On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

Liang, Sihua (avtor), Repovš, Dušan (avtor), Zhang, Binlin (avtor)

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 $\varepsilon \to 0$.

• Jezik: Angleški jezik
• Ključne besede: fractional Schrödinger-Kirchhoff equation, fractional magnetic operator, critical nonlinearity, variational methods
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: PEF - Pedagoška fakulteta; FMF - Fakulteta za matematiko in fiziko
• Leto izida: 2018
• Št. strani: Str. 1778-1794
• Številčenje: Vol. 75, iss. 1
• PID: 20.500.12556/RUL-109503
• UDK: 517.956
• ISSN pri članku: 0898-1221
• DOI: 10.1016/j.camwa.2017.11.033
• COBISS.SI-ID: 18207577
• Datum objave v RUL: 04.09.2019

LIANG, Sihua, REPOVŠ, Dušan in ZHANG, Binlin, 2018, On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity. Computers & mathematics with applications [na spletu]. 2018. Vol. 75, no. 1, p. 1778–1794. [Dostopano 7 julij 2025]. DOI 10.1016/j.camwa.2017.11.033. Pridobljeno s: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=slv&id=109503

Gradivo je del revije

• Naslov: Computers & mathematics with applications
• Skrajšan naslov: Comput. math. appl.
• Založnik: Elsevier
• ISSN: 0898-1221
• COBISS.SI-ID: 15336965