  In this paper we study the Nirenberg problem on standard half spheres
$(\mathbb{S}^n_+,g_0)$, which consists of finding conformal metrics of
prescribed scalar curvature and zero boundary mean curvature on the boundary
$\partial \mathbb{S}^n_+$. This problem amounts to solve the following boundary
value problem involving the critical Sobolev exponent: \begin{equation*}
(\mathcal{P}) \quad \begin{cases} -\Delta_{g_0} u \, + \, \frac{n(n-2)}{4} u \,
= K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+,\\
\frac{\partial u}{\partial \nu }\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+.
\end{cases} \end{equation*} where $K \in C^2(\mathbb{S}^n_+)$ is a positive
function. We construct, under generic conditions on the function $K$, finite
energy solutions of a subcritical approximation of $(\mathcal{P})$ on half
spheres of dimension $n \geq 5$, which exhibit multiple blow up at the same
boundary point. Such a blow up phenomenon contrasts with the case of the
Nirenberg problem on spheres, where non simple blow up for finite energy
subsolutions cannot occur and displays an unexpected connection with vortex
type problems arising in Euler equations in fluid dynamic and mean fields type
equations in mathematical physics.
