# How to show this weak formulation has unique solution?

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?

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If $\epsilon$ is small enough, $a+\epsilon b$ will be coercive.