Multiple positive solutions for a quasilinear elliptic system with critical exponent and sign-changing weight

In this paper, our main purpose is to establish the existence of multiple positive solutions of a quasilinear elliptic system with Sobolev critical nonlinearities and sign‐changing weight by some standard variational methods. By using the $\mathcal{N}$ ehari manifold, fibering maps and the Lusternik–Schnirelmann category, we study the topology of the global maximum set $F$ of $f\left(x\right)$ , and show the system has at least $ca{t}_{F_{ϵ}}\left(F\right)+1$ distinct positive solutions, where ${\text{cat}}_{F_{ϵ}}\left(F\right)$ denotes the Lusternik–Schnirelmann category of $F$ in $F_{ϵ}$ .