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 & Continuous Dynamical Systems - A, 2016, 36 (12) : 7081-7115. doi: 10.3934/dcds.2016109
References:
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W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potential,, Calc. Var. Partial Differential Equ., 51 (2014), 761. doi: 10.1007/s00526-013-0694-5. Google Scholar

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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. doi: 10.1080/03605309108820807. Google Scholar

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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. Google Scholar

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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. doi: 10.4171/RMI/92. Google Scholar

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

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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. doi: 10.1002/cpa.21410. Google Scholar

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

[9]

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

[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. doi: 10.1016/j.jde.2005.06.027. Google Scholar

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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. doi: 10.1007/BF01189950. Google Scholar

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

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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. doi: 10.1007/s00526-014-0756-3. Google Scholar

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

[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. Google Scholar

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

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B. Helffer, On spectral theory for Schrödinger operator with magnetic potentials, Spectral and scattering theory and applications,, Adv. Stud. Pure Math., 23 (1994), 113. Google Scholar

[18]

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

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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. 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. doi: 10.1016/S0362-546X(98)00308-3. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. doi: 10.4171/JEMS/351. Google Scholar

[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. doi: 10.1080/03605308908820631. Google Scholar

[27]

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

[28]

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

[29]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,, Applied Mathematical Sciences, (1999). Google Scholar

[30]

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

[31]

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

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. doi: 10.1007/s00526-013-0694-5. Google Scholar

[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. doi: 10.1080/03605309108820807. Google Scholar

[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. Google Scholar

[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. doi: 10.4171/RMI/92. Google Scholar

[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. doi: 10.1016/S0294-1449(97)80142-4. Google Scholar

[6]

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

[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. doi: 10.1002/cpa.21410. Google Scholar

[8]

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

[9]

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

[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. doi: 10.1016/j.jde.2005.06.027. Google Scholar

[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. doi: 10.1007/BF01189950. Google Scholar

[12]

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

[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. doi: 10.1007/s00526-014-0756-3. Google Scholar

[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. doi: 10.1007/BF00282336. Google Scholar

[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. Google Scholar

[16]

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

[17]

B. Helffer, On spectral theory for Schrödinger operator with magnetic potentials, Spectral and scattering theory and applications,, Adv. Stud. Pure Math., 23 (1994), 113. Google Scholar

[18]

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

[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. Google Scholar

[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. doi: 10.1016/S0362-546X(98)00308-3. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. doi: 10.4171/JEMS/351. Google Scholar

[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. doi: 10.1080/03605308908820631. Google Scholar

[27]

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

[28]

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

[29]

C. Sulem and P. L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse,, Applied Mathematical Sciences, (1999). Google Scholar

[30]

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

[31]

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

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