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, (i-A)^2u+V(x)u-μ xu=(1x* |u|^2)u-K(x)|u|^q-2u, xR^3, where A ∈ L l o c 2 ( ℝ 3, ℝ 3 ) AL_loc^2(R^3, R^3) denotes the magnetic potential such that the magnetic field B = curl A B=curl0.4em A is ℤ 3 Z^3 ‐periodic, μ ∈ ℝ, K ∈ L ∞ ℝ 3 μ R, KL^∞(R^3) is ℤ 3 Z^3 periodic and non‐negative, q ∈ ( 2, 4 ) q(2, 4) . Using the variational method, based on a profile decomposition of the Cerami sequence in H A 1 ℝ 3 H_A^1(R^3) , we obtain the existence of the ground state solution for suitable μ ≥ 0 μ 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^+ .