College of Science
East China Jiaotong University
Nanchang 330013, China
email: caizixing1229@163.com
College of Science
East China Jiaotong University
Nanchang 330013, China
email: wangli.423@163.com
Zixing Cai, Li Wang
Abstract:
This article studies normalized solutions for the
critical Schrodinger-Bopp-Podolsky system with singular coefficient
and critical growth,
$$\displaylines{
-\Delta u +\lambda u = \phi u + \dfrac{|u|^{s-2}u}{|x|^q}
+ ( I_{\alpha}*|u|^{2_\alpha^*})|u|^{2_\alpha^*-2}u, \quad x \in \mathbb{R}^3, \cr
-\Delta \phi+d^2\Delta ^2\phi = 4\pi u^2, \quad x \in \mathbb{R}^3
}$$
under the mass constraint
$$
\int_{\mathbb{R}^3} |u|^2 dx = a^2,
$$
where \(a >0\) is a constant, \( 0< \alpha < 3\),
\( 0 < q < 2< s < 10/3\) satisfying \( q+\frac{s}{2} < 3 < q+\frac{3s}{2}< 5 \), and
$$
I_\alpha(x) = \frac{A_\alpha}{|x|^{3 - \alpha}}, \quad \text{with }
A_\alpha = \frac{\Gamma\big(\frac{3 - \alpha}{2}\big)}{2^\alpha
\pi^{3/2}\Gamma(\frac{\alpha}{2})}
$$
which is the Riesz potential.
Here \( 2^*_{\alpha} = 3+\alpha\) is the Hardy Littlewood Sobolev upper critical
exponent. We establish the existence of multiple normalized solutions by using
truncation techniques and topological genus theory.
To address the non-compactness of the energy functional caused by singular and
critical growth, we use the concentration-compactness principle.
This paper also presents results regarding the stabilization of the
orbit and the asymptotic behavior of the ground-state solution as \(d\to 0\).
Submitted January 23, 2026. Published July 16, 2026.
Math Subject Classifications: 35J60, 35J50, 35B38.
Key Words: Schrodinger-Bopp-Podolsky system; normalized solutions;
orbitally stable; asymptotic behavior.
DOI: 10.58997/ejde.2026.56
Show me the PDF file (457 KB), TEX file for this article.
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Zixing Cai College of Science East China Jiaotong University Nanchang 330013, China email: caizixing1229@163.com |
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Li Wang College of Science East China Jiaotong University Nanchang 330013, China email: wangli.423@163.com |
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