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An Abstract Theorem in Nonlinear Analysis

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Book cover Analysis and Topology in Nonlinear Differential Equations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 85))

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Abstract

An elementary proof of the existence of multiple solutions of nonlinear operator equations is given. We show the existence, depending on a parameter in the equation, of either exactly two or at least four solutions for these equations.

Mathematics Subject Classification (2010). 35J60, (35J25, 47J30, 58E05).

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References

  1. Ambrosetti, A. and Prodi, G., On the inversion of some differentiable mappings with singularities between Banach Spaces, Ann. Math. Pura Appl. 93 (1973), 231–247

    Article  MathSciNet  Google Scholar 

  2. Amann, H. and Hess, P., A multiplicity result for a class of elliptic boundary value problems, Proc Roy. Soc. Edinb. 84 (1979), 145–151.

    Article  MATH  MathSciNet  Google Scholar 

  3. Berger, M. and Church, P., An application of singularity theory to nonlinear partial differential equations, Bull. A.M.S. 178 (1983) 119–126.

    Google Scholar 

  4. Berger, M.S. and Podolak, E., On the solutions of nonlinear Dirichlet problems, Indiana Univ. Math. Jour. 24 (1975) 837–846.

    Article  MATH  MathSciNet  Google Scholar 

  5. Breuer, B., McKenna, P.J., Plum, M., Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof. J. Differential Equations 195 (2003), 243–269.

    Article  MATH  MathSciNet  Google Scholar 

  6. Brezis, H. and Nirenberg, L., Characterization of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Scuola Norm. Sup. Pisa, Cl. Sci.,, 5 (1978) 225–326.

    Google Scholar 

  7. Cafagna, V. and Tarantello, G., Multiple solutions for some semilinear elliptic equations, Math. Ann. 276 (1987), 643–656.

    Article  MATH  MathSciNet  Google Scholar 

  8. Cal Neto, José Teixeira and Tomei, Carlos, Numerical analysis of semilinear elliptic equations with finite spectral interaction, J. Math. Anal. Appl. 395 (2012) 63–77.

    Article  MATH  MathSciNet  Google Scholar 

  9. Castro, A. and Gadam, S., The Lazer McKenna conjecture for radial solutions in the R N ball.Electron. J. Differential Equations 1993, approx. 6 pp.

    Google Scholar 

  10. Castro, A., and Shivaji, R., Multiple solutions for a Dirichlet problem with jumping nonlinearities, II Jour. Math. Anal. Appl., 133 (1988), 509–528.

    Google Scholar 

  11. Costa, D.G., de Figueiredo, D.G., Srikanth, P.N., The exact number of solutions for a class of ordinary differential equations through Morse index computation. J. Differential Equations 96 (1992), no. 1, 185–199.

    Article  MATH  MathSciNet  Google Scholar 

  12. Dancer, E.N., On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. Pures et Appl. 57 (1978), 351–366.

    MATH  MathSciNet  Google Scholar 

  13. Dalbono, Francesca and McKenna, P.J., Multiplicity results for a class of asymmetric weakly coupled systems of second-order differential equations, Boundary Value Problems 2 (2005) 129–151.

    Google Scholar 

  14. Dancer, E.N., A counterexample to the Lazer–McKenna conjecture, Nonlinear Analysis, T.M.A. 13 (1989) 19–22.

    Google Scholar 

  15. Dancer, E.N. and Yan, S., On the superlinear Lazer–McKenna conjecture, J. Differential Equations 210 (2005), 317–351.

    Article  MATH  MathSciNet  Google Scholar 

  16. Dancer, E.N. and Yan, S.S., On the superlinear Lazer–McKenna conjecture: Part II Comm. P.D.E. 30 (2005) 1331–1358.

    Google Scholar 

  17. Gallouet, T., and Kavian, O., Resultats d’existence et de non existence pour certain problems demilinearies a l’infini, Ann. Fac. Sci. Toulouse, 3 (1981), 201–246.

