Nonlinear resonant periodic problems

Abstract

We consider nonlinear periodic problems driven by the sum of a scalar $p$-Laplacian and a scalar Laplacian and a Carath\'{e}odory reaction, which at $\pm\infty$, is resonant with respect to any higher eigenvalue. Using variational methods, coupled with suitable perturbation and truncation techniques and Morse theory, we prove a three solutions theorem. For equations resonant with respect to the principal eigenvalue $\hat \lambda_0=0$, we establish the existence of nodal solutions.