## Abstract

This paper deals with the problem

$u\in {\mathcal{\mathcal{K}}}_{{u}_{*},\psi}(\mathrm{\Omega}),$$\forall v\in {\mathcal{\mathcal{K}}}_{{u}_{*},\psi}(\mathrm{\Omega}):{\displaystyle {\int}_{\mathrm{\Omega}}}{\displaystyle \sum _{i=1}^{n}}[{a}_{i}(x,Du)-{f}^{i}]{D}_{i}(u-v)dx\u2a7d{\displaystyle {\int}_{\mathrm{\Omega}}}f(u-v)\mathit{d}x,$

where

$\{\begin{array}{cc}& {\mathcal{\mathcal{K}}}_{{u}_{*},\psi}(\mathrm{\Omega})=\{v\in {u}_{*}+{W}_{0}^{1,({p}_{i})}(\mathrm{\Omega}):\sum _{i=1}^{n}{a}_{i}(x,Du){D}_{i}v\in {L}^{1}(\mathrm{\Omega})\text{and}v\u2a7e\psi ,\text{a.e.}\mathrm{\Omega}\},\hfill \\ & {u}_{*}\in {W}^{1,({p}_{i})}(\mathrm{\Omega}),\theta =\mathrm{max}\{{u}_{*},\psi \}\in {u}_{*}+{W}_{0}^{1,({p}_{i})}(\mathrm{\Omega}),\hfill \\ & f\in {L}^{{({\overline{p}}^{*})}^{\prime}}(\mathrm{\Omega}),{f}^{i}\in {L}^{{p}_{i}^{\prime}}(\mathrm{\Omega}),i=1,\mathrm{\dots},n,\hfill \end{array}$

and the Carathéodory functions ${a}_{i}:\mathrm{\Omega}\times {\mathbb{R}}^{n}\to \mathbb{R}$, $i=1,\mathrm{\dots},n$, satisfy some coercivity condition.
We assume that the function $\theta =\mathrm{max}\{{u}_{*},\psi \}$ makes ${a}_{i}(x,D\theta )$ to be more integrable than ${L}^{{p}_{i}^{\prime}}(\mathrm{\Omega})$, $i=1,\mathrm{\dots},n$, and then we prove that the solution *u* enjoys higher integrability.

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