Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

In this paper we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schr"{o}dinger equations with square root of the Laplacian $$\{\begin{array}{lrr}(-\mathrm{\Delta }{)}^{1\mathrm{/}2}u+V_1(x)u=f_1(u)+\lambda (x)v,& x\in \mathbb{R},(-\mathrm{\Delta }{)}^{1\mathrm{/}2}v+V_2(x)v=f_2(v)+\lambda (x)u,& x\in \mathbb{R},\end{array}$$ where the nonlinearities $f_1(s)$ and $f_2(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_1(x)$ and $V_2(x)$ are nonnegative and periodic. Moreover, we assume that there exists $\delta\in (0,1)$suchthat$\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is variational and based on minimization technique over the Nehari manifold.