Hamiltonian systems involving exponential growth in R2 with general nonlinearities

In this work, we establish the existence of ground state solution for Hamiltonian systems of the form - Δ u + V ( x ) u = H v ( x , u , v ) , x ∈ R 2 , - Δ v + V ( x ) v = H u ( x , u , v ) , x ∈ R 2 , where V ∈ C ( R 2 , ( 0 , ∞ ) ) and H ∈ C 1 ( R 2 × R 2 , R ) is allowed to have an exponential growth with respect to the Trudinger–Moser inequality. We study the case where V and H are periodic or asymptotically periodic. In the proof of the main results, we have used a reduction method involving the generalized Nehari manifold and also a linking theorem. In our approach, as we deal with general nonlinearities, it was necessary to obtain a new version of the Trudinger–Moser inequality.