Fractional Hamiltonian type system on $\mathbb R$ with critical growth nonlinearity

Abstract

This article investigates the existence and properties of ground state solutions to the following nonlocal Hamiltonian elliptic system:

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^\frac{1}{2} u +V_0 u =g(v),~x\in \mathbb R\\ (-\Delta )^\frac{1}{2} v +V_0 v =f(u),~x\in \mathbb R, \end{array}\right. } \end{aligned}$$

where $(-\Delta )^\frac{1}{2}$ 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 $H^\frac{1}{2}(\mathbb R) \times H^\frac{1}{2}(\mathbb R)$ up to translations. Furthermore under locally lipschitz continuity of f and g we obtain a suitable Pohožaev type identity for any $(u,v) \in \mathcal S$. We deduce the existence of semi-classical ground state solutions to the singularly perturbed system

$$\begin{aligned} {\left\{ \begin{array}{ll} \epsilon (-\Delta )^\frac{1}{2} \varphi +V(x) \varphi =g(\psi ),~x\in \mathbb R\\ \epsilon (-\Delta )^\frac{1}{2} \psi +V(x) \psi =f(\varphi ),~x\in \mathbb R, \end{array}\right. } \end{aligned}$$

where $\epsilon >0$ and $V \in C(\mathbb R)$ satisfy the assumption (V) given below (see Sect. 1). Finally as $\epsilon \rightarrow 0$, we prove the existence of minimal energy solutions which concentrate around the closest minima of the potential V.