On Nonlocal Equations under Symmetries

In this work, we establishes results of existence and regularity for solutions to nonlocal equations of the form $f(\mathrm{\Delta })u=U(x,u(x))\text{ and }f({\mathrm{\Delta }}_g)u=U(x,u(x))$ in the Euclidean context and Riemannian context respectively. We extend the results of \cite{Gorka 6} for the Euclidean space ${\mathbb{R}}^n$ under cylindrically symmetry and compact Riemannian manifolds $(M,g)$ under action of a closed Lie subgroup $G\le {\text{Isom}}_g(M)$. Our approach combines spectral theory, complex interpolation and fixed-point methods.