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We significantly improve the existence results of [Csat\\'o and Dacorogna, A Dirichlet problem involving the divergence operator, \\textit{Ann. Inst. H. Poincar\\'e Anal. Non Lin\\'eaire}, 33 (2016), 829--848], where this equation has been considered for the first time. 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We significantly improve the existence results of [Csat\\'o and Dacorogna, A Dirichlet problem involving the divergence operator, \\textit{Ann. Inst. H. Poincar\\'e Anal. Non Lin\\'eaire}, 33 (2016), 829--848], where this equation has been considered for the first time. 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