In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole $\mathbb R$ $$ \left\{ \begin{ array}{ ll} (-\Delta)^\frac12~ u +u=Q(x) g(v)&\quad\mbox{ in } \mathbb R,\\ (-\Delta)^\frac12~ v+v = P(x)f(u)&\quad\mbox{ in } \mathbb R, \end{ array}\right. $$ where $(-\Delta)^\frac12$ is { the} square root Laplacian operator. We assume that the nonlinearities $f, g$ have critical growth at $+\infty$ in the sense of Trudinger-Moser inequality and the nonnegative weights $P(x)$ and $Q(x)$ vanish at $+\infty$. Using suitable variational method combined with { the} generalized linking theorem, we obtain the existence of { at least one} positive solution for the above system.
Categories: math.AP