  We consider the Nelson model with variable coefficients and investigate the
problem of existence of a ground state and the removal of the ultraviolet
cutoff. Nelson models with variable coefficients arise when one replaces in the
usual Nelson model the flat Minkowski metric by a static metric, allowing also
the boson mass to depend on position. A physical example is obtained by
quantizing the Klein-Gordon equation on a static space-time coupled with a
non-relativistic particle. We investigate the existence of a ground state of
the Hamiltonian in the presence of the infrared problem, i.e. assuming that the
boson mass tends to 0 at infinity.
