Existence and multiplicity of solutions for nonlinear periodic problems with the scalar p-Laplacian and double resonance

We consider nonlinear periodic problems driven by the scalar p  –Laplacian and with a Caratheodory reaction which can exhibit double resonance at ±∞. Combining variational methods based on the critical point theory with Morse theoretic techniques, we show that we have existence when the double resonance occurs at any spectral interval and we have multiplicity with at least three nontrivial solutions, when the double resonance occurs at any spectral interval distinct from the “principal” one [λˆ0=0,λˆ1].