Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth

Abstract

We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. More precisely, for a fixed obstacle $\psi\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$, we consider $u\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ satisfying $u\geq\psi$ a.e. and $$ \langle A(u),v-u\rangle+\int_{\Omega}H(x,u,\nabla u)(v-u)\geq 0$$ for all $v\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ with $v\geq\psi$. 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 $|D u|^p$. 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.