Existence and homogenization of nonlinear elliptic systems in nonreflexive spaces

We consider a strongly nonlinear elliptic problem with the homogeneous Dirichletboundary condition. The growth and the coercivity of the elliptic operator isassumed to be indicated by an inhomogeneous anisotropicN-function. First, anexistence result is shown under the assumption that theN-function or its convexconjugate satisfies∆2-condition. The second result concerns the homogenizationprocess for families of strongly nonlinear elliptic problems with the homoge-neous Dirichlet boundary condition under above stated conditions on the ellipticoperator, which is additionally assumed to be periodic in the spatial variable.