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Multiple solutions for nonlinear discontinuous elliptic problems near resonance

Nikolaos KourogenisNikolaos Papageorgiou — 1999

Colloquium Mathematicae

We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when λ λ 1 from the left, λ 1 being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.

Discontinuous quasilinear elliptic problems at resonance

Nikolaos KourogenisNikolaos Papageorgiou — 1998

Colloquium Mathematicae

In this paper we study a quasilinear resonant problem with discontinuous right hand side. To develop an existence theory we pass to a multivalued version of the problem, by filling in the gaps at the discontinuity points. We prove the existence of a nontrivial solution using a variational approach based on the critical point theory of nonsmooth locally Lipschitz functionals.

Periodic solutions for quasilinear vector differential equations with maximal monotone terms

Nikolaos C. KourogenisNikolaos S. Papageorgiou — 1997

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider a quasilinear vector differential equation with maximal monotone term and periodic boundary conditions. Approximating the maximal monotone operator with its Yosida approximation, we introduce an auxiliary problem which we solve using techniques from the theory of nonlinear monotone operators and the Leray-Schauder principle. To obtain a solution of the original problem we pass to the limit as the parameter λ > 0 of the Yosida approximation tends to zero.

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