In this work, we investigate the existence of solutions for a fractional critical p-Kirchhoff system with non-local non-linearities. Specifically, we consider the following problem: \(\begin{aligned} {\left\{ \begin{array}{ll} \textrm{M}(\Vert u,v\Vert ^p)\mathcal {L}_\mathcal {K}^p(u) = \lambda \frac{\partial F}{\partial u}(x,u,v) + \frac{\alpha }{p^\star _s}|u|^{\alpha -2}u|v|^{\beta } & \text {in } \Omega ,\\ \textrm{M}(\Vert u,v\Vert ^p)\mathcal {L}_\mathcal {K}^p(v) = \lambda \frac{\partial F}{\partial v}(x,u,v) + \frac{\beta }{p^\star _s}|u|^{\alpha }|v|^{\beta -2}v & \text {in } \Omega ,\\ u = v = 0 & \text {in } \mathbb {R}^N {\setminus } \Omega , \end{array}\right. } \end{aligned}\) where \(\Omega \subset \mathbb {R}^N\) is a bounded domain with Lipschitz boundary, \(\mathcal {L}_\mathcal {K}^p\) is a nonlocal operator with singular kernel \(\mathcal {K}\) , and \(\Vert u,v\Vert ^p\) is defined by \(\Vert u,v\Vert ^p=\int _{\mathbb {R}^{2N}}|u(x)-u(y)|^p\mathcal {K}(x-y)dx\ dy+\int _{\mathbb {R}^{2N}}|v(x)-v(y)|^p\mathcal {K}(x-y)dx\ dy.\) The Kirchhoff function \(\textrm{M}\) is continuous, \(\lambda > 0\) is a parameter, and \(F: \Omega \times \mathbb {R}^2 \rightarrow \mathbb {R}\) is a Carathéodory function. The exponents \(\alpha , \beta > 1\) satisfy the critical condition \(\alpha + \beta = p^\star _s = \frac{Np}{N-sp}.\) Notably, our analysis covers the degenerate case where \(\textrm{M}\) may vanish at zero. By employing Ekeland’s variational principle, we prove the existence of a nontrivial weak solution with negative energy. Furthermore, using Kajikiya’s symmetric mountain pass lemma, we establish the existence of infinitely many small solutions converging to zero, all with negative energy. Our results extend and complement previous work on fractional Kirchhoff systems in critical settings.
