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A Multiplicity Property for a Class of Kirchhoff Problems with Magnetic Potential

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Abstract

This paper is concerned with the mathematical analysis of solutions to following class of magnetic Kirchhoff problems

$$\begin{aligned} \left\{ \begin{array}{ll} -K\Big (\displaystyle \int _\Omega |\nabla _A u|^2dx\Big )\Delta _Au = g(x,|u|^2)u &{} \quad \mathrm{in} \ \Omega \\ {\,u|_{\partial \Omega }=0}\\ \end{array} \right\} , \end{aligned}$$

where \(\Omega \) is a smooth bounded domain in \({{\mathbb {R}}}^N\) (\(N \ge 2\)), A is a magnetic potential, and \(\Delta _A :=(\nabla -iA)^2\) is the magnetic Laplace operator. Additionally, \(K:[0,+\infty )\mapsto (0,+\infty )\) and \(g:{{\bar{\Omega }}}\times {{\mathbb {R}}}\mapsto {{\mathbb {R}}}\) are appropriate continuous functions. Under some natural hypotheses, we obtain the existence of infinitely many solutions in a related magnetic Sobolev space.

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Acknowledgements

This research is partially supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2019zzts211). This paper has been completed while Youpei Zhang was visiting University of Craiova (Romania) with the financial support of China Scholarship Council (No. 201906370079). The author would like to thank the China Scholarship Council and the Embassy of the People’s Republic of China in Romania. The research was also supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project number PCE 137/2021, within PNCDI III.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, China

    Youpei Zhang

  2. Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585, Craiova, Romania

    Youpei Zhang

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Correspondence to Youpei Zhang.

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Zhang, Y. A Multiplicity Property for a Class of Kirchhoff Problems with Magnetic Potential. Results Math 76, 115 (2021). https://doi.org/10.1007/s00025-021-01426-1

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