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July  2010, 9(4): 1025-1040. doi: 10.3934/cpaa.2010.9.1025

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, Italy

Received  September 2009 Revised  January 2010 Published  April 2010

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.
Keywords: variational inequalities., Yosida approximation, singular $\Phi$-laplacian, usc and lsc multifunction, Dirichlet problem, maximal monotone map, Differential inclusion.
Mathematics Subject Classification: Primary: 34B15; Secondary: 34A6.
Citation: Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1025-1040. doi: 10.3934/cpaa.2010.9.1025
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