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

OriginalPaper

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140, 285-300 (1997)CrossRefMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Malchiodi, A., Secchi, S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159, 253-271 (2001)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Functional Anal. 14, 349-381 (1973)MATHGoogle Scholar
  4. 4.
    Berestycki, H., Lions, P.L.: Nonlinear scalar field equations I. Arch. Rat. Mech. Anal. 82, 313-346 (1983)MathSciNetMATHGoogle Scholar
  5. 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)CrossRefMATHGoogle Scholar
  6. 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)MathSciNetMATHGoogle Scholar
  7. 7.
    Byeon, J.: Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems. J. Differential Equations 163, 429-474 (2000)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Byeon, J.: Effect of symmetry to the structure of positive solutions in nonlinear elliptic problems, II. J. Differential Equations 173, 321-355 (2001)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Byeon, J.: Standing waves for nonlinear Schrödinger equations with a radial potential. Nonlinear Analysis 50, 1135-1151 (2002)CrossRefGoogle Scholar
  10. 10.
    Byeon, J., Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Rat. Mech. Anal. 65, 295-316 (2002)CrossRefMathSciNetGoogle Scholar
  11. 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)CrossRefMathSciNetGoogle Scholar
  12. 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)MATHGoogle Scholar
  13. 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)CrossRefMATHGoogle Scholar
  14. 14.
    Del Pino, M., Felmer, P.L.: Semi-classical states for nonlinear Schrödinger equations. J. Functional Analysis 149, 245-265 (1997)CrossRefMATHGoogle Scholar
  15. 15.
    Del Pino, M., Felmer, P.L.: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré 15, 127-149 (1998)MATHGoogle Scholar
  16. 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. 17.
    Esteban, M., Lions, P.L.: A compactness lemma. Nonlinear Anal. 7, 381-385 (1983)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential. J. Functional Analysis 69, 397-408 (1986)MATHGoogle Scholar
  19. 19.
    Gidas, B., Ni, W.N., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209-243 (1979)MathSciNetMATHGoogle Scholar
  20. 20.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second OrderGoogle Scholar
  21. 21.
    Gui, C.: Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method. Comm. in P.D.E. 21, 787-820 (1996)MathSciNetMATHGoogle Scholar
  22. 22.
    Jeanjean, L., Tanaka, K.: A remark on least energy solutions in \(\mathbf{R}^N\). Proc. Amer. Math. Soc. (to appear)Google Scholar
  23. 23.
    Jeanjean, L., Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. PreprintGoogle Scholar
  24. 24.
    Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differential Equations 5, 899-928 (2000)MATHGoogle Scholar
  25. 25.
    Li, Y.Y.: On a singularly perturbed elliptic equation. Adv. Differential Equations 2, 955-980 (1997)MathSciNetGoogle Scholar
  26. 26.
    Ladyzhenskaya, O.A., Ural'tseva, N.N.: Linear and quasilinear elliptic equations. Academic Press Inc. (1968)Google Scholar
  27. 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)MathSciNetGoogle Scholar
  28. 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)MathSciNetMATHGoogle Scholar
  29. 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)MATHGoogle Scholar
  30. 30.
    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, (1984)Google Scholar
  31. 31.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270-291 (1992)MATHGoogle Scholar
  32. 32.
    Strauss, W.: Existence of solitary waves in higher demensions. Comm. Math. Phys. 55, 149-162 (1977)MATHGoogle Scholar
  33. 33.
    Struwe, M.: Variational MethodsGoogle Scholar
  34. 34.
    Wang, X.: On concentration of positive bound states of nonlinear Schröinger equations. Comm. Math. Phys. 153, 229-244 (1993)MATHGoogle Scholar
  35. 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. 36.
    Wang, Z.-Q.: Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations. J. Differential Equations 159, 102-137 (1999)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsPOSTECHPohang KyungbukRepublic of Korea
  2. 2.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

Personalised recommendations