Three nontrivial solutions for Neumann problems resonant at any positive eigenvalue

We consider a semilinear Neumann problem with a parametric reaction which has a concave term and a perturbation which at ±∞ can be resonant with respect to any positive eigenvalue. Using variational methods based on the critical point theory and Morse theory, we show that there exists a critical parameter value λ ∗ > 0 such that if λ ∈(0, λ ∗ ), then the problem has at least three nontrivial smooth solutions