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

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.