Advertisement

Dancer–Fučik Spectrum

  • Zhitao Zhang
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 29)

Abstract

In Chap. 7, we show some results on the Dancer–Fučik spectrum for bounded domains. We are concerned with the Fučik point spectrum for Schrödinger operators, −Δ+V, in \(L^{2}(\mathbb {R}^{N})\) for certain types of potential, \(V:\mathbb {R}^{N}\to \mathbb {R}\). We use the Dancer–Fučik spectrum asymptotically to find linear elliptics to get one-sign solutions.

Keywords

Bounded Domain Point Spectrum Form Domain Critical Point Theory Compact Resolvent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 9.
    H. Amann, E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Sc. Norm. Super. Pisa 7, 539–603 (1980) MathSciNetzbMATHGoogle Scholar
  2. 16.
    M. Arias, J. Campos, M. Cuesta, J.-P. Gossez, Asymmetric elliptic problems with indefinite weights. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, 581–616 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 18.
    C. Azizieh, Ph. Clement, A priori estimates and continuation methods for positive solutions of p-Laplace equations. J. Differ. Equ. 179, 213–245 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 24.
    T. Bartsch, M. Degiovanni, Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 17, 69–85 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 25.
    T. Bartsch, S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28(3), 419–441 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 29.
    T. Bartsch, A. Pankov, Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 549–569 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 30.
    T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 31.
    T. Bartsch, Z. Wang, Z. Zhang, On the Fučik point spectrum for Schrödinger operators on R n. Fixed Point Theory Appl. 5(2), 305–317 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 37.
    N.P. Cac, On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue. J. Differ. Equ. 80, 379–404 (1989) zbMATHCrossRefGoogle Scholar
  10. 48.
    K.C. Chang, Solutions of asymptotically linear operator equations via Morse theory. Commun. Pure Appl. Math. 34, 693–712 (1981) zbMATHCrossRefGoogle Scholar
  11. 49.
    K.-C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems (Birkhäuser, Basel, 1993) CrossRefGoogle Scholar
  12. 61.
    M. Cuesta, Eigenvalue problems for the p-Laplacian with indefinite weights. Electron. J. Differ. Equ. 2001, 33 (2001) MathSciNetGoogle Scholar
  13. 62.
    M. Cuesta, D. de Figueiredo, J.P. Gossez, The beginning of the Fu\(\check{c}\)ik spectrum for the p-Laplacian. J. Differ. Equ. 159, 212–238 (1999) zbMATHCrossRefGoogle Scholar
  14. 64.
    E.N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations. Proc. R. Soc. Edinb., Sect. A 76, 283–300 (1977) MathSciNetzbMATHGoogle Scholar
  15. 66.
    E.N. Dancer, Multiple solutions of asymptotically homogeneous problems. Ann. Mat. Pura Appl. 152, 63–78 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 67.
    E.N. Dancer, Remarks on jumping nonlinearities, in Topics in Nonlinear Analysis, ed. by Escher, Simonett (Birkhäuser, Basel, 1999), pp. 101–116 CrossRefGoogle Scholar
  17. 71.
    E.N. Dancer, Z. Zhang, Fucik spectrum, sign-changing and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity. J. Math. Anal. Appl. 250, 449–464 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 78.
    D.G. de Figueiredo, J.-P. Gossez, On the first curve of the Fučik spectrum of an elliptic operator. Differ. Integral Equ. 7, 1285–1302 (1994) zbMATHGoogle Scholar
  19. 91.
    Y. Egorov, V. Kondratiev, On Spectral Theory of Elliptic Operators (Birkhäuser, Basel, 1996) zbMATHCrossRefGoogle Scholar
  20. 92.
    S. Fučik, Boundary value problems with jumping nonlinearities. Čas. Pěst. Math. Fys. 101, 69–87 (1976) zbMATHGoogle Scholar
  21. 93.
    S. Fučik, Solvability of Nonlinear Equations and Boundary Value Problems (Reidel, Dordrecht, 1980) zbMATHGoogle Scholar
  22. 127.
    S. Li, Z. Zhang, Fucik spectrum and sign-changing and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities at zero and infinity. Sci. China 44(7), 856–866 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 128.
    G. Li, H. Zhou, Asymptotically linear Dirichlet problem for the p-Laplacian. Nonlinear Anal., Theory Methods Appl. 43, 1043–1055 (2001) zbMATHCrossRefGoogle Scholar
  24. 130.
    C. Li, S. Li, Z. Liu, J. Pan, On the Fučík spectrum. J. Differ. Equ. 244, 2498–2528 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 133.
    P. Lindqvist, On the equation Δp u+div(|∇u|p−2u)=0. Proc. Am. Math. Soc. 109, 157–164 (1990). Addendum in Proc. Am. Math. Soc. 116, 583–584 (1992) MathSciNetzbMATHGoogle Scholar
  26. 142.
    Z. Liu, F.A. van Heerden, Z.-Q. Wang, Nodal type bound states of Schröedinger equations via invariant set and minimax methods. J. Differ. Equ. 214, 358–390 (2005) zbMATHCrossRefGoogle Scholar
  27. 159.
    P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 65 (Am. Math. Soc., Providence, 1986). Published for the Conference Board of the Mathematical Sciences, Washington, DC Google Scholar
  28. 160.
    M. Schechter, The Fucik spectrum. Indiana Univ. Math. J. 43(4), 1139–1157 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 174.
    A. Szulkin, Ljusternik–Schnirelman theory on C 1-manifolds. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 119–139 (1988) MathSciNetzbMATHGoogle Scholar
  30. 184.
    J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12, 191–202 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 205.
    Z. Zhang, S. Li, On sign-changing and multiple solutions of the p-Laplacian. J. Funct. Anal. 197(2), 447–468 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 208.
    Z. Zhang, S. Li, S. Liu, W. Feng, On an asymptotically linear elliptic Dirichlet problem. Abstr. Appl. Anal. 7(10), 509–516 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 210.
    H.S. Zhou, Existence of asymptotically linear Dirichlet problem. Nonlinear Anal., Theory Methods Appl. 44, 909–918 (2001) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Zhitao Zhang
    • 1
  1. 1.Academy of Mathematics & Systems ScienceThe Chinese Academy of SciencesBeijingP.R. China

Personalised recommendations