Existence and homogenization of nonlinear elliptic systems in nonreflexive spaces

TL;DR

This paper establishes the existence and homogenization results for strongly nonlinear elliptic systems with anisotropic growth conditions described by an inhomogeneous N-function, under certain regularity assumptions.

Contribution

It provides new existence and homogenization results for nonlinear elliptic systems with anisotropic growth in nonreflexive spaces, extending previous theories.

Findings

Existence of solutions under Δ₂-condition on N-function or its conjugate.

Homogenization results for periodic nonlinear elliptic systems.

Applicability to nonreflexive anisotropic growth conditions.

Abstract

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.