Multiple solutions for strongly resonant periodic systems

We considered a semilinear, second order periodic system. We assumed that the differential operator x→−x″−Axx→−x″−Ax has zero as an eigenvalue and has no negative eigenvalues. Also we imposed a strong resonance condition (with respect to the zero eigenvalue) on the potential function F(t,x)F(t,x). Using the second deformation theorem, we established the existence of at least two nontrivial solutions. To do this we needed to conduct a detailed analysis of the Cerami compactness condition, which is actually of independent interest.