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

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.