Divergence form nonlinear nonsmooth elliptic equations with locally arbitrary growth conditions and nonlinear maximal regularity

This is a simplification of our prior work on the existence theory for the Rosseland-type equations. Inspired by the Rosseland equation in the conduction-radiation coupled heat transfer, we use the locally arbitrary growth conditions instead of the common global restricted growth conditions. In the Lebesgue square integrable space, the solution to the linear elliptic equation depends continuously on the coefficients matrix. This is a simple version of the maximal regularity. There exists a fixed point for the linearized map (compact and continuous) in a closed convex set. We also consider the Galerkin method.