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 $$ \begin{aligned} \begin{cases} -\Delta_g u +a(x)u\sin u=0, &\forall \,x \in M, \\ u(x) =k\pi+f, &\forall\,x\in \partial M. \end{cases} \end{aligned} $$ 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 $u \in C^{2,\alpha}(M)$?
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