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  • Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities
    Liang, Sihua ; Repovš, Dušan, 1954- ; Zhang, Binlin
    In this paper, we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical ... nonlinearity: ▫$$\begin{cases} \varepsilon^{2s} N([u]^2_{s,A}) (-\Delta)^s_A u + V(x)u = (|x|^{-\alpha} \ast F(|u|^2)) f(|u|^2)u + |u|^{2^\ast_s-2}u, & x\in \mathbb{R}^N, \\ U(x) \to 0, & \text{as} \quad |x| \to \infty, \end{cases}$$▫ where ▫$(-\Delta)^s_A$▫ is the fractional magnetic operator with ▫$0<s<1$▫, ▫$2^\ast_s = 2N/(N-2s)$▫, ▫$\alpha < \min\{N, 4s\}$▫, ▫$M \colon \mathbb{R}^+_0 \to \mathbb{R}^+_0$▫ is a continuous function, ▫$A\colon \mathbb{R}^N \to \mathbb{R}^N$▫ is the magnetic potential, ▫$F(|u|) =\int^{|u|}_0f(t)dt$▫, and ▫$\varepsilon > 0$▫ is a positive parameter. The electric potential ▫$V \in C(\mathbb{R}^N, \mathbb{R}^+_0)$▫ satisfies ▫$V(x)=0$▫ in some region of ▫$\mathbb{R}^N$▫, which means that this is the critical frequency case. We first prove the ▫$(PS)_c$▫ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term ▫$M$▫ can vanish at zero.
    Vir: Mathematical methods in the applied sciences. - ISSN 0170-4214 (Vol. 43, iss. 5, March 2020, str. 2473-2490)
    Vrsta gradiva - članek, sestavni del ; neleposlovje za odrasle
    Leto - 2020
    Jezik - angleški
    COBISS.SI-ID - 18870617

    Povezava(-e):

    https://doi.org/10.1002/mma.6057
    Repozitorij Univerze v Ljubljani – RUL
    DOI

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