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Nonsmooth critical point theory and nonlinear elliptic equations at resonance

Part of: Variational problems in infinite-dimensional spaces Elliptic equations and systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  09 April 2009

Nikolaos C. Kourogenis  and
Nikolaos S. Papageorgiou
Nikolaos S. Papageorgiou
Affiliation:
National Technical UniversityDepartment of Mathematics Zografou Campus Athens 157 80Greece e-mail: npapg@math.ntua.gr
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Abstract

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In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

Keywords

Nonsmooth critical point theorylocally Lipschitz functionsubdifferentiallinkingnonsmooth Palais-Smale conditionnonsmooth Cerami conditionMountain pass theoremSaddle point theoremproblems at resonancep-Laplacianfirst eigenvalue

MSC classification

Secondary: 35J20: Variational methods for second-order elliptic equations 35R70: Partial differential equations with multivalued right-hand sides 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 2 , October 2000 , pp. 245 - 271
Copyright
Copyright © Australian Mathematical Society 2000

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