Sign-Changing Solutions

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


In Chap. 8, we introduce some results on sign-changing solutions of elliptic and p-Laplacian, including using Nehri manifold, invariant sets of descent flows, Morse theory, etc.


Nontrivial Solution Morse Index Smooth Bounded Domain Nontrivial Critical Point Standing Wave Solution 
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  1. 26.
    T. Bartsch, T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations. Topol. Methods Nonlinear Anal. 22(1), 1–14 (2003) MathSciNetzbMATHGoogle Scholar
  2. 28.
    T. Bartsch, K.C. Chang, Z. Wang, On the Morse indices of sign-changing solutions of nonlinear elliptic problems. Math. Z. 233, 655–677 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 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
  4. 44.
    A. Castro, J. Cossio, J.M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem. Rocky Mt. J. Math. 27, 1041–1053 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 47.
    G. Cerami, An existence criterion for the critical points on unbounded manifolds. Rend. - Ist. Lomb., Accad. Sci. Lett. A 112(2), 332–336 (1978) MathSciNetzbMATHGoogle Scholar
  6. 50.
    K.-C. Chang, Methods in Nonlinear Analysis (Springer, Berlin, 2005) zbMATHGoogle Scholar
  7. 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
  8. 68.
    E.N. Dancer, Y. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities. J. Differ. Equ. 114, 434–475 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 69.
    E.N. Dancer, Y. Du, Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero. Proc. R. Soc. Edinb., Sect. A 124, 1165–1176 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 70.
    E.N. Dancer, Y. Du, On sign-changing solutions of certain semilinear elliptic problems. Appl. Anal. 56, 193–206 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 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
  12. 72.
    E.N. Dancer, Z. Zhang, Dynamics of Lotka–Volterra competition systems with large interaction. J. Differ. Equ. 182(2), 470–489 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 86.
    G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian. Port. Math. 58(3), 339–377 (2001) MathSciNetzbMATHGoogle Scholar
  14. 95.
    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 2001) zbMATHGoogle Scholar
  15. 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
  16. 138.
    Z. Liu, A topological property of the boundary of bounded open sets in \(\mathbb{R}^{2}\). J. Shandong Univ. 29(3), 299–303 (1994) zbMATHGoogle Scholar
  17. 143.
    Z. Liu, Z.-Q. Wang, T. Weth, Multiple solutions of nonlinear Schrödinger equations via flow invariance and Morse theory. Proc. R. Soc. Edinb., Sect. A 136, 945–969 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 152.
    K. Perera, M. Schechter, Double resonance problems with respect to the Fucik spectrum. Indiana Univ. Math. J. 52(1), 1–17 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 153.
    K. Perera, M. Schechter, Computation of critical groups in Fucik resonance problems. J. Math. Anal. Appl. 279(1), 317–325 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 160.
    M. Schechter, The Fucik spectrum. Indiana Univ. Math. J. 43(4), 1139–1157 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 172.
    J. Sun, Z. Liu, Calculus of variations and super- and sub-solution in reverse order. Acta Math. Sin., 37(4), 512–514 (1994) (In Chinese) zbMATHGoogle Scholar
  22. 192.
    G.T. Whyburn, Topological Analysis (Princeton University Press, Princeton, 1958) zbMATHGoogle Scholar
  23. 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
  24. 209.
    Z. Zhang, J. Chen, S. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian. J. Differ. Equ. 201, 287–303 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 213.
    W. Zou, Sign-Changing Critical Point Theory (Springer, Berlin, 2008) zbMATHGoogle 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

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