Regularity for very weak solutions to elliptic equations of $p$-Laplacian type

We study the regularity problem with non-homogeneous terms of $p$-Laplacian type, which is a still unsolved problem for nonlinear elliptic equations. The main results of this work are obtained by three steps. First, we use the Hodge decomposition theorem to construct a suitable test function that satisfies the solution definition. Second, by combining the solution definition with the Hodge decomposition theorem, we establish a properly formulated inverse Hölder inequality to enhance the integrability of the very weak solutions. Finally, through an iterative process, we show that the considered very weak solutions can be improved to classical weak solutions.