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

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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.

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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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Authors and Affiliations

  1. Scientific Research Department, Changchun Normal University, Changchun, 130032, Jilin, P.R. China

    Yueqiang Song

  2. School of Mathematics, Jilin University, Changchun, 130012, P.R. China

    Yueqiang Song & Shaoyun Shi

  3. State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, 130012, P.R. China

    Shaoyun Shi

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  1. Yueqiang Song
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Song, Y., Shi, S. Existence and Multiplicity Solutions for the \(p\)-Fractional Schrödinger–Kirchhoff Equations with Electromagnetic Fields and Critical Nonlinearity. Acta Appl Math 165, 45–63 (2020). https://doi.org/10.1007/s10440-019-00240-w

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  • DOI: https://doi.org/10.1007/s10440-019-00240-w

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