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Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic field. (English) Zbl 1496.35430

Summary: In this paper, we consider the following critical fractional magnetic Choquard equation: \[ \begin{aligned} {\varepsilon }^{2s}{(-\Delta)}_{A/\varepsilon}^su+V(x)u \,= \; &{\varepsilon}^{\alpha-N} \left(\int_{\mathbb{R}^N} \frac{|u(y)|^{2_{s,\alpha}^\ast}}{|x-y|^\alpha}\mathrm{d}y\right) |u|^{2_{s,\alpha}^\ast-2}u \\ & +{\varepsilon }^{\alpha-N}\left(\int_{\mathbb{R}^N}\frac{F(y,| u(y)|^2)}{| x-y|^\alpha} \mathrm{d}y\right) f(x,|u|^2)u \quad\text{in }\mathbb{R}^N, \end{aligned} \] where \(\varepsilon > 0\), \(s\in (0,1)\), \(\alpha \in (0,N)\), \(N> \max \{2 \mu +4s,2s+\alpha /2\}\), \(2_{s,\alpha}^{\ast}=\frac{2N-\alpha}{N-2s}\) is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, \((-\Delta)_A^s\) stands for the fractional Laplacian with periodic magnetic field A of \(C^{0,\mu}\)-class with \(\mu \in (0,1]\) and \(V\) is a continuous potential and allows to be sign-changing. Under some mild assumptions imposed on \(V\) and \(f\), we establish the existence of at least one ground state solution.

MSC:

35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35J61 Semilinear elliptic equations
35R09 Integro-partial differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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