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

TL;DR

This paper develops a Nash-Moser iteration scheme to prove the existence and local uniqueness of solutions for nonlinear elliptic equations with singular perturbations on compact Lie groups and homogeneous spaces, including tori and spheres.

Contribution

It introduces a novel Nash-Moser-type method tailored for singular perturbation problems on compact Lie groups and homogeneous spaces, overcoming small divisor issues.

Findings

Established existence of solutions for nonlinear elliptic equations with singular perturbations.

Proved local uniqueness of spatially periodic solutions on tori and spheres.

Extended the applicability of Nash-Moser schemes to complex geometric settings.

Abstract

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 $\textbf{T}^n$ or the spheres $\textbf{S}^n$, our result shows that there is a local uniqueness of spatially periodic solutions for nonlinear elliptic equations with a singular perturbation.