  We consider a strongly nonlinear elliptic problem with the homogeneous
Dirichlet boundary condition. The growth and the coercivity of the elliptic
operator is assumed to be indicated by an inhomogeneous anisotropic
$\mathcal{N}$-function. First, an existence result is shown under the
assumption that the $\mathcal{N}$-function or its convex conjugate satisfies
$\Delta_2$-condition. The second result concerns the homogenization process for
families of strongly nonlinear elliptic problems with the homogeneous Dirichlet
boundary condition under above stated conditions on the elliptic operator,
which is additionally assumed to be periodic in the spatial variable.
