Nonlinear resonant periodic problems with concave terms

We consider a nonlinear periodic problem, driven by the scalar p-Laplacian with a concave term and a Caratheodory perturbation. We assume that this perturbation f ( t , x ) is ( p − 1 ) -linear at ±∞, and resonance can occur with respect to an eigenvalue λ m + 1 , m ⩾ 2 , of the negative periodic scalar p-Laplacian. Using a combination of variational techniques, based on the critical point theory, with Morse theory, we establish the existence of at least three nontrivial solutions. Useful in our considerations is an alternative minimax characterization of λ 1 > 0 (the first nonzero eigenvalue) that we prove in this work.