Acta mathematica scientia,Series A ›› 2024, Vol. 44 ›› Issue (4): 871-884.

### Multiplicity and Asymptotic Behavior of Normalized Solutions for Kirchhoff-Type Equation

Jin Zhenfeng1,2,Sun Hongrui2,*(),Zhang Weimin3

1. 1School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031
2School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000
3School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering Applications (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241
• Received:2023-07-24 Revised:2024-04-29 Online:2024-08-26 Published:2024-07-26
• Supported by:
NSF of Shanxi Province(202303021212160);NSFC(11671181);Science Technology Program of Gansu Province(21JR7RA535)

Abstract:

In this paper, we consider the following Kirchhoff-type equation

$\begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\ d x\right)\Delta u=\lambda u+|u|^{p-2}u \quad \mathrm{in}\ \mathbb{R}^{3},\\ \|u\|^2_{2}=\rho,\end{cases}$

where $a$, $b$, $\rho>0$ and $\lambda\in\mathbb{R}$ arises as Lagrange multiplier with respect to the mass constraint $\|u\|^2_{2}=\rho$. When $p\in\left(2,\frac{10}{3}\right)$ or $p\in\left(\frac{14}{3},6\right)$, we establish the existence of infinitely many radial $L^2$-normalized solutions by using the genus theory. Furthermore, we testify an asymptotic behavior of the above solutions with respect to the parameter $b\rightarrow 0^+$.

CLC Number:

• O175
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