Nirenberg problem on high dimensional spheres: blow up with residual mass phenomenon

In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in Malchiodi and Mayer(J Differ Equ 268(5):2089–2124, 2020; Int Math Res Not 18:14123–14203, 2021). 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 solutions involved. We also provide a partial converse, wherein we construct blowing up solutions when the weak limit is non-degenerate.