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

Abstract

This paper deals with the following nonlocal system of equations: \begin{equation}\tag{$\mathcal S$}\label{MAT1} (-\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{equation} where $0 sp$, $\alpha,\beta>1$, $\alpha+\beta=\frac{dp}{d-sp}$, 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 $\ker(f)=\ker(g)$, and $\|f\|_{(\mathcal{D}^{s,p})'}, \|g\|_{(\mathcal{D}^{s,p})'}$ are small.