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This paper is devoted to the study of a class of singular perturbationelliptic type problems on compact Lie groups or homogeneous spaces$\mathcal{M}$. By constructing a suitable Nash-Moser-type iteration scheme oncompact Lie groups and homogeneous spaces, we overcome the clusters of "smalldivisor" problem, then the existence of solutions for nonlinear ellipticequations with a singular perturbation is established. Especially, if$\mathcal{M}$ is the standard torus $\textbf{T}^n$ or the spheres$\textbf{S}^n$, our result shows that there is a local uniqueness of spatiallyperiodic solutions for nonlinear elliptic equations with a singularperturbation.

Nonlinear elliptic equations with a singular perturbation on compact Lie groups and Homogeneous spaces