Multiple positive solutions for p -Laplace elliptic equations involving concave–convex nonlinearities and a Hardy-type term

In this paper, the following problem is considered: { − Δ p u − μ | u | p − 2 u | x | p = λ f ( x ) | u | q − 2 u + g ( x ) | u | p ∗ − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded domain such that 0 ∈ Ω , 1 < q < p , λ > 0 , μ < μ ̄ , f and g are nonnegative functions, μ ̄ = ( N − p p ) p is the best Hardy constant and p ∗ = N p N − p is the critical Sobolev exponent. By extracting the Palais–Smale sequence in the Nehari manifold, the existence of multiple positive solutions to this equation is verified.