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  • On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity
    Liang, Sihua ; Repovš, Dušan, 1954- ; Zhang, Binlin
    In this paper, we consider the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity ▫$$\begin{cases} \varepsilon^{2s}M([u]^2_{s, ... A_{\varepsilon}})(-\Delta)^s_{A_\varepsilon} u + V(x)u = |u|^{{2^\ast_s}-2}u + h(x, |u|^2)u, \quad x \in \mathbb{R}^N ,\\ u(x) \to 0, \quad \text{as} \; |x| \to \infty, \end{cases}$$▫ where ▫$(-\Delta)^s_{A_\varepsilon}$▫ is source is the fractional magnetic operator with ▫$0 < s < 1$▫, ▫$2^\ast_s = 2N/(N - 2s)$▫, ▫$M \colon \mathbb{R}^+_0 \to \mathbb{R}^+$▫ is a continuous nondecreasing function, ▫$V \colon \mathbb{R}^N \to \mathbb{R}^+_0$▫ and ▫$A \colon \mathbb{R}^N \to \mathbb{R}^N$▫ are the electric and magnetic potentials, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ▫$\varepsilon < \mathcal{E}$▫; and (ii) for any ▫$m^\ast \in \mathbb{N}$▫, has ▫$m^\ast$▫ pairs of solutions if ▫$\varepsilon < \mathcal{E}_{m^\ast}$▫, where ▫$\mathcal{E}$▫ and $▫\mathcal{E}_{m^\ast}$▫ are sufficiently small positive numbers. Moreover, these solutions ▫$u_\varepsilon \to 0$▫ as ▫$\varepsilon \to 0$▫.
    Vir: Computers & mathematics with applications. - ISSN 0898-1221 (Vol. 75, iss. 1, March 2018, str. 1778-1794)
    Vrsta gradiva - članek, sestavni del
    Leto - 2018
    Jezik - angleški
    COBISS.SI-ID - 18207577

    Povezava(-e):

    http://doi.org/10.1016/j.camwa.2017.11.033
    Repozitorij Univerze v Ljubljani – RUL
    DOI

vir: Computers & mathematics with applications. - ISSN 0898-1221 (Vol. 75, iss. 1, March 2018, str. 1778-1794)
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