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A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent. (arXiv:1810.05548v1 [math.DG])
来源于:arXiv
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 查看全文>>