On noncoercive periodic systems with vector $p$-Laplacian

Abstract

We consider nonlinear periodic systems driven by the vector $p$-Laplacian. An existence and a multiplicity theorem are proved. In the existence theorem the potential function is $p$-superlinear, but in general does not satisfy the AR-condition. In the multiplicity theorem the problem is strongly resonant with respect to the principal eigenvalue $\lambda_0=0$. In both of the cases the Euler-Lagrange functional is noncoercive and the method is variational.