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

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

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 ▫$\varepsilon \to 0$▫.

• Language: English
• Keywords: fractional Schrödinger-Kirchhoff equation, fractional magnetic operator, critical nonlinearity, variational methods
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2018
• Number of pages: Str. 1778-1794
• Numbering: Vol. 75, iss. 1
• PID: 20.500.12556/RUL-109503
• UDC: 517.956
• ISSN on article: 0898-1221
• DOI: 10.1016/j.camwa.2017.11.033
• COBISS.SI-ID: 18207577
• Publication date in RUL: 04.09.2019

Record is a part of a journal

• Title: Computers & mathematics with applications
• Shortened title: Comput. math. appl.
• Publisher: Elsevier
• ISSN: 0898-1221
• COBISS.SI-ID: 15336965