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Our prototype is:\n  $ \\left\\{ \\begin{array}{ll} -\\mathcal Q_{p}u =[H(Du)]^{q}+f(x) &\\text{in }\\Omega,\\\\ u=0&\\text{on }\\partial\\Omega. \\end{array} \\right. $\n  Here $\\Omega$ is a bounded open set of $\\mathbb R^{N}$, $N\\ge 2$, $0<p-1<q\\le p<N$, and $\\mathcal Q_{p}$ is the anisotropic operator $ \\mathcal Q_{p} u ={\\rm div}\\left( [H(Du)]^{p-1}H_{\\xi}(Du) \\right)$, where $H$ is a suitable norm of $\\mathbb R^{N}$. 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