Multiple nodal solutions for nonlinear nonhomogeneous elliptic problems with a superlinear reaction

We consider nonlinear Dirichlet and Neumann problems driven by a nonlinear nonhomogeneous differential operator and with a (p−1)-superlinear Carathéodory reaction which does not satisfy the Ambrosetti-Rabinowitz condition. We prove two multiplicity theorems producing two nodal solutions, and answer two open questions posed by Aizicovici et al. (2013) and by Barletta and Papageorgiou (2014). Our approach uses variational methods together with suitable truncation techniques and flow invariance arguments.