Advertisement

The Fučík Spectrum for the Negative p-Laplacian with Different Boundary Conditions

  • Dumitru Motreanu
  • Patrick Winkert
Part of the Springer Optimization and Its Applications book series (SOIA, volume 68)

Abstract

This chapter represents a survey on the Fučík spectrum of the negative p-Laplacian with different boundary conditions (Dirichlet, Neumann, Steklov, and Robin). The close relationship between the Fučík spectrum and the ordinary spectrum is briefly discussed. It is also pointed out that for every boundary condition there exists a first nontrivial curve Open image in new window in the Fučík spectrum which has important properties such as Lipschitz continuity, being decreasing and a certain asymptotic behavior depending on the boundary condition. As a consequence, one obtains a variational characterization of the second eigenvalue λ 2 of the negative p-Laplacian with the corresponding boundary condition. The applicability of the abstract results is illustrated to elliptic boundary value problems with jumping nonlinearities.

Key words

Fučík spectrum p-Laplacian Boundary conditions Elliptic boundary value problems 

Mathematics Subject Classification

47A10 35J91 35K92 35J58 

References

  1. 1.
    Anane, A.: Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305(16), 725–728 (1987) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arias, M., Campos, J.: Radial Fučik spectrum of the Laplace operator. J. Math. Anal. Appl. 190(3), 654–666 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Arias, M., Campos, J., Cuesta, M., Gossez, J.-P.: An asymmetric Neumann problem with weights. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25(2), 267–280 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Arias, M., Campos, J., Gossez, J.-P.: On the antimaximum principle and the Fučik spectrum for the Neumann p-Laplacian. Differ. Integral Equ. 13(1–3), 217–226 (2000) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Các, N.P.: On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue. J. Differ. Equ. 80(2), 379–404 (1989) zbMATHCrossRefGoogle Scholar
  6. 6.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer Monographs in Mathematics. Springer, New York (2007) zbMATHGoogle Scholar
  7. 7.
    Carl, S., Motreanu, D.: Constant-sign and sign-changing solutions for nonlinear eigenvalue problems. Nonlinear Anal. 68(9), 2668–2676 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Carl, S., Motreanu, D.: Multiple and sign-changing solutions for the multivalued p-Laplacian equation. Math. Nachr. 283(7), 965–981 (2010) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Carl, S., Motreanu, D.: Multiple solutions of nonlinear elliptic hemivariational problems. Pac. J. Appl. Math. 1(4), 381–402 (2008) Google Scholar
  10. 10.
    Carl, S., Motreanu, D.: Sign-changing and extremal constant-sign solutions of nonlinear elliptic problems with supercritical nonlinearities. Commun. Appl. Nonlinear Anal. 14(4), 85–100 (2007) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Carl, S., Motreanu, D.: Sign-changing solutions for nonlinear elliptic problems depending on parameters. Int. J. Differ. Equ. 2010, 536236 (2010), pp. 33 MathSciNetGoogle Scholar
  12. 12.
    Carl, S., Perera, K.: Sign-changing and multiple solutions for the p-Laplacian. Abstr. Appl. Anal. 7(12), 613–625 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cuesta, M., de Figueiredo, D.G., Gossez, J.-P.: The beginning of the Fučik spectrum for the p-Laplacian. J. Differ. Equ. 159(1), 212–238 (1999) zbMATHCrossRefGoogle Scholar
  14. 14.
    Dancer, E.N.: Generic domain dependence for nonsmooth equations and the open set problem for jumping nonlinearities. Topol. Methods Nonlinear Anal. 1(1), 139–150 (1993) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dancer, E.N.: On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc. R. Soc. Edinb. A 76(4), 283–300 (1976/77) MathSciNetGoogle Scholar
  16. 16.
    de Figueiredo, D.G., Gossez, J.-P.: On the first curve of the Fučik spectrum of an elliptic operator. Differ. Integral Equ. 7(5–6), 1285–1302 (1994) zbMATHGoogle Scholar
  17. 17.
