  In this paper we perform a refined blow up analysis of finite energy
approximated solutions to a Nirenberg type problem on half spheres. The later
consists of prescribing, under minimal boundary conditions, the scalar
curvature to be a given function. In particular we give a precise location of
blow up points and blow up rates. Such an analysis shows that the blow up
picture of the Nirenberg problem on half spheres is far more complicated that
in the case of closed spheres. Indeed besides the combination of interior and
boundary blow ups, there are non simple blow up points for subcritical
solutions having zero or nonzero weak limit. The formation of such non simple
blow ups is governed by a vortex problem, unveiling an unexpected connection
with Euler equations in fluid dynamic and mean fields type equations in
mathematical physics.
