Anisotropic nonlinear Neumann problems

We consider nonlinear Neumann problems driven by the p(z)-Laplacian differential operator and with a p-superlinear reaction which does not satisfy the usual in such cases Ambrosetti?CRabinowitz condition. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). In the process, we also prove two results of independent interest. The first is about the L ^??-boundedness of the weak solutions. The second relates W ^1,p(z) and C ¹ local minimizers.