# 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 $$u in C^{2,alpha}(M)$$?