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Existence and Multiplicity of Normalized Solutions with Positive Energy for the Kirchhoff Equation

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Abstract

In this paper, we investigate the existence and multiplicity of normalized solutions for the following Kirchhoff equation,

$$\begin{aligned} \left\{ \begin{array}{l} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u-\lambda u=f(u), \quad \text{ in } \mathbb {R}^{3}, \\ \int _{\mathbb {R}^{3}}|u|^{2} d x=c, \end{array}\right. \end{aligned}$$
(P)

where a, b, c are positive constants and \(\lambda \in \mathbb {R}\) is an unknown parameter that appears as a Lagrange multiplier. Two normal solutions, manifesting as a local minimizer or mountain pass solution, have been obtained under the mass subcritical conditions on the nonlinearity f and some suitable mass c. Additionally, we employ the Symmetric Mountain Pass Theorem to establish the multiplicity of normalized solutions for problem (P). To the best of our knowledge, we extend and complement the research success in the Kirchhoff equation for a general nonlinearity with weaker subcritical mass growth.

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Acknowledgements

Supported by Guangdong Basic and Applied Basic Research Foundation (Nos. 2021A1515010383, 2022A1515010644), the Project of Science and Technology of Guangzhou (No.202102020730).

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  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510520, Guangdong, People’s Republic of China

    Lin Xu, Feng Li & Qilin Xie

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  1. Lin Xu
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QX gave this original ideal of this paper. LX and FL wrote the main manuscript text. All authors reviewed the manuscript.

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Correspondence to Qilin Xie.

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Xu, L., Li, F. & Xie, Q. Existence and Multiplicity of Normalized Solutions with Positive Energy for the Kirchhoff Equation. Qual. Theory Dyn. Syst. 23, 135 (2024). https://doi.org/10.1007/s12346-024-01001-3

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