We consider the following X‐ray free electron lasers Schrödinger equation false(i∇−Afalse)2u+Vfalse(xfalse)u−μfalse|xfalse|u=()1false|xfalse|∗false|ufalse|2u−Kfalse(xfalse)false|ufalse|q−2u,0.1em0.1emx∈ℝ3,$$ {\left(i\nabla -A\right)}^2u+V(x)u-\frac{\mu }{\mid x\mid }u=\left(\frac{1}{\mid x\mid}\ast {\left|u\right|}^2\right)u-K(x){\left|u\right|}^{q-2}u,x\in {\mathbb{R}}^3, $$ where A∈Lloc2false(ℝ3,ℝ3false)$$ A\in {L}_{loc}^2\left({\mathbb{R}}^3,{\mathbb{R}}^3\right) $$ denotes the magnetic potential such that the magnetic field B=curl0.4emA$$ B=\operatorname{curl}\kern0.4em A $$ is ℤ3$$ {\mathbb{Z}}^3 $$‐periodic, μ∈ℝ,0.1emK∈L∞()ℝ3$$ \mu \in \mathbb{R},K\in {L}^{\infty}\left({\mathbb{R}}^3\right) $$ is ℤ3$$ {\mathbb{Z}}^3 $$ periodic and non‐negative, q∈false(2,4false)$$ q\in \left(2,4\right) $$. Using the variational method, based on a profile decomposition of the Cerami sequence in HA1()ℝ3$$ {H}_A^1\left({\mathbb{R}}^3\right) $$, we obtain the existence of the ground state solution for suitable μ≥0$$ \mu \ge 0 $$. When μ<0$$ \mu <0 $$ is small, we also obtain the non‐existence. Furthermore, we give a description of the asymptotic behavior of the ground states as μ→0+$$ \mu \to {0}^{+} $$.
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 $\mu \to 0^+$.
We consider the following fractional Schrödinger-Poisson equation with combined nonlinearitieswhere s ∈ ( 3 4 , 1), µ > 0, p ∈ (3, s * ) and s * = 6 3−2s . By the perturbation approach we prove the existence of the ground state solution in fractional Coulomb-Sobolev space.
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