Nonlocal Equations on Domains and Manifolds Without Symmetries

In this work, we establish existence and regularity results for solutions to nonlinear nonlocal equations of the form $f(\mathrm{\Delta })u=U(x,u)$ and $f({\mathrm{\Delta }}_g)u=U(x,u)$ in Euclidean and Riemannian settings, respectively. We prove existence results across three distinct geometric scenarios: bounded domains in ${\mathbb{R}}^n$, perturbed equations on compact Riemannian manifolds, and noncompact Cartan-Hadamard manifolds under suitable geometric curvature conditions.