Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-20T12:53:34.628Z Has data issue: false hasContentIssue false

On the Dirichlet problem of elliptic type

Published online by Cambridge University Press:  17 April 2009

Marek Galewski
Affiliation:
Faculty of Mathematics, University of Lodz, Banacha 22, 90–238 Lodz, Poland, e-mail: galewski@math.uni.lodz.pl
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate the existence of solutions and their stability for elliptic Dirichlet problems with nonlinearity of a convex-concave type. By relating the primal action and the dual action functionals on certain subsets of their domains we get the existence of solutions which are further stable with respect to a numerical parameter. We allow also for the differential operator to depend on a numerical parameter.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Ekeland, I. and Temam, R., Convex analysis and variational problems (North-Holland, Amsterdam, 1976).Google Scholar
[2]Galewski, M., ‘Existence, stability and approximation of solutions for a certain class of Nonlinear BVP's’, Nonlinear Anal. 65 (2006), 159174.Google Scholar
[3]Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order, Classics in Mathematics, (Reprint of the 1998 edition) (Springer–Verlag, Berlin, 2001).Google Scholar
[4]Magrone, P., ‘On a class of semilinear elliptic equations with potential changing sign’, Dynam. Systems Appl. 9 (2000), 459467.Google Scholar
[5]Mawhin, J., Problems de Dirichlet variationnels non lineaires (Les Presses de l'Universite de Montreal, Canada, 1987).Google Scholar
[6]Orpel, A., ‘On the existence of positive solutions and their continuous dependence on functional parameters for some class of elliptic problems’, J. Differential Equations 204 (2004), 247264.Google Scholar
[7]Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS, Regional Conference Series in Mathematics 65 (American Mathematical Society, Providence, RI, 1986).Google Scholar
[8]Rockafellar, R.T., Contributions to Nonlinear Functional Analysis, in, (Zarantonello, E., Editor) (Academic Press, New York, 1971), pp. 215236.Google Scholar
[9]Walczak, S., ‘On the continuous dependance on parameters of solutions of the Dirichlet problem. Part I Coercive Case, Part II. The Case of Saddle Points’, Bulletin de la Case des Sciences de l'Academie Royale de Beligique 6 (1995), 247273.Google Scholar