Parameter continuity of variational regularization

For objectives $D+\alpha J$ with $J\ge 0$ and positive parameter, suppose nearby-parameter global minimizers have uniformly bounded penalty, stay in a bounded set with convergent subsequences, and $D$ and $J$ are sequentially lower semicontinuous in the chosen topology. Their optimality inequalities and lower semicontinuity show that every subsequential limit minimizes the limiting objective. Uniqueness makes the regularized solution continuous in the parameter. Existence alone does not make an arbitrary selection among multiple global minimizers continuous.