A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities

We consider a nonlinear parametric Dirichlet problem driven by the anisotropic p-Laplacian with the combined effects of “concave” and “convex” terms. The “superlinear” nonlinearity need not satisfy the Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory and the Ekeland variational principle, we show that for small values of the parameter, the problem has at least two nontrivial smooth positive solution