Strongly nonlinearmultivalued systems involving  singular $\Phi$-Laplacian operators

Francesca Papalini

doi:10.3934/cpaa.2010.9.1025

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2010, Volume 9, Issue 4: 1025-1040. Doi: 10.3934/cpaa.2010.9.1025

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Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators

Francesca Papalini^1,

1.
Polytechnic University of Marche, Department of Mathematical Sciences, Via Brecce Bianche, Ancona

Received: August 31, 2009

Revised: December 31, 2009

Published: June 2010

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Abstract

In this paper we study two vector problems with homogeneous Dirichlet boundary conditions for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is governed by a nonlinear differential operator of the form $x\mapsto (k(t)\Phi(x'))'$, where $k\in C(T, R_+)$ and $\Phi$ is an increasing homeomorphism defined on a bounded domain. In this problem the maximal monotone term need not be defined everywhere in the state space $R^N$, incorporating into our framework differential variational inequalities. The second problem is governed by the more general differential operator of the type $x\mapsto (a(t,x)\Phi(x'))'$, where $a(t,x)$ is a positive and continuous scalar function. In this case the maximal monotone term is required to be defined everywhere.

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Mathematics Subject Classification: Primary: 34B15; Secondary: 34A60.

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