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

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

Abstract

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 ▫$\varepsilon \to 0$▫.

Keywords

fractional Schrödinger-Kirchhoff equation;fractional magnetic operator;critical nonlinearity;variational methods;

Data

Language:	English
Year of publishing:	2018
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.956
COBISS:	18207577
ISSN:	0898-1221
Views:	515
Downloads:	386
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	str. 1778-1794
Volume:	ǂVol. ǂ75
Issue:	ǂiss. ǂ1
Chronology:	March 2018
DOI:	10.1016/j.camwa.2017.11.033
ID:	11215354