Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in $ \mathbb{R}^3 $

Zhongxiang Wang; Zhongxiang Wang

doi:10.3934/math.2022490

AIMS Mathematics

2022, Volume 7, Issue 5: 8774-8801. doi: 10.3934/math.2022490

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Research article

Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in $ \mathbb{R}^3 $

Zhongxiang Wang ^,

Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received: 01 October 2021 Revised: 27 January 2022 Accepted: 07 February 2022 Published: 04 March 2022
MSC : 35B38, 35B40, 35J62

This paper is concerned with the following modified Kirchhoff type problem

$ \begin{align*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u-u\Delta (u^2)-\lambda u=|u|^{p-2}u, \; \; \; x\in \mathbb{R}^3, \end{align*} $

where $ a, b > 0 $ are constants and $ \lambda\in \mathbb R $. When $ p=\frac{16}{3} $, we prove that the existence of normalized solution with a prescribed $ L^2 $-norm for the above equation by applying constrained minimization method. Moreover, when $ p\in\left(\frac{10}{3}, \frac{16}{3}\right) $, we prove the existence of mountain pass type normalized solution for the above modified Kirchhoff equation by using the perturbation method. And the asymptotic behavior of normalized solution as $ b\rightarrow 0 $ is analyzed. These conclusions extend some known ones in previous papers.
- modified Kirchhoff-type equations,
- $ L^2 $-normalized solutions,
- asymptotic behavior,
- constrained minimization method,
- perturbation method
Citation: Zhongxiang Wang. Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in $ \mathbb{R}^3 $[J]. AIMS Mathematics, 2022, 7(5): 8774-8801. doi: 10.3934/math.2022490

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Abstract

This paper is concerned with the following modified Kirchhoff type problem

$ \begin{align*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u-u\Delta (u^2)-\lambda u=|u|^{p-2}u, \; \; \; x\in \mathbb{R}^3, \end{align*} $

where $ a, b > 0 $ are constants and $ \lambda\in \mathbb R $. When $ p=\frac{16}{3} $, we prove that the existence of normalized solution with a prescribed $ L^2 $-norm for the above equation by applying constrained minimization method. Moreover, when $ p\in\left(\frac{10}{3}, \frac{16}{3}\right) $, we prove the existence of mountain pass type normalized solution for the above modified Kirchhoff equation by using the perturbation method. And the asymptotic behavior of normalized solution as $ b\rightarrow 0 $ is analyzed. These conclusions extend some known ones in previous papers.

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