Suppose $a$ is a bounded and coercive bilinear form on a Hilbert space $H$ and that $b$ is a bounded bilinear form on $H$ and $l$ is a bounded linear function also on $H$. Let $\epsilon > 0$ be small.
How do I show that
$a(u^\epsilon, v) + \epsilon b(u^\epsilon, v) = l(v)$ for all $v \in H$
has a unique solution $u^\epsilon$?
I can't apply Lax--Milgram because $b$ is not necessarily coercive. What's the general technique for these kinds of problems?