real analysis – Uniqueness for a certain semilinear equation

Suppose that $(M,g)$ is a smooth compact Riemannian manifold with smooth boundary $partial M$. Let $a in C^{infty}(M)$, let $k in mathbb Z$ and consider the equation
$$
begin{aligned}
begin{cases}
-Delta_g u +a(x)usin u=0,
&forall ,x in M,
\
u(x) =kpi+f,
&forall,xin partial M.
end{cases}
end{aligned}
$$

Is there some $epsilon>0$ depending on $k$ such that given any $f in C^{2,alpha}(partial M)$ with
$$|f|_{C^{2,alpha}(partial M)} leq epsilon,$$
the above equation admits a unique solution $u in C^{2,alpha}(M)$?