Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities

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

Abstract

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 ▫$0 0$▫ 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.

Keywords

Choquard-type equation;critical nonlinearity;fractional magnetic operator;variational method;

Data

Language:	English
Year of publishing:	2020
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.956
COBISS:	18870617
ISSN:	0170-4214
Views:	468
Downloads:	289
Average score:	0 (0 votes)
Metadata:

Other data

Type (COBISS):	Article
Pages:	str. 2473-2490
Volume:	ǂVol. ǂ43
Issue:	ǂiss. ǂ5
Chronology:	March 2020
DOI:	10.1002/mma.6057
ID:	11763917