Electron. J. Differential Equations, Vol. 2025 (2025), No. 22, pp. 1-23.

Existence and multiplicity of solutions to quasilinear Dirac-Poisson systems

Minbo Yang, Fan Zhou

Abstract:
In this article, we study the existence and multiplicity of solutions of the quasilinear Dirac-Poisson system i∑k=13αk∂ku−aβu−ωu−ϕu=h(x,|u|)u,x∈R3,−Δϕ−ε4Δ4ϕ=u2,x∈R3, where ∂k=∂/∂xk, k=1,2,3; a>0 is a constant; α1,α2,α3 and β are 4×4 Pauli-Dirac matrices; the operator Δ4 is the 4-Laplacian operator, defined as Δ4ϕ:=div⁡(|∇ϕ|2∇ϕ); 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 ε 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

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Minbo Yang
School of Mathematical Sciences
Zhejiang Normal University
Jinhua, Zhejiang 321004, China
email: mbyang@zjnu.edu.cn
Fan Zhou
School of Mathematical Sciences
Zhejiang Normal University
Jinhua, Zhejiang 321004, China
email: 826670487@zjnu.edu.cn

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