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We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacianand a reaction with gradient dependence (convection). The presence of thegradient in the source term excludes from consideration a variational approachin dealing with the qualitative analysis of this problem with unbalancedgrowth. Using the frozen variable method and eventually a fixed point theorem,the main result of this paper establishes that the problem has a positivesmooth solution.

Anisotropic $(p,q)$-equations with gradient dependent reaction