Semiclassical states of a type of Dirac–Klein–Gordon equations with nonlinear interacting terms

Yanheng Ding, Qi Guo () and Yuanyang Yu
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Yanheng Ding: Jilin University
Qi Guo: Renmin University of China
Yuanyang Yu: Tsinghua University

Partial Differential Equations and Applications, 2023, vol. 4, issue 5, 1-26

Abstract: Abstract In this paper, the concentration phenomena of the semiclassical states in the following nonlinear Dirac–Klein–Gordon system are studied $$\begin{aligned} \left\{ \begin{aligned}&-i\varepsilon \sum \limits _{k=1}^3\alpha _k\partial _k u+\omega u+m\beta u+V_1(x) u=F(x,u,\varphi ),\\&-\varepsilon ^2\Delta \varphi +\left( V_2(x)+M^2\right) \varphi =G(x,u,\varphi ), \end{aligned}\right. \end{aligned}$$ - i ε ∑ k = 1 3 α k ∂ k u + ω u + m β u + V 1 ( x ) u = F ( x , u , φ ) , - ε 2 Δ φ + V 2 ( x ) + M 2 φ = G ( x , u , φ ) , where $$u:\mathbb {R}^3\rightarrow \mathbb {C}^4$$ u : R 3 → C 4 , $$\varphi :\mathbb {R}^3\rightarrow \mathbb {R}$$ φ : R 3 → R , with cross nonlinearities $$\begin{aligned}&F(x,u,\varphi )=\frac{s_1}{2}|u|^{s_1-2}|\varphi |^{s_2}u+K(x) |u|^{p-2}u,\\&G(x,u,\varphi )=s_2|u|^{s_1}|\varphi |^{s_2-2}\varphi +Q(x) |\varphi |^{q-2}\varphi . \end{aligned}$$ F ( x , u , φ ) = s 1 2 | u | s 1 - 2 | φ | s 2 u + K ( x ) | u | p - 2 u , G ( x , u , φ ) = s 2 | u | s 1 | φ | s 2 - 2 φ + Q ( x ) | φ | q - 2 φ . Under certain conditions, we show that the systems with $$\varepsilon >0$$ ε > 0 small have semiclassical ground states that are concentrated around sets determined by the competing potential functions. Moreover, we also obtain some properties of these ground states as $$\varepsilon \rightarrow 0$$ ε → 0 , such as the exponentially decay estimate.

Keywords: Dirac–Klein–Gordon system; Solitary waves; Semiclassical states; Nonlinear interaction term; 35A15; 35M30; 35Q60 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00261-z

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