An Agmon Douglis-Nirenberg type result for some non linear equations

Abstract

We consider local distributional solutions $u\,\epsilon\, W_{loc}^{1,r}\left(\Omega\right)$of non linear elliptic equations of the type \[ -divA\left(x,Du\right)=-div\, f\left(x\right)+g\left(x\right) \] and we prove that $u\,\epsilon\, W_{loc}^{2,r}\left(\Omega\right)$ when r is sufficiently close to 2 which is the exponent related to the growth conditions of the operator (see assumptions (2) and (3)).