Dirichlet problem for $f$-minimal graphs

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with prescribed boundary behavior on a bounded domain $\Omega \subset M$ under suitable assumptions on $f$ and the boundary of $\Omega$. In the second part, we consider the asymptotic Dirichlet problem. Provided that $f$ decays fast enough, we construct solutions to the problem. Our assumption on the decay of $f$ is linked with the sectional curvatures of $M$. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp.