On the ground states for the X-ray free electron lasers Schrödinger equation

We consider the following X-ray free electron lasers Schr\”{o}dinger equation \begin{equation*} (i\nabla-A)^2u+V(x)u-\frac{\mu}{|x|} u=\left(\frac{1}{|x|}*|u|^2\right) u-K(x)|u|^{q-2} u, \,\, x\in \mathbb{R}^3, \end{equation*} where $A\in L_{loc}^2(\mathbb{R}^3,\mathbb{R}^3)$ denotes the magnetic potential such that the magnetic field $B=\text{curl} \, A$ is $\mathbb{Z}^{3}$-periodic, $\mu\in \mathbb{R}$, $K \in L^{\infty}\left(\mathbb{R}^3\right)$ is $\mathbb{Z}^{3}$ -periodic and non-negative, $q\in(2,4)$. Using the variational method, based on a profile decomposition of the Cerami sequence in $H^1_A\left(\mathbb{R}^3\right)$, we obtain the existence of the ground state solution for suitable $\mu\geq0$. When $\mu<0$ is small, we also obtain the non-existence. Furthermore, we give a description for the asymptotic behaviour of the ground states as $\mu \to 0^+$.