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 \[ \begin{cases} u \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), & u \geq \psi \quad \text{ in } \Omega, \\[8pt] \langle A(u), v - u \rangle + \displaystyle\int_{\Omega} H(x, u, D u)(v - u) \geq 0, & \\[14pt]\forall\; v \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), \quad &v \geq \psi \quad \text{ in } \Omega. \end{cases} \] \noindent Here, \( A \) is a Leray–Lions type operator, mapping \( W_0^{1,p}(\Omega) \) into its dual \( W^{-1, p'}(\Omega) \), while \( H(x, u, D u) \) grows like \( |D u|^p \). The obstacle \( \psi \) is a function in \( W_0^{1,p}(\Omega) \cap L^\infty(\Omega) \). 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.