Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric data

Let $(M,\textit{g},\sigma)$ be an $m$-dimensional closed spin manifold, with a fixed Riemannian metric $\textit{g}$ and a fixed spin structure $\sigma$; let $\mathbb{S}(M)$ be the spinor bundle over $M$. The spinorial Yamabe-type problems address the solvability of the following equation $$D_{\textit{g}}\psi = f(x)|\psi|_{\textit{g}}^{\frac2{m-1}}\psi, \quad \psi:M\to\mathbb{S}(M), \ x\in M$$ where $D_{\textit{g}}$ is the associated Dirac operator and $f:M\to\mathbb{R}$ is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when $m=2$, a solution corresponds to a conformal isometric immersion of the universal covering $\widetilde M$ into $\mathbb{R}^3$ with prescribed mean curvature $f$; meanwhile, for general dimensions and $f\equiv constant\neq0$, a solution provides an upper bound estimate for the B\"ar-Hijazi-Lott invariant. The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold $(S^m,\textit{g})$ where the solution set of the spinorial Yamabe-type problem is not compact: $1).$ the geometric potential $f$ is constant (say $f\equiv1$) with the background metric $\textit{g}$ being a $C^k$ perturbation of the canonical round metric $\textit{g}_{S^m}$, which is not conformally flat somewhere on $S^m$; $2).$ $f$ is a perturbation from constant and is of class $C^2$, while the background metric $\textit{g}\equiv\textit{g}_{S^m}$.