  In this article, we introduce an analogous problem to Yamabe type problem
considered by Case, J., which generalizes the Escobar-Riemann mapping problem
for smooth metric measure spaces with boundary. The last problem will be called
Escobar-Riemann mapping type problem. For this purpose, we consider the
generalization of Sobolev Trace Inequality deduced by Bolley at. al. This trace
inequality allows us to introduce an Escobar quotient and its infimum. This
infimum we call the Escobar weighted constant. The Escobar-Riemann mapping type
problem for smooth metric measure spaces in manifolds with boundary consists of
finding a function which attains the Escobar weighted constant. Furthemore, we
resolve the later problem when Escobar weighted constant is negative. Finally,
we get an Aubin type inequality connecting the weighted Escobar constant for
compact smooth metric measure space and the optimal constant for the trace
inequality due to Bolley at. al.
