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 $$ \left\{ \begin{array}{lr} (-\Delta)^{1/2}u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R}, (-\Delta)^{1/2}v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}, \end{array} \right. $$ 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 $\delta\in (0,1)$ such that $\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. 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.