A pair of positive solutions for $(p,q)$-equations with combined nonlinearities

We consider a parametric nonlinear Dirichlet problem driven by the (p, q)-differential operator, with a reaction consisting of a "concave" term perturbed by a (p -1)-superlinear perturbation, which need not satisfy the Ambrosetti-Rabinowitz condition (problem with combined or competing nonlinearities). Using variational methods we show that for small values of the parameter the problem has at least two nontrivial positive smooth solutions.