I am looking into the theory of non-coercive variational inequalities and I came accross the following approximation techinique:
Fix an infinite dimensional Hilbert space $H$ and let $K \subset V$ be a closed and convex set containing the origin. Consider the variational inequality to find $u \in K$:
$$ a(u, v - u) \ge (f , v-u)_H \qquad \forall v \in K, \qquad (*) $$
where $a(\cdot, \cdot): H \times H \rightarrow \mathbb{R}$ is a given non coercive bilinear form and $f \in H$.
The idea is now to approximate $(*)$ with a sequence of nonlinear equations: for $\lambda > 0$ consider the problem to find $u_{\lambda} \in H$ such that:
$$ a(u_{\lambda},v) + \frac{1}{\lambda}(u_{\lambda}, P_{K}v)_H = (f,v)_H \qquad \forall v \in H, \qquad (**) $$
where $P_K : H \rightarrow H$ is the orthogonal projection onto $K$. It is claimed that $u_{\lambda} \rightarrow u$ in $H$ as $\lambda \rightarrow 0$.
Rather than a proof of the claim (which I am not sure to be correct in this precise form), I am looking for an intuition behind the technique: why should we expect a similar type of converges as the coefficient in front of the nonlinear term explodes? When dealing with similar questions for equalities, there is the advantage that we can subtract the formulations and look for estimates, which here we don't have.
To long for a comment, and also to explain my comments above a bit.
For symmetric and coercive $a$, problem ( * ) is equivalent to $$ \min_{u\in K} \frac12 a(u,u) - f(u). $$ The constraint can be penalized by introducing a term that penalized the violation of the constraint $$ \min_{u\in H} \frac12 a(u,u) - f(u) + \frac1{2\lambda}\|u - P_K(u)\|_H^2. $$ This can be differentiated to yield something like ( ** ), but not equal to your suggestion ( ** ).