Pairs of nontrivial solutions for resonant Neumann problems

We study a semilinear Neumann problem which is resonant at ±∞ with respect to any eigenvalue different from the first and the second eigenvalue of −ΔN (the negative Neumann Laplacian). Using a combination of variational methods with Morse theoretic techniques, we show that the problem has at least two nontrivial smooth solutions.