  In this paper, we extend the analysis of the subcritical approximation of the
Nirenberg problem on spheres recently conducted in \cite{MM19, MM}.
Specifically, we delve into the scenario where the sequence of blowing up
solutions exhibits a non-zero weak limit, which necessarily constitutes a
solution of the Nirenberg problem itself. Our focus lies in providing a
comprehensive description of such blowing up solutions, including precise
determinations of blow-up points and blow-up rates. Additionally, we compute
the topological contribution of these solutions to the difference in topology
between the level sets of the associated Euler-Lagrange functional. Such an
analysis is intricate due to the potential degeneracy of the involved
solutions. We also provide a partial converse, wherein we construct blowing up
solutions when the weak limit is non-degenerate.
