Existence of translating solutions to the flow by powers of mean curvature on unbounded domains

In this paper, we prove the existence of classical solutions to the Dirichlet problem of a class of quasi-linear elliptic equations on an unbounded cone and a U-type domain in Rn (n 2). This problem comes from the study of mean curvature flow or its generalization, the flow by powers of mean curvature. Our approach is a modified version of the classical Perron method, where the solutions to the minimal surface equation are used as sub-solutions and a family auxiliary functions are constructed as super-solutions.