On the ground states for the X‐ray free electron lasers Schrödinger equation. (31st October 2022) - Hu, Hangzhou; Li, Yuan; Zhao, Dun
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, $$ {\left(i\nabla -A\right)}^2u+V(x)u-\frac{μ }{\mid x\mid }u=\left(\frac{1}{\mid x\mid}* {\left|u\right|}^2\right)u-K(x){\left|u\right|}^{q-2}u, x\in {\mathbb{R}}^3, $$ where A ∈ L l o c 2 ( ℝ 3, ℝ 3 ) $$ A\in {L}_{loc}^2\left({\mathbb{R}}^3, {\mathbb{R}}^3\right) $$ denotes the magnetic potential such that the magnetic field B = curl A $$ B=\operatorname{curl}\kern0.4em A $$ is ℤ 3 $$ {\mathbb{Z}}^3 $$ ‐periodic, μ ∈ ℝ, K ∈ L ∞ ℝ 3 $$ μ \in \mathbb{R}, K\in {L}^{∞}\left({\mathbb{R}}^3\right) $$ is ℤ 3 $$ {\mathbb{Z}}^3 $$ 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 $$ H_A^1\left({\mathbb{R}}^3\right) $$, we obtain the existence of the ground state solution for suitable μ ≥ 0 $$ μ \ge 0 $$ . When μ < 0 $$ μ <0 $$ is small, we also obtain the non‐existence. Furthermore, we give a description of the asymptotic behavior of the ground states as μ → 0 + $$ μ → {0}^{+} $$ .