It's the last question in my exam.
Prove that there is a neighborhood of $0$, denoted by $[-\delta,\delta]$, $\forall \epsilon \in [-\delta,\delta]$ the following equation has a unique solution in the sense of distribution. $$-\Delta u+\epsilon \sin u=f$$
Energy inequality shows that the homogeneous equation for $u$ has a unique solution. But I have no idea the how to prove the existence.
I'm poor at functional analysis, so if any theorem applied, please let me know.