Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic field

Abstract

Abstract In this paper, we consider the following critical fractional magnetic Choquard equation: ε 2 s ( − Δ ) A ∕ ε s u + V ( x ) u = ε α − N ∫ R N ∣ u ( y ) ∣ 2 s , α ∗ ∣ x − y ∣ α d y ∣ u ∣ 2 s , α ∗ − 2 u + ε α − N ∫ R N F ( y , ∣ u ( y ) ∣ 2 ) ∣ x − y ∣ α d y f ( x , ∣ u ∣ 2 ) u in R N , \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 { 2 μ + 4 s , 2 s + α ∕ 2 } N\gt {\rm{\max }}\left\{2\mu +4s,2s+\alpha /2\right\} , 2 s , α ∗ = 2 N − α N − 2 s {2}_{s,\alpha }^{\ast }=\frac{2N-\alpha }{N-2s} is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, ( − Δ ) A s {\left(-\Delta )}_{A}^{s} stands for the fractional Laplacian with periodic magnetic field A A of C 0 , μ {C}^{0,\mu } -class with μ ∈ ( 0 , 1 ] \mu \in (0,1] and V V is a continuous potential and allows to be sign-changing. Under some mild assumptions imposed on V V and f f , we establish the existence of at least one ground state solution.