    Article  MATH  MathSciNet  Google Scholar 

  18. Hammerstein, A., Nichtlineare Integralgleichungen nebst Anwendungen, Acta Mat. 54 (1930), 117–176.

    Article  MATH  MathSciNet  Google Scholar 

  19. Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, Math Ann., (1982), 493–514.

    Google Scholar 

  20. Kazdan, J.L. and Warner, F., Remarks on quasilinear elliptic equations., Comm. Pure and Appl. Math., 28 (1975), 567–597.

    MATH  MathSciNet  Google Scholar 

  21. Lazer, A.C. and McKenna, P.J., On the number of solutions of a nonlinear Dirichlet problem, Jour. Math. Anal. Appl., 84 (1981), 282–294.

    Article  MATH  MathSciNet  Google Scholar 

  22. Lazer, A.C. and McKenna, P.J., Some multiplicity results for a class of semilinear elliptic and parabolic boundary value problems, J. Math. Anal. Appl., 107 (1985), 371–395.

    Article  MATH  MathSciNet  Google Scholar 

  23. Lazer, A.C. and McKenna, P.J., On a conjecture related to the number of solutions of a nonlinear Dirichlet problem, Proc Roy. Soc. Edin., 95 A (1983), 275–283.

    Google Scholar 

  24. Lazer, A.C. and McKenna, P.J., Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues II, Comm. P.D.E., 11 (1986), 1653–1676.

    Google Scholar 

  25. Lazer, A.C., and McKenna, P.J., A symmetry theorem and applications to nonlinear partial differential equations, J. Diff. Equations, 71 (1988).

    Google Scholar 

  26. Lazer, A.C. and McKenna, P.J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990), 537–578.

    Article  MATH  MathSciNet  Google Scholar 

  27. Lupo, D., Solimini, S., Srikanth, P.N., Multiplicity results for an ODE problem with even nonlinearity. Nonlinear Anal. T.M.A. 12 (1988), 657–673.

    Google Scholar 

  28. Manes, A., and Micheletti, A.M., Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine. Boll. Un. Mat. Ital. 7 (1973), 285–301.

    MATH  MathSciNet  Google Scholar 

  29. Molle, Riccardo and Passaseo, Donato, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities, J. Funct. Anal., 259, (2010), 2253–2295.

    Google Scholar 

  30. Nirenberg, L., Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc., 4 (1981), 267–302.

    Article  MATH  MathSciNet  Google Scholar 

  31. Ruf, B., Remarks and generalizations related to a recent multiplicity result of A. Lazer and P. McKenna, Nonlinear Anal. T.M.A. 9, (1985), 1325–1330.

    Google Scholar 

  32. Ruf, B., On nonlinear elliptic problems with jumping nonlinearities, Annali di Matematica pura ed applicata 78, (1980), 133–151.

    Google Scholar 

  33. Ruf, B. and Solimini, S. On a class of superlinear Sturm–Liouville problems with arbitrarily many solutions. SIAM J. Math. Anal., 17 (1986), 761–77.

    Article  MATH  MathSciNet  Google Scholar 

  34. Solimini, S., Some remarks on the number of solutions of some nonlinear elliptic equations, Anal. Nonlin., I.H.P., 2 (1985), 143–156.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33124, USA

    A. C. Lazer

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    P. J. McKenna

  3. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    P. J. McKenna

Authors
  1. A. C. Lazer
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  2. P. J. McKenna
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Editor information

Editors and Affiliations

  1. IMECC-UNICAMP, Cidade Universitaria Instituto de Matematica, Estatistica e C, Campinas, São Paulo, Brazil

    Djairo G de Figueiredo

  2. Departamento de Matemática - CCEN, Universidade Federal da Paraíba, João Pessoa, Paraíba, Brazil

    João Marcos do Ó

  3. Departamento de Matemática, Pontifícia Universidade Católica do Rio, Rio de Janeiro, Rio de Janeiro, Brazil

    Carlos Tomei

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Lazer, A.C., McKenna, P.J. (2014). An Abstract Theorem in Nonlinear Analysis. In: de Figueiredo, D., do Ó, J., Tomei, C. (eds) Analysis and Topology in Nonlinear Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-04214-5_18

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