  We apply the vertical Jeans equation to the kinematics of Milky Way stars in
the solar neighbourhood to measure the local dark matter density. More than
90,000 G- and K-type dwarf stars are selected from the cross-matched sample of
LAMOST DR5 and Gaia DR2 for our analyses. The mass models applied consist of a
single exponential stellar disc, a razor thin gas disc and a constant dark
matter density. We first consider the simplified vertical Jeans equation which
ignores the tilt term and assumes a flat rotation curve. Under a Gaussian prior
on the total stellar surface density, the local dark matter density inferred
from Markov Chain Monte Carlo simulations is $0.0133_{-0.0022}^{+0.0024}\ {\rm
M}_{\odot}\,{\rm pc}^{-3}$. The local dark matter densities for subsamples in
an azimuthal angle range of $-10^{\circ} < \phi < 5^{\circ}$ are consistent
within their 1$\sigma$ errors. However, the northern and southern subsamples
show a large discrepancy due to plateaux in the northern and southern vertical
velocity dispersion profiles. These plateaux may be the cause of the different
estimates of the dark matter density between the north and south. Taking the
tilt term into account has little effect on the parameter estimations and does
not explain the north and south asymmetry. Taking half of the difference of
$\sigma_{z}$ profiles as unknown systematic errors, we then obtain consistent
measurements for the northern and southern subsamples. We discuss the influence
of the vertical data range, the scale height of the tracer population, the
vertical distribution of stars and the sample size on the uncertainty of the
determination of the local dark matter density.
