Existence and multiplicity of solutions for variational inequality problems involving the p-Laplacian

This paper is concerned with the existence and the multiplicity for the following variational inequality problem involving the p -Laplacian $$ \{ {array}{l} u∈ K ∫ Ω|∇ u|ᵖ⁻²∇ u∇ (v-u)dx -λ ∫ Ω |u|p^*-2u(v-u)dx ≥ ∫ Ω f(x,u)(v-u)dx ∀ v ∈ K {array} . $$ u ∈ K ∫ Ω | ∇ u | p - 2 ∇ u ∇ ( v - u ) d x - λ ∫ Ω | u | p ∗ - 2 u ( v - u ) d x ≥ ∫ Ω f ( x , u ) ( v - u ) d x ∀ v ∈ K where $$Ω ⊂ R^N, N≥ 3$$ Ω ⊂ R N , N ≥ 3 is a smooth bounded domain, $$K=\{v∈ W₀1,p(Ω ), v≥ φ \; a.e.\; in \; Ω \}$$ K = { v ∈ W 0 1 , p ( Ω ) , v ≥ φ a . e . in Ω }