a magnetic Schrödinger equation with critical growth, Adv. Differential Equ., 11 (2006), 1135-1166.

[7]

G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with non-symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413. doi: 10.1002/cpa.21410.

[8]

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.

[9]

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

[10]

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

[11]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950.

[12]

M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.

[13]

M. del Pino, J. Wei and W. Yao, Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equ., 53 (2015), 473-523. doi: 10.1007/s00526-014-0756-3.

[14]

W. Ding and W. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.

[15]

M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Progr. Nonlinear Differential Equations Appl., 1 (1989), 401-449.

[16]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.

[17]

B. Helffer, On spectral theory for Schrödinger operator with magnetic potentials, Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 113-141.

[18]

B. Helffer, Semiclassical analysis for Schrödinger operator with magnetic wells, Quasiclassical methods (Minneapolis, MN, 1995), IMA Vol. Math. Appl., Springer, New York, 95 (1997), 99-114. doi: 10.1007/978-1-4612-1940-8_4.

[19]

B. Helffer and J. Sjöstrand, The tunnel effect for the Schrödinger equation with magnetic field, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 625-657.

[20]

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.

[21]

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.

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincare, 1 (1984), 109-145.

[23]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincare, 1 (1984), 223-283.

[24]

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

[25]

M. Musso, F. Pacard and J. Wei, Finite-energy sigh-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953. doi: 10.4171/JEMS/351.

[26]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equ., 14 (1989), 833-834. doi: 10.1080/03605308908820631.

[27]

H. Pi and C. Wang, Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields, ESAIM Control Optim. Calc. Var., 19 (2013), 91-111. doi: 10.1051/cocv/2011207.

[28]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[29]

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

[30]

X. Wang, On a concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.

[31]

C. Wang and J. Yang, Infinitely many solutions to linearly coupled Schrödinger equations with non-symmetric potential, J. Math. Phys., 56 (2015), 051505, 25pp. doi: 10.1063/1.4921637.

show all references

References:
[1]

W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential, Calc. Var. Partial Differential Equ., 51 (2014), 761-798. doi: 10.1007/s00526-013-0694-5.

[2]

R. Brummelhuis, Expotential decay in the semi-classical limit for eigenfunctions of Schrödinger operators with magnetic fields and potentials which degenerate at infinity, Comm. Partial Differential Equ., 16 (1991), 1489-1502. doi: 10.1080/03605309108820807.

[3]

T. Bartsch, E. N. Dancer and S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields, Adv. Differential Equ., 11 (2006), 781-812.

[4]

A. Bahri and Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbbR^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15. doi: 10.4171/RMI/92.

[5]

A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincare, 14 (1997), 365-413. doi: 10.1016/S0294-1449(97)80142-4.

[6]

S. Barile, S. Cingolani and S. Secchi, Single-peaks for a magnetic Schrödinger equation with critical growth, Adv. Differential Equ., 11 (2006), 1135-1166.

[7]

G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equations with non-symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413. doi: 10.1002/cpa.21410.

[8]

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.

[9]

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

[10]

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

[11]

M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950.

[12]

M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.

[13]

M. del Pino, J. Wei and W. Yao, Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equ., 53 (2015), 473-523. doi: 10.1007/s00526-014-0756-3.

[14]

W. Ding and W. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.

[15]

M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Progr. Nonlinear Differential Equations Appl., 1 (1989), 401-449.

[16]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.

[17]

B. Helffer, On spectral theory for Schrödinger operator with magnetic potentials, Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 113-141.

[18]

B. Helffer, Semiclassical analysis for Schrödinger operator with magnetic wells, Quasiclassical methods (Minneapolis, MN, 1995), IMA Vol. Math. Appl., Springer, New York, 95 (1997), 99-114. doi: 10.1007/978-1-4612-1940-8_4.

[19]

B. Helffer and J. Sjöstrand, The tunnel effect for the Schrödinger equation with magnetic field, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 625-657.

[20]

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.

[21]

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.

[22]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincare, 1 (1984), 109-145.

[23]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. Inst. H. Poincare, 1 (1984), 223-283.

[24]

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

[25]

M. Musso, F. Pacard and J. Wei, Finite-energy sigh-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation, J. Eur. Math. Soc., 14 (2012), 1923-1953. doi: 10.4171/JEMS/351.

[26]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equ., 14 (1989), 833-834. doi: 10.1080/03605308908820631.

[27]

H. Pi and C. Wang, Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields, ESAIM Control Optim. Calc. Var., 19 (2013), 91-111. doi: 10.1051/cocv/2011207.

[28]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[29]

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

[30]

X. Wang, On a concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.

[31]

C. Wang and J. Yang, Infinitely many solutions to linearly coupled Schrödinger equations with non-symmetric potential, J. Math. Phys., 56 (2015), 051505, 25pp. doi: 10.1063/1.4921637.

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