Multiplicity results for p -laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions

The multiple results of positive solutions for the following quasilinear elliptic equation: - Δ p u=λf(x) | u | q-2 u+g(x) | u | p -2 u in , u=0 on , are established. Here, 0 is a bounded smooth domain in N, Δ p denotes the p -Laplacian operator, 1≤q<p<N, p = Np / (N-p),λ is a positive real parameter, and f,g are continuous functions on which are somewhere positive but which may change sign on . The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.