Proyecciones
Vol. 21, N o 3, pp. 199-224, December 2002.
Universidad Católica del Norte
Antofagasta - Chile

CRITICAL POINT THEOREMS AND
APPLICATIONS

HAFIDA BOUKHRISSE
and
MIMOUN MOUSSAOUI
University Mohamed I, Morocco

Abstract

We Consider the nonlinear Dirichlet problem:

where . W Î R ^N is a bounded open domain, F : W C R ® R is a carath´eodory function and D_uF(x; u) is the partial derivative of F. We are interested in the resolution of problem (1) when F is concave. Our tool is absolutely variational. Therefore, we state and prove a critical point theorem which generalizes many other results in the literature and leads to the resolution of problem (1). Our theorem allows us to express our assumptions on the nonlinearity in terms of F and not of Ñ F. Also, we note that our theorem doesn’t necessitate the verification of the famous compactness condition introduced by Palais-Smale or any of its variants.

Key words: Critical point theory, convexity conditions, Elliptic semilinear problem.

References

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Recieved : September, 2001.

Hafida Boukhrisse
Mathematical Department
Faculty of Sciences
University Mohamed I
P. O. Box 524
60000 Oujda
Morocco
e-mail : boukhrisse@operamail.com

and

Mimoun Moussaoui
Mathematical Department
Faculty of Sciences
University Mohamed I
P. O. Box 524
60000 Oujda
Morocco