School of Mathematical Sciences
Zhejiang Normal University
Jinhua, Zhejiang 321004, China
email: mbyang@zjnu.edu.cn
School of Mathematical Sciences
Zhejiang Normal University
Jinhua, Zhejiang 321004, China
email: 826670487@zjnu.edu.cn
Minbo Yang, Fan Zhou
Abstract:
In this article, we study the existence and multiplicity of solutions
of the quasilinear Dirac-Poisson system
$$\displaylines{
i\sum^3_{k=1}\alpha_k\partial_k u-a\beta u-\omega u-\phi u
=h(x,|u|)u ,\quad x\in R^3,\cr
-\Delta\phi-\varepsilon^4\Delta_4\phi=u^2,\quad x\in R^3,
}$$
where \(\partial_k=\partial/\partial x_k\), \(k=1,2,3\);
\(a>0\) is a constant; \(\alpha_1, \alpha_2, \alpha_3\) and \(\beta\) are
\(4\times 4\) Pauli-Dirac matrices; the operator \(\Delta_4\) is the
4-Laplacian operator, defined as
\(\Delta_4\phi:=\operatorname{div}(|\nabla\phi|^2\nabla\phi)\); and
\(h(x,|u|)u\) describes the self-interaction. We prove the existence of the
least energy solutions for the critical case and obtained that there exist
finitely many critical points under certain conditions by variational
methods. Additionally, we demonstrate the convergence behavior of solutions
as \(\varepsilon\) tends to zero.
Submitted November 26, 2024. Published March 3, 2025.
Math Subject Classifications: 35Q40, 35J92, 49J35.
Key Words: Quasilinear Dirac-Poisson system; strongly indefinite problem; least energy solutions; asymptotic behavior
DOI: 10.58997/ejde.2025.22
Show me the PDF file (466 KB), TEX file for this article.
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Minbo Yang School of Mathematical Sciences Zhejiang Normal University Jinhua, Zhejiang 321004, China email: mbyang@zjnu.edu.cn |
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Fan Zhou School of Mathematical Sciences Zhejiang Normal University Jinhua, Zhejiang 321004, China email: 826670487@zjnu.edu.cn |
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