Fixed-spinor radial solutions of a perturbed Euclidean spinorial Yamabe equation

Abstract

We study the perturbed Euclidean spinorial Yamabe equation \[ D_{g_{\mathbb R^n}}ψ = \frac n2\bigl(1+\varepsilon K(x)\bigr) |ψ|^{2^*-2}ψ \qquad\text{on }\mathbb R^n,\quad n\ge2, \]where $2^*:=2n/(n-1)$, $\varepsilon>0$ is a small parameter and $K$ is a prescribed radial function. We establish a finite-dimensional reduction within a radial spinor space determined by a fixed unit spinor, reducing the problem to the analysis of a function of the dilation parameter. Under suitable assumptions on $K$, we obtain explicit existence criteria for solutions with this prescribed spinorial structure, together with a multiplicity criterion yielding solutions associated with distinct scales. These criteria are expressed in terms of the behavior of $K$ at the origin and weighted integral quantities.