On the ground states for the X‐ray free electron lasers Schrödinger equation. (31st October 2022)

• Record Type: Journal Article
• Authors: Hu, Hangzhou; Li, Yuan; Zhao, Dun
• Is Part Of: Mathematical methods in the applied sciences. Volume 46:Number 5(2023)
• Page Start: 5099
• Page End: 5118
• Publication Date: 2022-10-31
• Subjects: asymptotic behavior -- ground states -- NLS equations -- periodic magnetic field -- variational methods for second‐order elliptic equations -- XFEL Schrödinger equation; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8820
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We consider the following X‐ray free electron lasers Schrödinger equation ( i ∇ − A ) 2 u + V ( x ) u − μ | x | u = 1 | x | ∗ | u | 2 u − K ( x ) | u | q − 2 u, x ∈ ℝ 3, $$ {\left(i\nabla -A\right)}&#x0005E;2u&#x0002B;V(x)u-\frac{\mu }{\mid x\mid }u&#x0003D;\left(\frac{1}{\mid x\mid}\ast {\left&#x0007C;u\right&#x0007C;}&#x0005E;2\right)u-K(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{q-2}u, x\in {\mathbb{R}}&#x0005E;3, $$ where A ∈ L l o c 2 ( ℝ 3, ℝ 3 ) $$ A\in {L}_{loc}&#x0005E;2\left({\mathbb{R}}&#x0005E;3, {\mathbb{R}}&#x0005E;3\right) $$ denotes the magnetic potential such that the magnetic field B = curl A $$ B&#x0003D;\operatorname{curl}\kern0.4em A $$ is ℤ 3 $$ {\mathbb{Z}}&#x0005E;3 $$ ‐periodic, μ ∈ ℝ, K ∈ L ∞ ℝ 3 $$ \mu \in \mathbb{R}, K\in {L}&#x0005E;{\infty}\left({\mathbb{R}}&#x0005E;3\right) $$ is ℤ 3 $$ {\mathbb{Z}}&#x0005E;3 $$ periodic and non‐negative, q ∈ ( 2, 4 ) $$ q\in \left(2, 4\right) $$ . Using the variational method, based on a profile decomposition of the Cerami sequence in H A 1 ℝ 3 $$ {H}_A&#x0005E;1\left({\mathbb{R}}&#x0005E;3\right) $$, we obtain the existence of the ground state solution for suitable μ ≥ 0 $$ \mu \ge 0 $$ . When μ < 0 $$ \mu &lt;0 $$ is small, we also obtain the non‐existence. Furthermore, we give a description of the asymptotic behavior of the ground states as μ → 0 + $$ \mu \to {0}&#x0005E;{&#x0002B;} $$ .