Periodic solution for Hamiltonian type systems with critical growth

We consider an elliptic system of Hamiltonian type in a strip in \({\mathbb {R}}^N\), satisfying the periodic boundary condition for the first k variables. In the superlinear case with critical growth, we prove the existence of a single bubbling solution for the system under an optimal condition on k. The novelty of the paper is that all the estimates needed in the proof of the existence result can be obtained once the Green’s function of the Laplacian operator in a strip with periodic boundary conditions is found.