On the Dirichlet boundary value problem on Cartan-Hadamard manifolds

Abstract

In this paper, we investigate the Dirichlet boundary value problem on Cartan-Hadamard manifolds, focusing on the non-existence of bounded (viscosity) solutions to semi-linear elliptic equations of the form $\Delta u + f(u) = 0$ in domains with prescribed asymptotic boundary, extending previous results by Bonorino and Klaser originally established for hyperbolic spaces. Using a novel comparison technique based on convex hypersurfaces inspired by Choi, G\'alvez, and Lozano, we overcome the absence of totally geodesic foliations, which are instrumental in the hyperbolic space. Our results highlight the interplay between curvature, the spectrum of the Laplacian, and the geometry of the asymptotic boundary.