全空间中一类非局部问题非平凡解的存在性  

Existence of Nontrivial Solutions for a Class of Nonlocal Problem in R^N

作　　者：杨莲 郭祖记 刘进生 

机构地区：[1]太原理工大学数学学院,山西太原030024

出　　处：《数学的实践与认识》2017年第24期278-288,共11页Mathematics in Practice and Theory

基　　金：国家自然科学青年基金(11601363);山西省青年科技基金(201601D021011)

摘　　要：利用变分方法研究了R^N上一类带有临界非线性项的p-Kirchhoff型问题非平凡解的存在性.首先得到了该问题的能量泛函并证明了其具有山路引理的几何结构.其次给出了山路值c的一个上界并且证明了相应的(PS)_c序列是有界的.最终利用集中紧性原理及其它相关知识证明了能量泛函满足(PS)_c条件,从而表明了能量泛函存在非零的临界点,即证明了该问题至少存在一个非平凡解.This paper is concerned with the existence of nontrivial solutions for p-Kirchhoff type problem in R-N with critical nonlinearities by using variational methods. Firstly, the energy functional of the problem was obtained and it was proved to satisfy mountain pass geometry. Then, an upper bound of mountain pass value c was given and it was proved that the（PS）c sequence of the energy functional is bounded. Consequently, it was proved that the（PS）c sequence contains a strong convergent subsequence by using the concentrated compactness principle and other related knowledge, thus indicated that the energy functional exists nonzero critical points, it was proved that the problem has at least a nontrivial solution.

关 键 词：Kirchhoff型问题 非平凡解 变分方法 集中紧性 

分 类 号：O175[理学—数学]