Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity

We consider a semilinear Neumann problem with an indefinite and unbounded potential and an asymmetric reaction that crosses at least the principal eigenvalue of the operator − Δ+ βI in H 1 (Ω), β being the po- tential function. Using a combination of variational methods, with truncation and perturbation techniques and Morse theory, we prove multiplicity theorems providing precise sign information for all the solutions.