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Standing waves with a critical frequency for nonlinear Schrödinger equations, II

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Abstract.

For elliptic equations of the form \(\Delta u -V(\varepsilon x) u + f(u)=0, x\in {\bf R}^N\), where the potential V satisfies \(\liminf_{\vert x\vert\to \infty} V(x) > \inf_{{\bf R}^N} V(x) =0\), we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrödinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.

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References

  1. Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140, 285-300 (1997)

    Article  MATH  Google Scholar 

  2. Ambrosetti, A., Malchiodi, A., Secchi, S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159, 253-271 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Functional Anal. 14, 349-381 (1973)

    MATH  Google Scholar 

  4. Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I. Arch. Rat. Mech. Anal. 82, 313-346 (1983)

    MathSciNet  MATH  Google Scholar 

  5. Byeon, J.: Existence of many nonequivalent non-radial positive solutions of semilinear elliptic equations on three dimensional annuli. J. Differential Equations 136, 136-165 (1997)

    Article  MATH  Google Scholar 

  6. Byeon, J.: Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains. Comm. in P.D.E. 22, 1731-1769 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Byeon, J.: Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems. J. Differential Equations 163, 429-474 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Byeon, J.: Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, II. J. Differential Equations 173, 321-355 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byeon, J.: Standing waves for nonlinear Schrödinger equations with a radial potential. Nonlinear Analysis 50, 1135-1151 (2002)

    Article  Google Scholar 

  10. Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Rat. Mech. Anal. 65, 295-316 (2002)

    Article  MathSciNet  Google Scholar 

  11. Dancer, E.N., Lam, K.Y., Yan, S.: The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations. Abstr. Appl. Anal. 3, 293-318 (1998)

    Article  MathSciNet  Google Scholar 

  12. Dancer, E.N., Yan, S.: On the existence of multipeak solutions for nonlinear field equations on \({\bf R}^N\). Discrete Contin. Dynam. Systems 6, 39-50 (2000)

    MATH  Google Scholar 

  13. Del Pino, M., Felmer, P.L.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calculus of Variations and PDE 4, 121-137 (1996)

    Article  MATH  Google Scholar 

  14. Del Pino, M., Felmer, P.L.: Semi-classical states for nonlinear Schrödinger equations. J. Functional Analysis 149, 245-265 (1997)

    Article  MATH  Google Scholar 

  15. Del Pino, M., Felmer, P.L.: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 15, 127-149 (1998)

    MATH  Google Scholar 

  16. Del Pino, M., Felmer, P.L., Miyagaki, O.H.: Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential. Nonlinear Analysis, TMA 34, 979-989 (1998)

    Google Scholar 

  17. Esteban, M., Lions, P.L.: A compactness lemma. Nonlinear Anal. 7, 381-385 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  18. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential. J. Functional Analysis 69, 397-408 (1986)

    MATH  Google Scholar 

  19. Gidas, B., Ni, W.N., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209-243 (1979)

    MathSciNet  MATH  Google Scholar 

  20. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order

  21. Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Comm. in P.D.E. 21, 787-820 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \(\mathbf{R}^N\). Proc. Amer. Math. Soc. (to appear)

  23. Jeanjean, L., Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Preprint

  24. Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differential Equations 5, 899-928 (2000)

    MATH  Google Scholar 

  25. Li, Y.Y.: On a singularly perturbed elliptic equation. Adv. Differential Equations 2, 955-980 (1997)

    MathSciNet  Google Scholar 

  26. Ladyzhenskaya, O.A., Ural'tseva, N.N.: Linear and quasilinear elliptic equations. Academic Press Inc. (1968)

  27. Oh, Y.G.: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class \((V)\sb a\). Comm. P.D.E. 13, 1499-1519 (1988)

    MathSciNet  Google Scholar 

  28. Oh, Y.G.: Correction to: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class \((V)_a\). Comm. P.D.E. 14, 833-834 (1989)

    MathSciNet  MATH  Google Scholar 

  29. Oh, Y.G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Comm. Math. Phys. 131, 223-253 (1990)

    MATH  Google Scholar 

  30. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, (1984)

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

    MATH  Google Scholar 

  32. Strauss, W.: Existence of solitary waves in higher demensions. Comm. Math. Phys. 55, 149-162 (1977)

    MATH  Google Scholar 

  33. Struwe, M.: Variational Methods

  34. Wang, X.: On concentration of positive bound states of nonlinear Schröinger equations. Comm. Math. Phys. 153, 229-244 (1993)

    MATH  Google Scholar 

  35. Wang, Z.-Q.: Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent. Nonlinear Analysis, TMA 27, 1281-1306 (1996)

    Google Scholar 

  36. Wang, Z.-Q.: Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations. J. Differential Equations 159, 102-137 (1999)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, POSTECH, 790-784, Pohang Kyungbuk, Republic of Korea

    Jaeyoung Byeon

  2. Department of Mathematics and Statistics, Utah State University, UT 84322, Logan, USA

    Zhi-Qiang Wang

Authors
  1. Jaeyoung Byeon
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  2. Zhi-Qiang Wang
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Correspondence to Jaeyoung Byeon.

Additional information

Received: 5 July 2002, Accepted: 24 October 2002, Published online: 14 February 2003

The research of the first author was supported by Grant No. 1999-2-102-003-5 from the Interdisciplinary Research Program of the KOSEF.

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Byeon, J., Wang, ZQ. Standing waves with a critical frequency for nonlinear Schrödinger equations, II. Cal Var 18, 207–219 (2003). https://doi.org/10.1007/s00526-002-0191-8

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  • DOI: https://doi.org/10.1007/s00526-002-0191-8

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