Existence and multiplicity of nontrivial solutions for‎ ‎$p$-Laplacian system with nonlinearities of concave-convex type and‎ ‎sign-changing weight functions

This paper is concerned with the existence of multiple positive‎ ‎solutions for a quasilinear elliptic system involving concave-convex‎ ‎nonlinearities‎ ‎and sign-changing weight functions‎. ‎With the help of the Nehari manifold and Palais-Smale condition‎, ‎we prove that the system has at least two nontrivial positive‎ ‎solutions‎, ‎when the pair of parameters $(lambda,mu)$ belongs to a certain subset of $mathbb{R}^2$‎.