|
| 摘要: |
| 设~$D$~是~Dirac~算子,~$u:S^{N}\rightarrow \Sigma S^{N}$~是一个旋量.本文研究了具有临界指数的~p-Dirac~方程
\begin{equation}\label{eq0.1}
D_{p} u = {|u|}^{p^{*}-2} u + f(u)
\end{equation}
解的存在性和多解性.首先,因为方程~(\ref{eq0.1})~含有临界增长的非线性项,使得~Sobolev~嵌入失去紧性,所以本文利用球面~$S^{N}$~上一个等距子群的作用,适当缩小所考虑的函数空间,使得~Sobolev~嵌入重新获得紧性;然后,利用山路定理证明方程~(\ref{eq0.1})~存在一个弱解;最后,利用双正交系理论对函数空间进行分解,结合喷泉定理证明方程~(\ref{eq0.1})~的多解性. |
| 关键词: $p-Dirac$~方程 山路定理 喷泉定理 群作用 |
| DOI： |
| 分类号:O175.25 |
| 基金项目:国家自然科学基金青年科学基金项目 |
|
| Existence and multiplicity of solutions for the p-Dirac equation with critical exponent on a sphere |
|
liu ying,yang xu
|
| Abstract: |
| \textbf{Abstract:}
~Assuming that~$D $~is the Dirac operator and $u:S^{N} \rightarrow \Sigma S^{N} $~is a spinor. This article investigates existence and multiplicity of solutions about the ~$p-Dirac $~equation with critical exponent
\begin{equation*}
D_{p} u = {|u|}^{p^{*}-2} u + f(u).
\end{equation*}
Firstly, because equation~(\ref {eq0.1})~contains a critical growth nonlinear term, the Sobolev~embedding loses its compactness. Therefore, in this paper, we utilize the action of an isometry subgroup on the sphere~$S^{N} $~to appropriately reduce the considered function space, so that the Sobolev~embedding regains compactness; Then, using the Mountain Road Theorem, we prove the existence of a weak solution to equation~(\ref{eq0.1})~; Finally, the function space is decomposed using the theory of orthogonal systems, and the multiple solutions of equation~(\ref{eq0.1})~are proved by combining the Fountain theorem. |
| Key words: P-Dirac equation Mountain Road theorem Fountain theorem Group action |
