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

Abstract

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.

Keywords

• choquard equation
• fractional magnetic laplacian
• ground state solution
• critical
• hardy-littlewood-sobolev inequality
• 35a15
• 35j60
• 58e05