Local bounds for nonlinear higher-order vector fields for the p-Laplace equation

We study higher regularity for weak solutions of the $p$-Laplace equation $-\Delta_p u = f$ in a domain $\Omega \subset \mathbb{R}^n$ for $p$ sufficiently close to 2. For $m \ge 3$, assuming that $f$ satisfies suitable Sobolev and H\"older regularity conditions, we prove that the nonlinear quantity $|\nabla u|^{m-2}\nabla u$ belongs to $W^{m-1,q}_{{loc}}(\Omega)$, and that $|\nabla u|^{m-2} D^2u$ belongs to $W^{m-2,q}_{{loc}}(\Omega)$, for any $q\ge 2$. Furthermore, we obtain uniform $L^\infty$ bounds for the weighted $(m-1)$-th derivatives of $|\nabla u|^{m-2}\nabla u$ and the weighted $(m-2)$-th derivatives of $|\nabla u|^{m-2} D^2u$, providing quantitative control even near critical points of $\nabla u$.