  In this paper, we establish a class of Hardy-Littlewood-Sobolev inequality
with partial variable weight functions on the upper half space using a weighted
Hardy type inequality. Overcoming the impact of weighted functions, the
existence of extremal functions is proved via the concentration compactness
principle, whereas Riesz rearrangement inequality is not available. Moreover,
the cylindrical symmetry with respect to $t$-axis and the explicit forms on the
boundary of all nonnegative extremal functions are discussed via the method of
moving planes and method of moving spheres, as well as, regularity results are
obtained by the regularity lift lemma and bootstrap technique. As applications,
we obtain some weighted Sobolev inequalities with partial variable weight
function for Laplacian and fractional Laplacian.
