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 $\ell$ is a bounded linear function also on $H$.
How do I show that: For sufficiently small $\varepsilon > 0$, the equation $a(u^\epsilon, v) + \varepsilon b(u^\varepsilon, v) = \ell(v), \quad \mathit{ for\,\,all}\,\, v \in H, $ has a unique solution $u^\varepsilon$.
I can't apply Lax-Milgram because $b$ is not necessarily coercive. What's the general technique for these kinds of problems?