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A Variational Approach for One-Dimensional Scalar Field Problems

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Abstract

In this paper, we are interested in the existence of infinitely many weak solutions for a onedimensional scalar field problem. By using variational methods, in an appropriate functional space which involves the potential V, we determine intervals of parameters such that our problem admits either a sequence of weak solutions strongly converging to zero provided that the nonlinearity has a suitable behavior at zero or an unbounded sequence of weak solutions if a similar behavior occurs at infinity.

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References

  1. T. Bartsch, Z. Liu, and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations, 29 (2004), 25–42.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Bartsch, A. Pankov, and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 4 (2001), 549–569.

    Article  MATH  Google Scholar 

  3. H. Berestycki and P.-L. Lions, Nonlinear scalar field equations I, II., Arch. Ration. Mech. Anal., 82 (1983), 313–345, 347–375.

    Article  MATH  Google Scholar 

  4. F. Faraci and A. Kristály, One-dimensional scalar field equations involving an oscillatory nonlinear term, Discrete Contin. Dyn. Syst., 18 (2007), 107–120.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397–408.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Kristály, V. Rădulescu, and Cs. Varga, Variational principles in mathematical physics, geometry, and economics: Qualitative analysis of nonlinear equations and unilateral problems, Encyclopedia of mathematics and its applications, 136, Cambridge University Press, Cambridge, 2010.

    Book  MATH  Google Scholar 

  7. J. B. McLeod, C. A. Stuart, and W. C. Troy, Stability of standing waves for some nonlinear Schrödinger equations, Differential Integral Equations, 16 (2003), 1025–1038.

    MathSciNet  MATH  Google Scholar 

  8. P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270–291.

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401–410.

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Zeidler, Nonlinear functional analysis and its applications, vol. II/B, Berlin-Heidelberg-New York, 1990.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, 94531, Iran

    Armin Hadjian

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  1. Armin Hadjian
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Correspondence to Armin Hadjian.

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Hadjian, A. A Variational Approach for One-Dimensional Scalar Field Problems. Indian J Pure Appl Math 49, 621–632 (2018). https://doi.org/10.1007/s13226-018-0290-7

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  • DOI: https://doi.org/10.1007/s13226-018-0290-7

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