Existence of solutions for a nonlocal elliptic system with critical and subcritical exponential growth

In this paper, the existence of nontrivial solutions to the following nonlocal elliptic system: −Δu=1|x|μ∗G(v)g(v),v>0inΩ,−Δv=1|x|μ∗F(u)f(u),u>0inΩ,u=0,v=0on∂Ω$$ \left\{\begin{array}{l}-\Delta u=\left[\frac{1}{{\left|x\right|}^{\mu }}\ast G(v)\right]g(v),v>0\kern0.5em \mathrm{in}\kern0.5em \Omega, \\ {}-\Delta v=\left[\frac{1}{{\left|x\right|}^{\mu }}\ast F(u)\right]f(u),u>0\kern0.5em \mathrm{in}\kern0.5em \Omega, \\ {}u=0,v=0\kern0.5em \mathrm{on}\kern0.5em \mathrm{\partial \Omega}\end{array}\right. $$ is studied by variational method, where Ω$$ \Omega $$ is a bounded open subset of ℝ2$$ {\mathrm{\mathbb{R}}}^2 $$ with smooth boundary ∂Ω,0