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 NNehari manifold, fibering maps and the Lusternik–Schnirelmann category, we study the topology of the global maximum set FF of f(x)f(x), and show the system has at least catFϵ(F)+1catFϵ(F)+1 distinct positive solutions, where catFϵ(F)catFϵ(F) denotes the Lusternik–Schnirelmann category of FF in FϵFϵ.