Continuous dependence of solutions to variational inequality wrt. right hand side

Question

Given a closed convex subset $K$ of $H^1_0(\Omega)$, if $u \in K$ solves the variational inequality $$\langle -\Delta u - f, w-u \rangle \geq 0\quad\forall w \in K$$ and $v \in K$ solves the variational inequality as above but with $g$ instead of $f$, and both $f$ and $g$ are bounded a.e. and non-negative, do we get $$\lVert u-v \rVert_{L^\infty} \leq C\lVert f-g \rVert_X$$ for some norm $X$ on the right hand side? This ought to be a standard result but I can't find a reference. Note that I am looking for $L^\infty$ on the LHS, and not $H^1_0$ norm.