Existence and uniqueness of solutions for singular double phase equations

We consider a nonautonomous (Formula presented.) -equation with unbalanced growth (double phase problem) with a nonparametric reaction which exhibits the combined effects of a singular term and of a Carathéodory perturbation which can be sign changing. The hypotheses on the perturbation are minimal. Using the solutions of some auxiliary double phase equations and the theory of nonlinear operators of monotone type, we prove the existence of a bounded positive solution. By imposing an additional condition on the perturbation, we show that this solution is unique.