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| 摘要: |
| 设D是Dirac算子,u:SN→ΣSN是一个旋量.本文研究了具有临界指数的p-Dirac方程Dpu = |u|p*-2u + f(u) (0.1)解的存在性和多解性.首先,因为方程（0.1）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SN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（0.1）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（0.1）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:国家自然科学基金资助(11801499). |
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| EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR THE P-DIRAC EQUATION WITH CRITICAL EXPONENT ON A SPHERE |
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LIU Ying,YANG Xu
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| Abstract: |
| Assuming that D is the Dirac operator and u:SN→ΣSN is a spinor.This article investigates existence and multiplicity of solutions about the p-Dirac equation with critical exponent Dpu = |u|p*-2u + f(u) (0.1) Firstly,because equation （0.1） 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 SN 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 （0.1）;finally,the function space is decomposed using the theory of orthogonal systems,and the multiple solutions of equation （0.1） are proved by combining the Fountain theorem. |
| Key words: P-Dirac equation mountain road theorem fountain theorem group action |
