Multiple positive solutions for a critical elliptic system with concave and convex nonlinearities

• We get a relation about the positive solutions of our problem.
• Our problem is involving critical nonlinearities.
• Our result is available for sub-linear term |u|q−2u|u|q−2u with all 1

In this paper, we study the multiplicity results of positive solutions for a semi-linear elliptic system involving both concave–convex and critical growth terms. With the help of Nehari manifold and Lusternik–Schnirelmann category, we prove that the problem admits at least cat(Ω)+1cat(Ω)+1 distinct positive solutions.