Nonlinear elliptic equations with a singular perturbation on compact Lie groups and Homogeneous spaces
Abstract
This paper is devoted to the study of a class of singular perturbation elliptic type problems on compact Lie groups or homogeneous spaces
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
M
is the standard torus
T
n
or the spheres
S
n
, our result shows that there is a local uniqueness of spatially periodic solutions for nonlinear elliptic equations with a singular perturbation.