Abstract
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 $\mu
\to 0^+$.
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