中文摘要：

考虑如下Kirchhoff型问题 {-（a＋b∫R^N｜△↓u｜^2dx）△u＋V（x）u=f（x,u）在R^N上， u∈H^1（R^N）. 通过山路引理，喷泉定理和对称山路引理得到问题非平凡解的存在性和多解性．

英文摘要：

The existence and multiplicity of nontrivial solutions are obtained for the Kirchhoff- type problem {-（a＋b∫R^N｜△↓u｜^2dx）△u＋V（x）u=f（x,u） in R^N, u∈H^1（R^N）. by using the Mountain Pass Lemma, the Fountain Theorem and the Symmetric Mountain Pass Lemma.