On constrained minimizers for Kirchhoff type equations with Berestycki-Lions type mass subcritical conditions

Jing Hu; Jijiang Sun$ ^{} $; Jing Hu; Jijiang Sun$ ^{} $

doi:10.3934/era.2023131

Electronic Research Archive

2023, Volume 31, Issue 5: 2580-2594. doi: 10.3934/era.2023131

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On constrained minimizers for Kirchhoff type equations with Berestycki-Lions type mass subcritical conditions

Jing Hu ,
Jijiang Sun$ ^{} $ ^,

School of Mathematics and Computer Sciences, Nanchang University, Nanchang, Jiangxi 330031, China

Received: 23 January 2023 Revised: 24 February 2023 Accepted: 27 February 2023 Published: 07 March 2023

In this paper, for given mass $ m > 0 $, we focus on the existence and nonexistence of constrained minimizers of the energy functional

$ \begin{equation*} I(u): = \frac{a}{2}\int_{\mathbb{R}^3}\left|\nabla u\right|^2dx+\frac{b}{4}\left(\int_{\mathbb{R}^3}\left|\nabla u\right|^2dx\right)^2-\int_{\mathbb{R}^3}F(u)dx \end{equation*} $

on $ S_m: = \left\{u\in H^1(\mathbb{R}^3):\, \|u\|^2_2 = m\right\}, $where $ a, b > 0 $ and $ F $ satisfies the almost optimal mass subcritical growth assumptions. We also establish the relationship between the normalized ground state solutions and the ground state to the action functional $ I(u)-\frac{\lambda}{2}\|u\|_2^2 $. Our results extend, nontrivially, the ones in Shibata (Manuscripta Math. 143 (2014) 221–237) and Jeanjean and Lu (Calc. Var. 61 (2022) 214) to the Kirchhoff type equations, and generalize and sharply improve the ones in Ye (Math. Methods. Appl. Sci. 38 (2015) 2603–2679) and Chen et al. (Appl. Math. Optim. 84 (2021) 773–806).
- Kirchhoff type equations,
- constrained minimizers,
- $ L^2 $-subcritical,
- Berestycki-Lions type conditions
Citation: Jing Hu, Jijiang Sun$ ^{} $. On constrained minimizers for Kirchhoff type equations with Berestycki-Lions type mass subcritical conditions[J]. Electronic Research Archive, 2023, 31(5): 2580-2594. doi: 10.3934/era.2023131

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Abstract

In this paper, for given mass $ m > 0 $, we focus on the existence and nonexistence of constrained minimizers of the energy functional

$ \begin{equation*} I(u): = \frac{a}{2}\int_{\mathbb{R}^3}\left|\nabla u\right|^2dx+\frac{b}{4}\left(\int_{\mathbb{R}^3}\left|\nabla u\right|^2dx\right)^2-\int_{\mathbb{R}^3}F(u)dx \end{equation*} $

on $ S_m: = \left\{u\in H^1(\mathbb{R}^3):\, \|u\|^2_2 = m\right\}, $where $ a, b > 0 $ and $ F $ satisfies the almost optimal mass subcritical growth assumptions. We also establish the relationship between the normalized ground state solutions and the ground state to the action functional $ I(u)-\frac{\lambda}{2}\|u\|_2^2 $. Our results extend, nontrivially, the ones in Shibata (Manuscripta Math. 143 (2014) 221–237) and Jeanjean and Lu (Calc. Var. 61 (2022) 214) to the Kirchhoff type equations, and generalize and sharply improve the ones in Ye (Math. Methods. Appl. Sci. 38 (2015) 2603–2679) and Chen et al. (Appl. Math. Optim. 84 (2021) 773–806).

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