Parameter-dependent variational–hemivariational inequalities and an unstable degenerate elliptic free boundary problem

Abstract

The goal of the present paper is twofold. First, we study the λ -dependence of the solution set F ( λ ) of the variational–hemivariational inequality depending on a parameter λ of the form 〈 − Δ p u , v − u 〉 + ∫ Ω j o ( λ , u ; v − u ) d x ≥ 〈 h , v − u 〉 , ∀ v ∈ K , where Δ p is the p -Laplacian, and K is a nonempty closed, convex set of the Sobolev space W 1 , p ( Ω ) . Assuming an ordered pair ( λ ¯ , u ¯ ) ≤ ( λ ¯ , u ¯ ) of appropriately defined sub–supersolutions, we are going to show by variational methods that λ ↦ F ( λ ) is compact-valued and possesses extremal single-valued selections, which depend monotonously on λ provided that λ ↦ j o ( λ , s ; ± 1 ) satisfies a certain monotonicity assumption. Second, the results of the first part along with regularity results for the p -Laplacian allow us to characterize the solution behavior of an unstable degenerate elliptic free boundary problem (for λ > 0 and 2 ≤ p < ∞ ) of the form: − Δ p u = λ χ { u > 0 } in Ω , u = g on ∂ Ω .