Non simple blow ups for the Nirenberg problem on half spheres

In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting 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} -\D_{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 of \emph{cluster-type} at the same boundary point. These solutions may have zero or non zero weak limit and may develop clusters at different boundary points. Such blow up phenomena on half spheres drastically contrast with the case of the Nirenberg problem on spheres, where non simple blow up for finite energy subsolutions cannot occur and unveils an unexpected connection with vortex type problems arising in Euler equations in fluid dynamic and mean fields type equations in mathematical physics. We construct also, under suitable conditions on the restriction of $K$ on $\partial \mathbb{S}^n_+$, approximate solutions of arbitrarily large energy and Morse index