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 $$\begin{cases} u \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), & u \geq \psi \quad \text{ in } \Omega, \\[1mm] \langle A(u), v - u \rangle + \displaystyle\int_{\Omega} H(x, u, D u)(v - u) \geq 0, & \\[3mm] \forall\; v \in W_0^{1,p}(\Omega) \cap L^\infty(\Omega), \quad &v \geq \psi \quad \text{ in } \Omega. \end{cases}$$ 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