Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 22, pp. 1-23.

Title: Existence and multiplicity of solutions to quasilinear Dirac-Poisson systems

Authors: Minbo Yang (Zhejiang Normal Univ., Jinhua, Zhejiang, China)
         Fan Zhou   (Zhejiang Normal Univ., Jinhua, Zhejiang, China)
  
Abstract:
In this article, we study the existence and multiplicity of solutions 
of the quasilinear Dirac-Poisson system
$${
 i\sum^3_{k=1}\alpha_k\partial_k u-a\beta u-\omega u-\phi u
  =h(x,|u|)u ,\quad x\in R^3,\cr
 -\Delta\phi-\varepsilon^4\Delta_4\phi=u^2,\quad x\in R^3,
}$$
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 03, 2025.
Math Subject Classifications: 35Q40, 35J92, 49J35.
Key Words: Quasilinear Dirac-Poisson system; strongly indefinite problem; least energy solutions; asymptotic behavior