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 $-{\mathrm{\Delta }}_{p}u=f$ in a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ for $p$ sufficiently close to 2. For $m\ge 3$, assuming that $f$ satisfies suitable Sobolev and Hölder regularity conditions, we prove that the nonlinear quantity $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{m-2}\mathrm{\nabla }u$ belongs to $W_{loc}^{m-1,q}(\mathrm{\Omega })$, and that $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{m-2}{D}^{2}u$ belongs to $W_{loc}^{m-2,q}(\mathrm{\Omega })$, for any $q\ge 2$. Furthermore, we obtain uniform $L^{\mathrm{\infty }}$ bounds for the weighted $(m-1)$-th derivatives of $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{m-2}\mathrm{\nabla }u$ and the weighted $(m-2)$-th derivatives of $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{m-2}{D}^{2}u$, providing quantitative control even near critical points of $\mathrm{\nabla }u$.