Singular non-autonomous (p,q)-equations with competing nonlinearities

We consider a parametric non-autonomous $(p,q)$-equation with a singular term and competing nonlinearities, a parametric concave term and a Carathéodory perturbation. We consider the cases where the perturbation is $(p-1)$-linear and where it is $(p-1)$-superlinear (but without the use of the Ambrosetti–Rabinowitz condition). We prove an existence and multiplicity result which is global in the parameter $\lambda >0$ (a bifurcation type result). Also, we show the existence of a smallest positive solution and show that it is strictly increasing as a function of the parameter. Finally, we examine the set of positive solutions as a function of the parameter (solution multifunction). First, we show that the solution set is compact in $C_0^1\left(\bar{\Omega }\right)$ and then we show that the solution multifunction is Vietoris continuous and also Hausdorff continuous as a multifunction of the parameter.