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Applicable Analysis
An International Journal
Volume 97, 2018 - Issue 12
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Articles

On the least energy solutions for semilinear Schrödinger equation with electromagnetic fields involving critical growth and indefinite potentials

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Pages 2157-2169 | Received 30 Dec 2016, Accepted 04 Jul 2017, Published online: 03 Aug 2017

References

  • Esteban MJ, Lions PL. Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. In: Colombini F, Marino A, Modica L, Spagnolo S, editors. Partial differential equations and the calculus of variations. Vol. 1. Birkhäuser; 1989. p. 401–449.
  • Kurata K. Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger eqaution with electromagnetic fields. Nonlinear Anal TMA. 2000;41:763–778.
  • Cingolani S. Semiclassical stationary states of nonlinear Schrödinger eqaution with external magnetic field. J Differ Equ. 2003;188:52–79.
  • Cingolani S, Secchi S. Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields. J Math Anal Appl. 2002;275:108–130.
  • Cao D, Tang Z. Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields. J Differ Equ. 2006;222:381–424.
  • Liang S, Zhang J. Solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity. Proc Edinb Math Soc. 2011;54:131–147.
  • Tang Z. On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields. Comm Math Appl. 2007;54:627–637.
  • Tang Z. Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency. J. Differ Equ. 2008;245:2723–2748.
  • Fu S, Jiao Y, Tang Z. Multi-bump bound states for a nonlinear Schröinger system with electromagnetic fields. J Math Anal Appl. 2013;404:239–259.
  • Tang Z, Wang Y. Least energy solutions for semilinear Schrödinger equations with electromagnetic fields and critical growth. Sci China Math. 2015;58:2317–2328.
  • Floer A, Weinstein A. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J Funct Anal. 1986;69:397–408.
  • Ambrosetti A, Badiale M, Cingolani S. Semiclassical states of nonlinear Schrödinger equations. Arch Rat Mech Anal. 1997;140:285–300.
  • Ambrosetti A, Malchiodi A, Secchi S. Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch Rat Mech Anal. 2001;159:253–271.
  • Cingolani S, Lazzo M. Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J Differ Equ. 2000;160:118–138.
  • Cingolani S, Nolasco M. Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc Royal Soc Edinburgh. 1998;128:1249–1260.
  • Del Pino M, Felmer P. Semi-classical states for nonlinear Schrödinger equations. Ann Inst Henri Poincaré. 1998;15:127–149.
  • Del Pino M, Felmer P. Multi-peak bound states for nonlinear Schrödinger equations. J Funct Anal. 1997;149:245–265.
  • Oh Y-G. On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm Math Phys. 1990;131:223–253.
  • Oh Y-G. Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class (V)\textsubscript{a}. Comm Part Differ Equ. 1988;13:1499–1519.
  • Tang Z. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Comm Pure Appl. Anal. 2014;13:237–248.
  • Ding Y, Wei J. Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. J Funct Anal. 2007;251:546–572.
  • Szulkin A, Weth T. Ground state solutions for some indefinite variational problems. J Funct Anal. 2009;257:3802–3822.
  • Bartsch T, Tang Z. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete Cont Dyn Syst. 2013;33:7–26.
  • Guo Y, Tang Z. Multi-bump solutions for Schrödinger equation involving critical growth and potential wells. Discrete Cont Dyn Syst. 2015;35:3393–3415.
  • Arioli G, Szulkin A. A semilinear Schrödinger equation in the presence of a maganetic field. Arch Rat Mech Anal. 2003;170:277–295.
  • Pankov A. Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J Math. 2005;73:563–574.
  • Tang Z, Wang L. Solutions for the problems involving fractional Laplacian and indefinite potentials. Adv Nonlinear Stud. 2017;17:551–580.
  • Capozzi A, Fortunato D, Palmieri G. An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann Inst Henri Poincaré. 1985;2:463–470.
  • Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical sobolev exponents. Cummuni Pure Appl Math. 1983;36:437–477.

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