## A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent. (arXiv:1810.05548v1 [math.DG])

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 $2&lt;p&lt;\frac{2m}{m-1}$, $\alpha\in\mathbb{R}$. 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 nontrivial solutions $(\lambda,\psi)$ of (BND) for every $\lambda&gt;0$, ev查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 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 nontrivial solutions $(\lambda,\psi)$ of (BND) for every $\lambda>0$, ev
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