Skip to main content

Part of the book series: Developments in Mathematics ((DEVM,volume 29))

  • 1227 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. 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)

    MathSciNet  MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Castro, J. Cossio, J.M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem. Rocky Mt. J. Math. 27, 1041–1053 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Cerami, An existence criterion for the critical points on unbounded manifolds. Rend. - Ist. Lomb., Accad. Sci. Lett. A 112(2), 332–336 (1978)

    MathSciNet  MATH  Google Scholar 

  6. K.-C. Chang, Methods in Nonlinear Analysis (Springer, Berlin, 2005)

    MATH  Google Scholar 

  7. 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)

    Article  MATH  Google Scholar 

  8. E.N. Dancer, Y. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities. J. Differ. Equ. 114, 434–475 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. E.N. Dancer, Y. Du, On sign-changing solutions of certain semilinear elliptic problems. Appl. Anal. 56, 193–206 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. E.N. Dancer, Z. Zhang, Dynamics of Lotka–Volterra competition systems with large interaction. J. Differ. Equ. 182(2), 470–489 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian. Port. Math. 58(3), 339–377 (2001)

    MathSciNet  MATH  Google Scholar 

  14. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 2001)

    MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Z. Liu, A topological property of the boundary of bounded open sets in \(\mathbb{R}^{2}\). J. Shandong Univ. 29(3), 299–303 (1994)

    MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Perera, M. Schechter, Double resonance problems with respect to the Fucik spectrum. Indiana Univ. Math. J. 52(1), 1–17 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. K. Perera, M. Schechter, Computation of critical groups in Fucik resonance problems. J. Math. Anal. Appl. 279(1), 317–325 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Schechter, The Fucik spectrum. Indiana Univ. Math. J. 43(4), 1139–1157 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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)

    MATH  Google Scholar 

  22. G.T. Whyburn, Topological Analysis (Princeton University Press, Princeton, 1958)

    MATH  Google Scholar 

  23. Z. Zhang, S. Li, On sign-changing and multiple solutions of the p-Laplacian. J. Funct. Anal. 197(2), 447–468 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. W. Zou, Sign-Changing Critical Point Theory (Springer, Berlin, 2008)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Zhang, Z. (2013). Sign-Changing Solutions. In: Variational, Topological, and Partial Order Methods with Their Applications. Developments in Mathematics, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30709-6_8

Download citation

Publish with us

Policies and ethics