Mosco-Convergence of Convex Sets and Unilateral Problems for Differential Operators with Lower Order Terms Having Natural Growth

We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. Specifically, we consider problems of the form {u∈W01,p(Ω)∩L∞(Ω),u≥ψ in Ω,⟨A(u),v−u⟩+∫ΩH(x,u,Du)(v−u)≥0,∀ v∈W01,p(Ω)∩L∞(Ω),v≥ψ in Ω. Here, A is a Leray–Lions type operator, mapping W01,p(Ω) into its dual W−1,p′(Ω), while H(x,u,Du) grows like ∣Du∣p. The obstacle ψ is a function in W01,p(Ω)∩L∞(Ω). Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems