A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent

Abstract

Let $(M,\textit{g},\sigma)$ be a compact Riemannian spin manifold of dimension $m\geq2$, 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 study the existence of solutions of the nonlinear Dirac equation with critical exponent \[ D\psi = \lambda\psi + f(|\psi|)\psi + |\psi|^{\frac2{m-1}}\psi \tag{NLD} \] where $\lambda\in\mathbb{R}$ and $f(|\psi|)\psi$ is a subcritical nonlinearity in the sense that $f(s)=o\big(s^{\frac2{m-1}}\big)$ as $s\to\infty$. A model nonlinearity is $f(s)=\alpha s^{p-2}$ with $2<p<\frac{2m}{m-1}$, $\alpha\in\mathbb{R}$. In particular we study the nonlinear Dirac equation \[ D\psi=\lambda\psi+|\psi|^{\frac2{m-1}}\psi, \quad \lambda\in\mathbb{R}. \tag{BND} \] This equation is a spinorial analogue of the Brezis-Nirenberg problem. As corollary of our main results we obtain the existence of least energy solutions $(\lambda,\psi)$ of (BND) and (NLD) for every $\lambda>0$, even if $\lambda$ is an eigenvalue of $D$. For some classes of nonlinearities $f$ we also obtain solutions of (NLD) for every $\lambda\in\mathbb{R}$, except for non-positive eigenvalues. If $m\not\equiv3$ (mod 4) we obtain solutions of (NLD) for every $\lambda\in\mathbb{R}$, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it. The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD).