  A Dirichlet problem driven by the $(p,q)$-Laplace operator and an asymmetric
concave reaction with positive parameter is investigated. Four nontrivial
smooth solutions (two positive, one negative, and the remaining nodal) are
obtained once the parameter turns out to be sufficiently small. Under a oddness
condition near the origin for the perturbation, a whole sequence of
sign-changing solutions, which converges to zero, is produced.
