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

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