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Existence and Multiplicity Solutions for the \(p\)-Fractional Schrödinger–Kirchhoff Equations with Electromagnetic Fields and Critical Nonlinearity

  • Yueqiang SongEmail author
  • Shaoyun Shi
Article
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Abstract

This paper is devoted to the study of the \(p\)-fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity. By using the variational methods, we obtain the existence of mountain pass solutions \(u_{\varepsilon }\) which tend to the trivial solutions as \(\varepsilon \rightarrow 0\). Moreover, we get \(m^{\ast }\) pairs of solutions for the problem in absence of magnetic effects under some extra assumptions.

Keywords

Fractional Schrödinger–Kirchhoff equations Fractional magnetic operator Critical nonlinearity Variational methods 

Mathematics Subject Classification (2010)

35J10 35B99 35J60 47G20 

Notes

Acknowledgements

Y.Q. Song was supported by NSFC (No. 11301038), the Natural Science Foundation of Jilin Province (No. 20160101244JC), Research Foundation during the 13th Five-Year Plan Period of Department of Education of Jilin Province, China (JJKH20170648KJ). S.Y. Shi was supported by NSFC grant (No. 11771177), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20).

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Scientific Research DepartmentChangchun Normal UniversityChangchunP.R. China
  2. 2.School of MathematicsJilin UniversityChangchunP.R. China
  3. 3.State Key Laboratory of Automotive Simulation and ControlJilin UniversityChangchunP.R. China

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