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

Abstract This paper is concerned with the existence and the multiplicity for the following variational inequality problem involving the p -Laplacian $$ \left\{ \begin{array}{l} u\in \mathbb {K} \displaystyle \int _{\Omega }|\nabla u|^{p-2}\nabla u\nabla (v-u)dx -\lambda \displaystyle \int _{\Omega } |u|^{p^*-2}u(v-u)dx \ge \displaystyle \int _{\Omega } f(x,u)(v-u)dx \quad \forall v \in \mathbb {K} \end{array} \right. $$ 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 $$\Omega \subset \mathbb {R}^N, N\ge 3$$ Ω ⊂ R N , N ≥ 3 is a smooth bounded domain, $$\mathbb {K}=\{v\in W_{0}^{1,p}(\Omega ), v\ge \varphi \; a.e.\; \text{ in }\; \Omega \}$$ K = { v ∈ W 0 1 , p ( Ω ) , v ≥ φ a . e . in Ω }