Let $\Omega$ be a domain of ${\mathbb R}^N$, $N>2.$ We establish higher integrability for solutions $u \in W^{1,p}_{\text{loc}}(\Omega)$ of nonlinear PDEs whose prototype~is \begin{equation*} \text{div\,}[|\nabla u|^{p-2}\nabla u +B(x)|u|^{p-2}u]=\text{div\,}(|F|^{p-2}F) \end{equation*} with $1 < p < N$. We assume that the vector field $B\colon\Omega\to{\mathbb R}^N$ belongs to the Marcinkiewicz space $L^{\frac N{p-1},\infty}$. We prove that $F\in L^r\text{loc}(\Omega,{\mathbb R}^N)$, $p < r < N$, implies $u\in L^{r^*} \text{loc}(\Omega).$