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 …