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Asymptotic properties of standing waves for Maxwell-Schrödinger-Poisson system
Journal of Mathematical Analysis and Applications ( IF 1.3 ) Pub Date : 2020-06-01 , DOI: 10.1016/j.jmaa.2019.123835
Tingxi Hu , Lu Lu

Abstract In this paper, we study the asymptotic properties of minimizers for a class of constraint minimization problems derived from the Maxwell-Schrodinger-Poisson system − Δ u − ( | u | 2 ⁎ | x | − 1 ) u − α | u | 2 p u − μ p u = 0 , x ∈ R 3 on the L 2 -spheres A λ = { u ∈ H 1 ( R 3 ) : ∫ R 3 | u | 2 d x = λ } , where α , p > 0 . Let λ ⁎ = ‖ Q 2 3 ‖ 2 2 , and Q 2 3 is the unique (up to translations) positive radial solution of − 3 p 2 Δ u + 2 − p 2 u − | u | 2 p u = 0 in R 3 with p = 2 3 . We prove that if λ α − 3 2 λ ⁎ , then minimizers are relatively compact in A λ as p ↗ 2 3 . On the contrary, if λ > α − 3 2 λ ⁎ , by directly using asymptotic analysis, we prove that all minimizers must blow up and give the detailed asymptotic behavior of minimizers.

中文翻译:

麦克斯韦-薛定谔-泊松系统驻波的渐近性质

摘要 在本文中,我们研究了从 Maxwell-Schrodinger-Poisson 系统导出的一类约束最小化问题的最小化器的渐近性质 − Δ u − ( | u | 2 ⁎ | x | − 1 ) u − α | 你| 2 pu − μ pu = 0 , x ∈ R 3 在 L 2 球面上 A λ = { u ∈ H 1 ( R 3 ) : ∫ R 3 | 你| 2 dx = λ } ,其中 α , p > 0 。令 λ ⁎ = ‖ Q 2 3 ‖ 2 2 ,Q 2 3 是 − 3 p 2 Δ u + 2 − p 2 u − |的唯一(直到平移)正径向解。你| 2 pu = 0 在 R 3 中 p = 2 3 。我们证明,如果 λ α − 3 2 λ ⁎ ,那么极小值在 A λ 中相对紧凑,因为 p ↗ 2 3 。相反,如果 λ > α − 3 2 λ ⁎ ,通过直接使用渐近分析,我们证明所有的极小化器必定爆炸,并给出极小化器的详细渐近行为。
更新日期:2020-06-01
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