Zhewen Chen, Muzi Li
Abstract:
This article shows the existence of normalized solutions for Kirchhoff type
system with van der Waals type potentials,
$$\displaylines{
-(a+b\int_{\mathbb{R}^N}|\nabla u_1|^2dx)\Delta u_1=\lambda_1 u_1+\mu_1(I_\alpha\ast|u_1|^{p_1})|u_1|^{p_1-2}u_1+
\Theta r_1(I_\beta\ast|u_2|^{r_2})|u_1|^{r_1-2}u_1, \cr
-(a+b\int_{\mathbb{R}^N}|\nabla u_2|^2dx)\Delta u_2=\lambda_2 u_2+\mu_2(I_\alpha\ast|u_2|^{p_2})|u_2|^{p_2-2}u_2+
\Theta r_2(I_\beta\ast|u_1|^{r_1})|u_2|^{r_2-2}u_2, \cr
\int_{\mathbb{R}^N}|u_1|^2dx=d_1>0,\quad
\int_{\mathbb{R}^N}|u_2|^2dx=d_2>0,
}$$
where \(N=3,4\), \(\mu_1,\mu_2,\Theta >0\), \(\frac{N+\alpha}{N}< p_1,p_2< \frac{N+\alpha+2}{N}\),
\(2\cdot\frac{N+\beta}{N}< r_1+r_2< 2\cdot2_\beta^*=2\cdot\frac{N+\beta}{N-2}\),
\(0< \alpha,\beta< N\), \(I_\alpha\) and \(I_\beta\) are the Riesz potentials.
We show that the system has a positive least energy solution at negative energy
level for \(\Theta\) small. In addition, we also prove that the system admits a
high energy positive solution at positive energy level in the special case.
Submitted July 2, 2025. Published January 19, 2026.
Math Subject Classifications: 35J20, 35J47, 35J50.
Key Words: Ground state; normalized solution; Van der Waals type potential; Kirchhoff system.
DOI: 10.58997/ejde.2026.06
Show me the PDF file (460 KB), TEX file for this article.
| Zhewen Chen College of Science Jimei University Xiamen 361021, China email: zhewenchen@jmu.edu.cn |
| Muzi Li College of Mathematics and Statistics Fujian Normal University Qishan Campus, Fuzhou 350117, China email: mutoudededemuzi@163.com |
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