Degenerate fractional Kirchhoff-type system with magnetic fields and upper critical growth

Sun, Mingzhe (author), Shi, Shaoyun (author), Repovš, Dušan (author)

This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: ▫$$\begin{aligned} \left\{ \begin{array}{lll} -\mathfrak{M}(\Vert u\Vert _{s,A}^2)[(-\Delta )^s_Au+u] = G_u(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal{I}_\mu *|u|^{p^*}\right) |u|^{p^*-2}u \ &{}\text{ in }\,\,\mathbb {R}^N,\\ \mathfrak{M}(\Vert v\Vert _{s,A})[(-\Delta )^s_Av+v] = G_v(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal{I}_\mu *|v|^{p^*}\right) |v|^{p^*-2}v \ &{}\text{ in }\,\,\mathbb{R}^N, \end{array}\right. \end{aligned}$$▫ where ▫$$\begin{aligned}\Vert u\Vert _{s,A}=\left( \iint _{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}{\text{d}}x {\text{d}}y+\int _{\mathbb{R}^N}|u|^2{\text {d}}x\right) ^{1/2},\end{aligned}$$▫ and ▫$(-\Delta )_{A}^s$▫ and ▫$A$▫ are called magnetic operator and magnetic potential, respectively, ▫$\mathfrak{M}: \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}_0$▫ is a continuous Kirchhoff function, ▫$\mathcal{I}_\mu (x) = |x|^{N-\mu }$▫ with ▫$0<\mu <N$▫-function ▫$G$▫ satisfies some suitable conditions, and ▫$p^* =\frac{N+\mu }{N-2s}$▫. We prove the multiplicity results for this problem using the limit index theory. The novelty of our work is the appearance of convolution terms and critical nonlinearities. To overcome the difficulty caused by degenerate Kirchhoff function and critical nonlinearity, we introduce several analytical tools and the fractional version concentration-compactness principles which are useful tools for proving the compactness condition.

• Language: English
• Keywords: fractional Kirchhoff-type system, upper critical exponent, concentration-compactness principle, variational method, multiple solutions
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication version: Author Accepted Manuscript
• Publication date: 01.08.2022
• Year: 2022
• Number of pages: 23 str.
• Numbering: Vol. 19, iss. 4, art. 170
• PID: 20.500.12556/RUL-137708
• UDC: 517.956
• ISSN on article: 1660-5446
• DOI: 10.1007/s00009-022-02076-5
• COBISS.SI-ID: 112532995
• Publication date in RUL: 29.06.2022

Record is a part of a journal

• Title: Mediterranean journal of mathematics
• Shortened title: Mediterr. j. math.
• Publisher: Springer Nature, Birkhäuser
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433