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Semiclassical states for Dirac-Klein-Gordon system with critical growth
Journal of Mathematical Analysis and Applications ( IF 1.2 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.jmaa.2020.124092
Yanheng Ding , Qi Guo , Yuanyang Yu

Abstract In this paper, we study the following critical Dirac-Klein-Gordon system in R 3 : { i e ∑ k = 1 3 α k ∂ k u − a β u + V ( x ) u − λ ϕ β u = P ( x ) f ( | u | ) u + Q ( x ) | u | u , − e 2 Δ ϕ + M ϕ = 4 π λ ( β u ) ⋅ u , where e > 0 is a small parameter, a > 0 is a constant. We prove the existence and concentration of solutions under suitable assumptions on the potential V ( x ) , P ( x ) and Q ( x ) . We also show the semiclassical solutions ω e with maximum points x e concentrating at a special set H P characterized by V ( x ) , P ( x ) and Q ( x ) , and for any sequence x e → x 0 ∈ H P , v e ( x ) : = ω e ( e x + x e ) converges in H 1 ( R 3 , C 4 ) to a least energy solution u of { i ∑ k = 1 3 α k ∂ k u − a β u + V ( x 0 ) u − λ ϕ β u = P ( x 0 ) f ( | u | ) u + Q ( x 0 ) | u | u , − Δ ϕ + M ϕ = 4 π λ ( β u ) ⋅ u .

中文翻译:

具有临界增长的 Dirac-Klein-Gordon 系统的半经典态

摘要 在本文中,我们研究了 R 3 中的以下临界 Dirac-Klein-Gordon 系统: { ie ∑ k = 1 3 α k ∂ ku − a β u + V ( x ) u − λ ϕ β u = P ( x ) f ( | u | ) u + Q ( x ) | 你| u , − e 2 Δ ϕ + M ϕ = 4 π λ ( β u ) ⋅ u ,其中 e > 0 是一个小参数,a > 0 是一个常数。我们证明了在对电位 V ( x ) 、 P ( x ) 和 Q ( x ) 进行适当假设的情况下解的存在和浓度。我们还展示了最大点 xe 集中在以 V ( x ) , P ( x ) 和 Q ( x ) 为特征的特殊集合 HP 的半经典解 ω e ,并且对于任何序列 xe → x 0 ∈ HP , ve ( x ) := ω e ( ex + xe ) 在 H 1 ( R 3 , C 4 ) 中收敛到 { i ∑ k = 1 3 α k ∂ ku − a β u + V ( x 0 ) u − 的最小能量解 u - λ ϕ β u = P ( x 0 ) f ( | u | ) u + Q ( x 0 ) | 你| u , − Δ ϕ + M ϕ = 4 π λ ( β u ) ⋅ u 。
更新日期:2020-08-01
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