We prove an existence result for radial solutions of a Neumann elliptic problem whose
nonlinearity asymptotically lies between the first two eigenvalues. To this aim, we
introduce an alternative nonresonance condition with respect to the second eigenvalue
which, in the scalar case, generalizes the classical one, in the spirit of Fonda et al. (1991).
Our approach also applies for nonlinearities which do not necessarily satisfy a subcritical
growth assumption.
