R^N上一类临界p-Kirchhoff型问题非平凡解的存在性

Existence of Nontrivial Solutions for a class p-Kirchhoff type Problems with Critical Nonlinearities in R^N

主要通过变分方法研究了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.首先得到了问题的能量泛函并证明了其具有山路引理的几何结构,由此获得了能量泛函的一个(PS)_c序列.其次证明了此(PS)_c序列有界并且给出了c的一个上界.最终利用相关知识证明了此(PS)_c序列存在收敛子列,从而证明了问题至少存在一个非平凡解.

The existence of nontrivial solutions for a class p-Kirchhoff type problems in R^N with critical nonlinearities was investigated by using variational methods. Firstly,the energy functional of the problem was obtained and it was proved to satisfy mountain pass geometry, and thus a（PS）_c sequence of the energy functional was obtained. Then, it was proved that the sequence is bounded and an upper bound of c was given. Consequently, it was proved that the（PS）_c sequence has convergent subsequence by using related knowledge,thus indicated that the energy functional exists nonzero critical points, it was proved that the problem has at least a nontrivial solution.