Superlinear Neumann problems with the p-Laplacian plus an indefinite potential

We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. First, we prove an existence theorem, and then, under stronger conditions on the reaction, we prove a multiplicity theorem producing three nontrivial solutions. Then, we examine parametric problems with competing nonlinearities (concave and convex terms). We show that for all small values of the parameter (Formula presented.), the problem has five nontrivial solutions and if (Formula presented.) (semilinear equation), there are six nontrivial solutions. Finally, we prove a bifurcation result describing the set of positive solutions as the parameter (Formula presented.) varies.