Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T11:16:38.389Z Has data issue: false hasContentIssue false
  • English
  • Français

Solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  28 October 2010

Sihua Liang  and
Jihui Zhang
Sihua Liang
Affiliation:
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, 210097 Jiangsu, People's Republic of China ([email protected]) College of Mathematics, Changchun Normal University, Changchun, 130032 Jilin, People's Republic of China ([email protected])
Jihui Zhang
Affiliation:
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, 210097 Jiangsu, People's Republic of China ([email protected]) The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the existence and multiplicity of standing-wave solutions

of nonlinear Schrödinger equations with electromagnetic fields and critical nonlinearity

Under suitable assumptions, we prove that it has at least one solution and that, for any m ∈ ℕ, it has at least m pairs of solutions.

Keywords

nonlinear Schrödinger equationcritical nonlinearitymagnetic fieldsvariational methods

MSC classification

Secondary: 35J60: Nonlinear elliptic equations 35B33: Critical exponents
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 131 - 147
Copyright
Copyright © Edinburgh Mathematical Society 2010

References

1.Ambrosetti, A., Badiale, M. and Cingolani, S., Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Analysis 140 (1997), 285–300.CrossRefGoogle Scholar
2.Ambrosetti, A., Malchiodi, A. and Secchi, S., Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Analysis 159 (2001), 253–271.CrossRefGoogle Scholar
3.Arioli, G. and Szulkin, A., A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Analysis 170 (2003), 277–295.CrossRefGoogle Scholar
4.Bartsch, T., Dancer, E. N. and Peng, S., On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields, Adv. Diff. Eqns 11 (2006), 781–812.Google Scholar
5.Benci, V., On critical point theory of indefinite functionals in the presence of symmetries, Trans. Am. Math. Soc. 274 (1982), 533–572.CrossRefGoogle Scholar
6.Byeon, J. and Wang, Z. Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Analysis 165 (2002), 295–316.CrossRefGoogle Scholar
7.Cao, D. and Noussair, E. S., Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, J. Diff. Eqns 203 (2004), 292–312.CrossRefGoogle Scholar
8.Cao, D. and Tang, Z., Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields, J. Diff. Eqns 222 (2006), 381–424.CrossRefGoogle Scholar
9.Chabrowski, J. and Szulkin, A., On the Schrödinger equation involving a critical Sobolev exponent and magnetic field, Topolog. Meth. Nonlin. Analysis 25 (2005), 3–21.CrossRefGoogle Scholar
10.Cingolani, S., Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field, J. Diff. Eqns 188 (2003), 52–79.CrossRefGoogle Scholar
11.Cingolani, S. and Lazzo, M., Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff. Eqns 160 (2000), 118–138.CrossRefGoogle Scholar
12.Cingolani, S. and Nolasco, M., Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. R. Soc. Edinb. A 128 (1998), 1249–1260.CrossRefGoogle Scholar
13.Cingolani, S. and Secchi, S., Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields, J. Math. Analysis Applic. 275 (2002), 108–130.CrossRefGoogle Scholar
14.Cingolani, S. and Secchi, S., Semiclassical states for NLS equations with magnetic potentials having polynomial growths, J. Math. Phys. 46 (2005), 053503.CrossRefGoogle Scholar
15.Clapp, M. and Ding, Y. H., Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential, Diff. Integ. Eqns 16 (2003), 981–992.Google Scholar
16.Del Pino, M. and Felmer, P., Multi-peak bound states for nonlinear Schrödinger equations, J. Funct. Analysis 149 (1997), 245–265.CrossRefGoogle Scholar
17.Del Pino, M. and Felmer, P., Semi-classical states for nonlinear Schrödinger equations, Annales Inst. H. Poincaré 15 (1998), 127–149.CrossRefGoogle Scholar
18.Ding, Y. H. and Lin, F. H., Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. PDEs 30 (2007), 231–249.CrossRefGoogle Scholar
19.Ding, Y. H. and Wei, J. C., Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Analysis 251 (2007), 546–572.CrossRefGoogle Scholar
20.Esteban, M. and Lions, P. L., Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, in Partial differential equations and the calculus of variations, essays in honor of Ennio De Giorgi, pp. 369–408 (Birkhäuser, 1989).Google Scholar
21.Floer, A. and Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Analysis 69 (1986), 397–408.CrossRefGoogle Scholar
22.Han, P., Solutions for singular critical growth Schrödinger equation with magnetic field, Portugaliae Math. 63 (2006), 37–45.Google Scholar
23.Kurata, K., Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlin. Analysis 41 (2000), 763–778.CrossRefGoogle Scholar
24.Oh, Y.-G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys. 131 (1990), 223–253.CrossRefGoogle Scholar
25.Tang, Z., On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl. 54 (2007), 627–637.CrossRefGoogle Scholar
26.Tang, Z., Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency, J. Diff. Eqns 245 (2008), 2723–2748.CrossRefGoogle Scholar
27.Wang, F., On an electromagnetic Schrödinger equation with critical growth, Nonlin. Analysis 69 (2008), 4088–4098.CrossRefGoogle Scholar
28.Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys. 153 (1993), 229–244.CrossRefGoogle Scholar
29.Willem, M., Minimax theorems (Birkhäuser, Boston, MA, 1996).CrossRefGoogle Scholar