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

The goal of the present paper is twofold. First, we study the λλ-dependence of the solution set F(λ)F(λ) of the variational–hemivariational inequality depending on a parameter λλ of the form 〈−Δpu,v−u〉+∫Ωjo(λ,u;v−u)dx≥〈h,v−u〉,∀v∈K, where ΔpΔp is the pp-Laplacian, and KK is a nonempty closed, convex set of the Sobolev space W1,p(Ω)W1,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)λ↦F(λ) is compact-valued and possesses extremal single-valued selections, which depend monotonously on λλ provided that λ↦jo(λ,s;±1)λ↦jo(λ,s;±1) satisfies a certain monotonicity assumption. Second, the results of the first part along with regularity results for the pp-Laplacian allow us to characterize the solution behavior of an unstable degenerate elliptic free boundary problem (for λ>0λ>0 and 2≤p<∞2≤p<∞) of the form: −Δpu=λχ{u>0}in Ω,u=g on ∂Ω.