    Deng, S.-G.: Positive solutions for Robin problem involving the p(x)-Laplacian. J. Math. Anal. Appl. 360(2), 548–560 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Drábek, P.: Solvability and Bifurcations of Nonlinear Equations. Longman Scientific & Technical, Harlow (1992) zbMATHGoogle Scholar
  19. 19.
    Fernández Bonder, J., Rossi, J.D.: Existence results for the p-Laplacian with nonlinear boundary conditions. J. Math. Anal. Appl. 263(1), 195–223 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fučík, S.: Boundary value problems with jumping nonlinearities. Čas. Pěst. Mat. 101(1), 69–87 (1976) zbMATHGoogle Scholar
  21. 21.
    Fučík, S.: Solvability of Nonlinear Equations and Boundary Value Problems. Reidel, Dordrecht (1980) zbMATHGoogle Scholar
  22. 22.
    Lê, A.: Eigenvalue problems for the p-Laplacian. Nonlinear Anal. 64(5), 1057–1099 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Lindqvist, P.: On the equation \(\operatorname{div}(\vert\nabla u\vert ^{p-2}\nabla u)+ \lambda\vert u\vert^{p-2}u=0\). Proc. Am. Math. Soc. 109(1), 157–164 (1990) MathSciNetzbMATHGoogle Scholar
  24. 24.
    Margulies, C.A., Margulies, W.: An example of the Fučik spectrum. Nonlinear Anal. 29(12), 1373–1378 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Marino, A., Micheletti, A.M., Pistoia, A.: A nonsymmetric asymptotically linear elliptic problem. Topol. Methods Nonlinear Anal. 4(2), 289–339 (1994) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Martínez, S.R., Rossi, J.D.: Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition. Abstr. Appl. Anal. 7(5), 287–293 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Martínez, S.R., Rossi, J.D.: On the Fučik spectrum and a resonance problem for the p-Laplacian with a nonlinear boundary condition. Nonlinear Anal. 59(6), 813–848 (2004) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Micheletti, A.M.: A remark on the resonance set for a semilinear elliptic equation. Proc. R. Soc. Edinb. A 124(4), 803–809 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Micheletti, A.M., Pistoia, A.: A note on the resonance set for a semilinear elliptic equation and an application to jumping nonlinearities. Topol. Methods Nonlinear Anal. 6(1), 67–80 (1995) MathSciNetzbMATHGoogle Scholar
  30. 30.
    Motreanu, D., Tanaka, M.: Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities. J. Differ. Equ. 249(11), 3352–3376 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Motreanu, D., Tanaka, M.: Existence of solutions for quasilinear elliptic equations with jumping nonlinearities under the Neumann boundary condition. Calc. Var. Partial Differ. Equ. 43(1–2), 231–264 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Motreanu, D., Winkert, P.: On the Fuc̆ik spectrum for the p-Laplacian with a robin boundary condition. Nonlinear Anal. 74(14), 4671–4681 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Pardalos, P.M., Rassias, T.M., Khan, A.A. (Guest eds.): Nonlinear Analysis and Variational Problems. In Honor of George Isac. Springer Optimization and Its Applications, vol. 35. Springer, New York (2010), xxviii+490 pp. zbMATHGoogle Scholar
  34. 34.
    Pistoia, A.: A generic property of the resonance set of an elliptic operator with respect to the domain. Proc. R. Soc. Edinb. A 127(6), 1301–1310 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Schechter, M.: The Fučík spectrum. Indiana Univ. Math. J. 43(4), 1139–1157 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Winkert, P.: Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values. Adv. Differ. Equ. 15(5–6), 561–599 (2010) MathSciNetzbMATHGoogle Scholar
  37. 37.
    Winkert, P.: Multiple solution results for elliptic Neumann problems involving set-valued nonlinearities. J. Math. Anal. Appl. 377(1), 121–134 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Winkert, P.: Sign-changing and extremal constant-sign solutions of nonlinear elliptic Neumann boundary value problems. Bound. Value Probl. 2010, 139126 (2010), pp. 22 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de PerpignanPerpignan CedexFrance
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations