In this paper, we establish a weighted Trudinger–Moser type inequality and as the application of this result by using the Galerkin methods and a linking theorem, we prove the existence of weak solutions for the following class of elliptic systems: $\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{1/2}u+V(x)u=g(x,v),& v>0\mathrm{in}\mathbb{R},\\ (-\mathrm{\Delta }{)}^{1/2}v+V(x)v=f(x,u),& u>0\mathrm{in}\mathbb{R},\end{array}$when the potential V is neither bounded away from zero, nor bounded from above. The nonlinear terms $g(x,s)$ and $f(x,s)$ are superlinear at infinity and have subcritical or critical exponential growth.

中文翻译：

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一类指数增长的分数哈密顿系统的解

在本文中，我们建立了一个加权的 Trudinger-Moser 型不等式，并作为该结果的应用，通过使用 Galerkin 方法和链接定理，我们证明了以下类椭圆系统的弱解的存在性：$\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{1/2}你+五(X)你=G(X,v),& v>0\mathrm{在}\mathbb{R},\\ (-\mathrm{\Delta }{)}^{1/2}v+五(X)v=F(X,你),& 你>0\mathrm{在}\mathbb{R},\end{array}$当势V既不受零限制，也不受上限制。非线性项$G(X,s)$和$F(X,s)$在无穷远处是超线性的并且具有亚临界或临界指数增长。