We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system (Formula presented.) where a,b are numbers belonging to the interval [0, 2), V is a continuous potential bounded below on R2 by a positive constant and the functions f and g possess exponential growth range established by Trudinger–Moser inequalities in Lorentz–Sobolev spaces. The proof involves linking theorem and a finite-dimensional approximation.