On $p$-fractional weakly-coupled system with critical nonlinearities

This paper deals with the following nonlocal system of equations: \begin{align}\tag{$\mathcal{S}$}\label{MAT1} (-Δ_p)^s u = \fracα{p_s^}|u|^{α-2}u|v|^β+f(x) $\text{ in }$ $\mathbb{R}$^{d}, , (-Δ_p)^s v = \fracβ{p_s^}|v|^{β-2}v|u|^α+g(x) $\text{ in }$ $\mathbb{R}$^{d},; u,v >0 \mbox{ in } $\mathbb{R}$^{d}, \end{align} where $0<s<1<p<\mathrm{\infty }$, $d>sp$, $\alpha ,\beta >1$, $\alpha +\beta =\frac{dp}{d-sp}$, and $f,g$ are nontrivial nonnegative functionals in the dual space of ${\mathcal{D}}^{s,p}({\mathbb{R}}^d)$. 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 \eqref{MAT1}. Using this characterization, within a certain range of $s$, we establish the existence of a solution with negative energy for \eqref{MAT1} when $\ker (f)=\ker (g)$.