I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this line for the Dirichlet problem at infinity for harmonic functions (see M. T. Anderson's work on Laplace-Beltrami operator) but I need similar results for more general elliptic equations. I am also aware of some similar results on compact manifolds (for example, the ones in Taylor's 'Partial Differential Equations') but nothing for noncompact ones. Concretely, I am trying to solve the inhomogeneous Helmholtz equation: $\Delta_g u+\lambda u=f$, possibly with restrictions over $\lambda$ or the regularity of $f$, with homogeneous Dirichlet conditions at the ideal boundary $\mathbb{S}^{d-1}(\infty)$.

I tend to believe that properties of the ends of this kind of manifold play an important role in solving elliptic PDEs and would really appreciate any reference on this topic that could help me understand PDE in this specific kind of noncompact manifolds. My final aim is to prove the existence of a Green's function for the Dirichlet equation on an Asymptotically Hyperbolic Manifold, which I am trying by changing a little Li and Tam's construction for the Laplacian. Therefore, I will also appreciate any literature about Green functions for general elliptic equations on manifolds with negative curvature (outside a compact set).