This article investigates the existence and properties of ground state solutions to the following nonlocal Hamiltonian elliptic system: \begin{ align*} \begin{ cases} (-\Delta)^\frac12 u +V_0 u =g(v),~x\in \mathbb{ R} (-\Delta)^\frac12 v +V_0 v =f(u),~x\in \mathbb{ R}, \end{ cases} \end{ align*} where $(-\Delta)^\frac12$ is the square root Laplacian operator, $V_0 >0$ and $f,~g$ have critical exponential growth in $\mathbb{ R}$. Using minimization technique over some generalized Nehari manifold, we show that the set $\mathcal{ S}$ of ground state solutions is non empty. Moreover for $(u,v) \in \mathcal{ S}$, $u,~v$ are uniformly bounded in $L^\infty(\mathbb{ R})$ and uniformly decaying at infinity. We also show that the set $\mathcal{ S}$ is compact in $H^\frac12(\mathbb{ R}) \times H^\frac12(\mathbb{ R})$ up to translations. Furthermore under locally lipschitz continuity of $f$ and $g$ we obtain a suitable Poho\v{ z}aev type identity for any $(u,v) \in \mathcal{ S}$. We deduce the existence of semi-classical ground state solutions to the singularly perturbed system \begin{ align*} \begin{ cases} \epsilon(-\Delta)^\frac12 \varphi +V(x) \varphi =g(\psi),~x\in \mathbb{ R} \epsilon (-\Delta)^\frac12 \psi +V(x) \psi =f(\varphi),~x\in \mathbb{ R}, \end{ cases} \end{ align*} where $\epsilon>0$ and $V \in C(\mathbb{ R})$ satisfy the assumption $(V)$ given below (see Section 1). Finally as $\epsilon \rightarrow 0$, we prove the existence of minimal energy solutions which concentrate around the closest minima of the potential $V$.
Categories: math.AP