广西师范大学学报（自然科学版） ›› 2023, Vol. 41 ›› Issue (6): 105-112.doi: 10.16088/j.issn.1001-6600.2023030402

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一类具有临界指数和不定位势的Kirchhoff型问题解的存在性

郭洁, 索洪敏*, 朱怡颖, 郭加超   

  1. 贵州民族大学 数据科学与信息工程学院, 贵州 贵阳 550025
  • 收稿日期:2023-03-04 修回日期:2023-05-05 发布日期:2023-12-04
  • 通讯作者: 索洪敏(1965—), 男(布依族), 贵州都匀人, 贵州民族大学教授。E-mail: gzmysxx88@sina.com
  • 基金资助:
    国家自然科学基金(11661021, 11861021); 贵州省科技计划项目(黔科合基础〔2017〕1084)

Existence of Solutions for a Class of Kirchhoff Type Problems with Critical Exponent and Indefinite Potential

GUO Jie, SUO Hongmin*, ZHU Yiying, GUO Jiachao   

  1. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang Guizhou 550025, China
  • Received:2023-03-04 Revised:2023-05-05 Published:2023-12-04

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摘要： 本文研究一类具有临界指数和不定位势的Kirchhoff型问题解的存在性。首先证明该问题的能量泛函满足山路结构; 其次证明能量泛函满足局部(PS)c条件,从而获得泛函的紧性条件; 最后通过Ekeland变分原理和山路引理,得到该问题2个非平凡解的存在性。

关键词: Kirchhoff型方程, 临界指数, 不定位势, 变分法

Abstract: In this paper, the existence of solutions for a class of Kirchhoff type problems with critical exponent and indefinite potential is studied. Firstly, the energy functional is proved to satisfy the mountain pass geometry. Secondly, the compactness condition of function is obtained by proving that the energy functional satisfies the local (PS)c condition. Finally, the existence of two nontrivial solutions is obtained via Ekeland’s variational principle and the mountain pass lemma.

Key words: Kirchhoff type equations, critical exponent, indefinite potential, variational method

中图分类号:  O177.9

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