Abstract
This paper contains a variational treatment of the Ambrosetti-Prodi problem, including the superlinear case. The main result extends previous ones by Kazdan-Warner, Amann-Hess, Dancer, K. C. Chang and de Figueiredo. The required abstract results on critical point theory of functionals in Hilbert space are all proved using Ekeland’s variational principle. These results apply as well to other superlinear elliptic problems provided an ordered pair of a sub- and a supersolution is exhibited.
Received January 1984.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ekeland, I. - “Non convex minimization problems”. Bull. AMS 1 (1979), pp. 443-474.
Hofer, H. - “Variational and Topological Methods in Partially Ordered Hilbert Spaces”. Math. Ann. 261 (1982), pp. 493-514.
Rabinowitz, P. - “Some aspects of critical point theory”. MRC Tech. Rep. #2465 (Jan. 1983).
Dancer, E. N. - “On the ranges of certain weakly nonlinear elliptic partial differential equations”. J. Math. Pures et Appl. 57 (1978), pp. 351-366.
Chang, K. C. - personal communication.
de Figueiredo, D. G. - “On the superlinear Ambrosetti-Prodi problem”. MRC Tech. Rep. #2522 (May 1983).
Gidas, B. and Spruck, J. - “A priori bounds for positive solutions of nonlinear elliptic equations”. Comm. PDE 6 (1981), pp. 883-901.
Brezis, H. and Turner, R.E.L. - “On a class of superlinear elliptic problems”. Comm. PDE 2 (1977), pp. 601-614.
Brezis, H. and Kato, T. - “Remarks on the Schrodinger operator with singular complex potentials”. J. Math. Pures et Appl. 58 (1979), pp. 137-151.
Kazdan, J. and Warner, F. W. - “Remarks on some quasilinear elliptic equations”. Comm. Pure Appl. Math. XXVIII (1975), pp. 567-597.
Amann, H. and Hess, P. - “A multiplicity result for a class of elliptic boundary value problems”. Proc. Royal Soc. Edinburgh 84A (1979), pp. 145-151.
Berestycki, H. - “Le nombre de solutions de certains problemes semilineaires elliptiques”. J. Fct. Anal.
Ambrosetti, A. and Rabinowitz, P. - “Dual variational methods in critical point theory and applications”. J. Fctl. Anal. 14 (1973), pp. 349-381.
Pucci, P. and Serrin, J. - “A mountain pass theorem”. To appear.
Willem, M. - “Lectures on Critical Point Theory”. Trabalho de Matematica n°199. Universidade de Brasilia (Feb. 1983).
Gilbarg, D. and Trudinger, N. S. - “Elliptic Partial Differential Equations”. Springer Verlag (1977).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Marcel Dekker, Inc.
About this chapter
Cite this chapter
de Figueiredo, D.G., Solimini, S. (1984). A Variational Approach to Superlinear Elliptic Problems. In: Costa, D. (eds) Djairo G. de Figueiredo - Selected Papers. Springer, Cham. https://doi.org/10.1007/978-3-319-02856-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-02856-9_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02855-2
Online ISBN: 978-3-319-02856-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)