Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

Abstract

In this paper, a class of Kirchhoff type equations in {\mathbb{R}}^N(N⩾3) with zero mass and a critical term is studied. Under some appropriate conditions, the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem. The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem. Compared to the usual Kirchhoff-type problems, we only require the nonlinearity to satisfy the classical superquadratic condition (Ambrosetti-Rabinowitz condition).

Key words： Kirchhoff type equations with a critical term; variational methods; Symmetric Mountain Pass theorem; Second Concentration Compactness lemma

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11701346, 11671239, 11801338), the Natural Science Foundation of Shanxi Province (Grant No. 201801D211001), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2019L0024), and the Research Project Supported by Shanxi Scholarship Council of China (Grant No. 2020-005).

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