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Degenerate elliptic boundary value problems with asymptotically linear nonlinearity

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Abstract

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems with asymptotically linear nonlinearity which include as particular cases the Dirichlet and Robin problems. Our approach is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the variational method, we prove existence and uniqueness theorems for our problem. The results here extend three earlier theorems due to Ambrosetti and Prodi to the degenerate case.

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References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)

    MATH  Google Scholar 

  2. Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620–709 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosetti, A., Prodi, G.: On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. 93, 231–247 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  5. Berger, M.S., Podolak, E.: On the solutions of a nonlinear Dirichlet problem. Indiana Univ. Math. J. 24, 837–846 (1974/75)

    Article  MathSciNet  Google Scholar 

  6. Castro, A., Lazer, A.C.: Critical point theory and the number of solutions of a nonlinear Dirichlet problem. Ann. Mat. Pura Appl. 120, 113–137 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang, K.C.: Methods in Nonlinear Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  8. de Figueiredo, D.G.: Positive solutions of semilinear elliptic problems. In: Differential Equations, Lecture Notes in Mathematics, vol. 957, pp. 34–87. Springer, Berlin (1982)

    Chapter  Google Scholar 

  9. Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1969)

    MATH  Google Scholar 

  10. Dolph, C.L.: Nonlinear integral equations of the Hammerstein-type. Trans. Am. Math. Soc. 66, 289–307 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hammerstein, A.: Nichtlineare Integralgleichungen nebst Anwendungen. Acta Math. 54, 117–176 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1985)

    MATH  Google Scholar 

  13. Leray, J., Schauder, J.: Topologie et équations fonctionelles. Ann. Sci. Ec. Norm. Super. 51, 45–78 (1934)

    MathSciNet  MATH  Google Scholar 

  14. Mawhin, J.: Topological degree methods in nonlinear boundary value problems. In: CBMS Regional Conference Series in Mathematics, vol. 40. American Mathematical Society, Providence (1979)

    Google Scholar 

  15. Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Institute of Mathematical Sciences, New York University, New York (1974)

    MATH  Google Scholar 

  16. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin (1996)

    Book  MATH  Google Scholar 

  17. Taira, K.: Semigroups, Boundary Value Problems and Markov Processes. Springer, Berlin (2004)

    MATH  Google Scholar 

  18. Taira, K.: Degenerate elliptic boundary value problems with indefinite weights. Mediterr. J. Math. 5, 133–162 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Taira, K.: Degenerate elliptic boundary value problems with asymmetric nonlinearity. J. Math. Soc. Jpn. 62, 431–465 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Taira, K.: Semilinear degenerate elliptic boundary value problems at resonance. Ann. Univ. Ferrara 56, 369–392 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Taira, K.: Multiple solutions of semilinear degenerate elliptic boundary value problems. Math. Nachr. 284, 105–123 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Thom, R.: Les singularités des applications différentiables. Ann. Inst. Fourier 6, 43–87 (1955–1956)

    MathSciNet  Google Scholar 

  23. Zeidler, E.: Nonlinear Functional Analysis and Its Applications I. Springer, New York (1986)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan

    Kazuaki Taira

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  1. Kazuaki Taira
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Correspondence to Kazuaki Taira.

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Dedicated to Professor Hiroshi Fujita on the occasion of his 80th birthday.

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Taira, K. Degenerate elliptic boundary value problems with asymptotically linear nonlinearity. Rend. Circ. Mat. Palermo 60, 283–308 (2011). https://doi.org/10.1007/s12215-011-0052-4

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  • DOI: https://doi.org/10.1007/s12215-011-0052-4

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