  This paper studies a nonlinear Dirichlet problem for the $p$-Laplacian
operator with nonlinearity consisting of power components. The problem under
consideration can be though of as a perturbation of the
Ambrosetti-Brezis-Cerami problem with concave-convex nonlinearity. The combined
effect of power components in the perturbed nonlinearity allows to establish a
higher order multiplicity of positive solutions. We study properties of the
perturbed energy functional and prove the existence of three positive solutions
to the problem at hand.
