Existence and multiplicity of solutions for a critical system involving a biharmonic and p-Laplacian operator

This paper is concerned with the existence of a ground state solution for the following class of elliptic Kirchhoff–Boussinesq-type problems given by $$ \left\{ \begin{array}{l} \Delta (a(\varepsilon x) \Delta u) - \Delta _p u + u = Q_{u}(u,v) + \dfrac{1}{2_{*}} K_{u}(u,v) \;\; \text {in} \;\; \mathbb {R}^N,\\ \Delta (b(\varepsilon x) \Delta v) - \Delta _p v + v = Q_{v}(u,v) + \dfrac{1}{2_{*}} K_{v}(u,v) \;\; \text {in} \;\; \mathbb {R}^N,\\ u, v \in H^{2}(\mathbb {R}^N), \end{array} \right. $$ where $$\varepsilon >0$$ , $$2< p < 2^{*}= \frac{2N}{N-2}$$ , $$2_{*}= \frac{2N}{N-4}$$ and $$N\ge 5$$ . We establish the existence of a ground state solution for the critical system by employing variational methods, and we prove the multiplicity of solutions through the Ljusternik–Schnirelmann category.