On <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-fractional weakly-coupled system with critical nonlinearities

This paper deals with the following nonlocal system of equations:$ \left\{\begin{aligned} &(-\Delta_p)^s u = \frac{\alpha}{p_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{d},\\ &(-\Delta_p)^s v = \frac{\beta}{p_s^*}|v|^{\beta-2}v|u|^{\alpha}+g(x)\;\;\text{in}\;\mathbb{R}^{d},\\ & u, \, v >0\, \mbox{ in }\,\mathbb{R}^{d}, \end{aligned} \right. \quad\quad\quad\quad (\mathcal{S})$where $ 0<s<1<p< \infty $, $ d>sp $, $ \alpha,\beta>1 $, $ \alpha+\beta = \frac{dp}{d-sp} $, and $ f,g $ are nontrivial nonnegative functionals in the dual space of $ \mathcal{D}^{s,p}(\mathbb{R}^{d}) $, i.e., $ f,g \neq 0 $, and $ { }_{(\mathcal{D}^{s,p})'}{\langle}f,u{\rangle}_{\mathcal{D}^{s,p}}\geq 0, \,{ }_{(\mathcal{D}^{s,p})'}{\langle}g,u{\rangle}_{\mathcal{D}^{s,p}}\geq 0 $, whenever $ u $ is a nonnegative function in $ \mathcal{D}^{s,p}(\mathbb{R}^{d}) $. The primary objective of this paper is to present a global compactness result that offers a complete characterization of the Palais-Smale sequences of the energy functional associated with $(\mathcal S)$. Using this characterization, within a certain range of $ s $, we establish the existence of a solution with negative energy for $(\mathcal S)$ when $ \ker(f) = \ker(g) $.