Singular double phase problems with convection

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplacian and of a q-Laplacian (double phase equation). In the reaction we have the combined effects of a singular term and of a gradient dependent term (convection) which is locally defined. Using a mixture of variational and topological methods, together with suitable truncation and comparison techniques, we prove the existence of a positive smooth solution.