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

Weiming  Liu; Chunhua  Wang

doi:10.3934/dcds.2016109

Article Contents

2016, Volume 36, Issue 12: 7081-7115. Doi: 10.3934/dcds.2016109

This issue Previous Article Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$ Next Article Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping

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

Weiming Liu^1, and
Chunhua Wang^2,

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

Received: November 30, 2015

Revised: June 30, 2016

Published: May 2017

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35J10, 35B99; Secondary: 35J60.

Citation:

Related Papers

Cited by

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.