Skip to main content

A Priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations

  • Chapter
  • First Online:
Djairo G. de Figueiredo - Selected Papers

Abstract

We are interested here in the existence of positive solutions of the problem.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. S. Agmon, A. Douglis and L. Nirenberg, Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions, I {Comm. Pure Appl. Math., Vol. 12, 1959, pp. 623-727).

    Google Scholar 

  2. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces (S.I.A.M., Vol. 18, 1976, pp. 620-709).

    Google Scholar 

  3. A. Ambrosetti and P. H. Rabinowitz, Dual Variational Methods in Critical Point Theory and Applications (J. Functional Analysis, Vol. 14, 1973, pp. 369-381).

    Google Scholar 

  4. C. Bandle, Existence Theorems, Qualitative Results and a priori bounds for a Class of Nonlinear Dirichlet Problems (Arch. Rat. Mech. Anal., Vol. 58, 1975, pp. 219-238).

    Google Scholar 

  5. T. B. Benjamin, A Unified Theory of Conjugate Flows(Phil. Trans. Royal Soc., Vol. 269 A, 1971, pp. 587-643).

    Google Scholar 

  6. H. Berestycki and P. L. Lions, Une methode locale pour l’existence de solutions positives de problemes semilineaires dans R” (J. Analyse Mathematique, Vol. 38, 1980. pp. 144-187).

    Google Scholar 

  7. H. Brezis and T. Kato. Remarks on the Schrodinger Operator With Singular Complex Potentials (J. Math. pures et appl., Vol. 58, 1978. p. 137-151).

    Google Scholar 

  8. H. Brezis and R. E. L. Turner, On a Class of Superlinear Elliptic Problems (Comm. in P.D.E., Vol. 2, 1977, pp. 601-614).

    Google Scholar 

  9. D. S. Cohen, Positive Solutions of Nonlinear Eigenvalue Problems: Applications to Nonlinear Reactor Dynamics (Arch. Rat. Mech. Anal., Vol. 26, 1967, pp. 305-315).

    Google Scholar 

  10. M. G. Crandall and P. H. Rabinowitz, Some Continuation and Variational Methods for Positive Solutions of Nonlinear Elliptic Eigenvalue Problems (Arch. Rat. Mech. Anal., Vol. 58, 1975, pp. 207-218).

    Google Scholar 

  11. D. de Figueiredo, P. L. Lions and R. D. Nussbaum, Estimations a priori pour les solutions positives de problemes elliptiques semi-lineaires (C. R. Acad. Sc., Paris, serie A, 1980).

    Google Scholar 

  12. I. M. Gelfand, Some Problems in the Theory of Quasilinear Equations (Amer. Math. Soc. Transl., Vol. 29, 1963, pp. 295-381).

    Google Scholar 

  13. B. Gidas, Wei-Ming Ni and L. Nirenberg, Symmetry and Related Properties via the Maximum Principle (Comm. Math. Phys., Vol. 68, 1979, pp. 209-243).

    Google Scholar 

  14. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, New York, 1977.

    Google Scholar 

  15. D. D. JOSEPH and T. S. LUNDGREN. Quasilinear Dirichlet Problems Driven by Positive Sources (Arch. Rat. Mech. Anal., Vol. 49, 1972, pp. 241-269).

    Google Scholar 

  16. M. A. Krasnoselskii. Fixed Points of Cone-Compressing of Cone-Extending Operators (Soviet Math. Dokl., Vol. 1, 1960, pp. 1285-1288).

    Google Scholar 

  17. M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, the Netherlands, 1964.

    Google Scholar 

  18. O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated by Scripta Technica, Academic Press, New York, 1968.

    Google Scholar 

  19. J. Leray and J. Schauder, Topologie et equations fonctionnelles (Ann. Scient. Ec. Norm. Sup., Vol. 51, 1934, pp. 45-78).

    Google Scholar 

  20. J. L. Lions and G. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. I, Springer, Berlin, 1970.

    Google Scholar 

  21. P. L. Lions, A priori Estimates and Existence Results for Some Semilinear Elliptic Problems, M.R.C. Technical Summary Report, 1975, Madison, Wisconsin, 1979.

    Google Scholar 

  22. L. Nirenberg, Topics in Non Linear Functional Analysis, N. Y.U. Courant Institute Lecture Notes, New York, 1974.

    Google Scholar 

  23. R. D. Nussbaum, Positive Solutions of Nonlinear Elliptic boundary Value Problems (J. Math. Anal. Appl., Vol. 51, 1975, pp. 461-482).

    Google Scholar 

  24. R. D. NUSSBAUM, A Global Bifurcation Theorem With Applications to Functional Differential Equations (J. Functional Analysis, Vol. 19, 1975, pp. 319-338).

    Google Scholar 

  25. R. D. Nussbaum, Uniqueness and Nonuniqueness for Periodic Solutions of x’ (i) = -g(x(t–1)) (J. Differential Equations, Vol. 34, 1979, pp. 24-54).

    Google Scholar 

  26. R. D. NUSSBAUM, Periodic Solutions of Some Nonlinear Autonomous Functional Differential Equations, II (J. Differential Equations, Vol. 14, 1973, pp. 368-394).

    Google Scholar 

  27. S. I. Pohozaev, Eigenfunctions of Δu+f(u) = 0 (Soviet Math. Dokl., Vol. 6, 1965, pp. 1408-1411).

    Google Scholar 

  28. P. H. Rabinowitz, Variational Methods for Nonlinear Eigenvalue Problems (in Eigenvalues and Nonlinear Problems, C.I.M.E., Ediz. Cremonese, Rome, 1979.

    Google Scholar 

  29. P. Rabinowitz, Some Aspects of Nonlinear Eigenvalue Problems (Rocky Mountain J., Vol. 3, 1973, pp. 161-202).

    Google Scholar 

  30. R. E. L. TURNER, A Priori Bounds for Positive Solutions of Nonlinear Elliptic Equations in two Variables (Duke Math. J., Vol. 41, 1974, p. 759-774).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1982 Gauthier-Villars

About this chapter

Cite this chapter

De Figueiredo, D.G., Lions, P.L., Nussbaum, R.D. (1982). A Priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations. In: Costa, D. (eds) Djairo G. de Figueiredo - Selected Papers. Springer, Cham. https://doi.org/10.1007/978-3-319-02856-9_11

Download citation

Publish with us

Policies and ethics