A spinorial analogue of the Brezis-Nirenberg theorem.

Let $(M,g,\sigma)$ be a compact Riemannian spin manifold of dimension $m \ge 2,$ let $\mathbb S(M)$ denote the spinor bundle on $M$, and let $D$ be the Atiyah-Singer Dirac operator acting on spinors $\Psi:M\to \mathbb S(M)$. We present recent results on the existence of solutions of the nonlinear Dirac equation with critical exponent $$D\Psi=\lambda \Psi+f(|\Psi|)\Psi+|\Psi|^{2\over m-1}\Psi$$ where $\lambda\in\mathbb R$ and $f(|\Psi|)\Psi$ is a subcritical nonlinearity in the sense that $f(s)=o\left(s^{2\over m-1}\right)$ as $s\to\infty$. This is joint work with Tian Xu.Non UBCUnreviewedAuthor affiliation: Universität GiessenFacult