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{align*} \begin{cases} (-Δ)^\frac12 u +V_0 u =g(v),~x\in \mathbb{R} (-Δ)^\frac12 v +V_0 v =f(u),~x\in \mathbb{R}, \end{cases} \end{align*} 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{align*} \begin{cases} ε(-Δ)^\frac12 φ+V(x) φ=g(ψ),~x\in \mathbb{R} ε(-Δ)^\frac12 ψ+V(x) ψ=f(φ),~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 $\epsilon \to 0$, we prove the existence of minimal energy solutions which concentrate around the closest minima of the potential $V$.