Multiple and nodal solutions for nonlinear equations with a nonhomogeneous differential operator and concave-convex terms

In this paper we consider a nonlinear parametric Dirichlet problem driven by a nonhomogeneous differential operator (special cases are the p-Laplacian and the (p, q)-differential operator) and with a reaction which has the combined effects of concave ((p - 1)-sublinear) and convex ((p - 1)-superlinear) terms. We do not employ the usual in such cases AR-condition. Using variational methods based on critical point theory, together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small λ > 0 (λ is a parameter), the problem has at least five nontrivial smooth solutions (two positive, two negative and the fifth nodal). We also prove two auxiliary results of independent interest. The first is a strong comparison principle and the second relates Sobolev and Hölder local minimizers for C¹ functionals.