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In this paper, we consider the following critical fractional magnetic Choquard equation: ε2s(−Δ)A∕εsu+V(x)u=εα−N∫RN∣u(y)∣2s,α∗∣x−y∣αdy∣u∣2s,α∗−2u+εα−N∫RNF(y,∣u(y)∣2)∣x−y∣αdyf(x,∣u∣2)uinRN,\begin{array}{rcl}{\varepsilon }^{2s}{\left(-\Delta )}_{A/\varepsilon }^{s}u+V\left(x)u& =& {\varepsilon }^{\alpha -N}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{| u(y){| }^{{2}_{s,\alpha }^{\ast }}}{| x-y\hspace{-0.25em}{| }^{\alpha }}{\rm{d}}y\right)| u\hspace{-0.25em}{| }^{{2}_{s,\alpha }^{\ast }-2}u\\ & & +{\varepsilon }^{\alpha -N}\left(\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}\frac{F(y,| u(y){| }^{2})}{| x-y\hspace{-0.25em}{| }^{\alpha }}{\rm{d}}y\right)\hspace{0.08em}f\left(x,| u\hspace{-0.25em}{| }^{2})u\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array} where ε>0\varepsilon \gt 0, s∈(0,1)s\in \left(0,1), α∈(0,N)\alpha \in \left(0,N), N>max{2μ+4s,2s+α∕2}N\gt {\rm{\max }}\left\{2\mu +4s,2s+\alpha /2\right\}, 2s,α∗=2N−αN−2s{2}_{s,\alpha }^{\ast }=\frac{2N-\alpha }{N-2s} is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, (−Δ)As{\left(-\Delta )}_{A}^{s} stands for the fractional Laplacian with periodic magnetic field AA of C0,μ{C}^{0,\mu }-class with μ∈(0,1]\mu \in (0,1] and VV is a continuous potential and allows to be sign-changing. Under some mild assumptions imposed on VV and ff, we establish the existence of at least one ground state solution.

In this paper, we consider the following fractional Choquard equation: \[ \begin{cases} (-\Delta)^s u+V(x) u+\lambda u=(I_\alpha\ast F(u))f(u)\quad \mathrm{in}\ \mathbb{R}^{N},\\ \displaystyle\int_{\mathbb{R}^{N}} u^{2}\,\mathrm{d} x=\rho^{2},\ \rho>0, \end{cases} \] { (− Δ) s u + V (x) u + λu = (I α ∗ F (u)) f (u) in R N , ∫ R N u 2 d x = ρ 2 , ρ > 0 , where $ s\in (0,1) $ s ∈ (0 , 1) , $ \alpha \in (0,N) $ α ∈ (0 , N) , N>2s and $ V:\mathbb {R}^{N}\rightarrow \mathbb {R} $ V : R N → R is a continuous nonnegative potential and vanishes at infinity. Under some mild assumptions imposed on V and f, we establish the existence of $ L^2 $ L 2 -normalized solution $ (u,\lambda)\in H^{s}(\mathbb {R}^{N})\times \mathbb {R}^{+} $ (u , λ) ∈ H s (R N) × R + . [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fourth‐order equation of Kirchhoff type Δ2u−(a+b‖∇u‖22)Δu=λu+|u|p−2uinℝd$$ {\Delta}^2u-\left(a+b{\left\Vert \nabla u\right\Vert}_2^2\right)\Delta u=\lambda u+{\left|u\right|}^{p-2}u\kern0.30em \mathrm{in}\kern6.05pt {\mathrm{\mathbb{R}}}^d $$ under the normalized constraint ‖u‖2=m$$ {\left\Vert u\right\Vert}_2=m $$, where a,b,m>0$$ a,b,m>0 $$ are constants, d≥1$$ d\ge 1 $$ and 2