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Existence of Ground State Solutions for Kirchhoff Problems with Hardy Potential

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Abstract

This study is concerned with the following Kirchhoff problem:

$$\begin{aligned} -\Bigg (a + b\int _{{\mathbb {R}}^{3}}|\nabla u|^{2}\text{ d }x\Bigg )\Delta u -\frac{\mu }{|x|^{2}} u = g(u)\,\,\,\,\,\text{ in }\,\, {\mathbb {R}}^{3}\backslash \{0\},\,\,\,\;\;\;\;\;\;\;\;\;\;\;\text{(A) } \end{aligned}$$

where \(a, b > 0 \) are constants, \(\mu < \frac{1}{4}\). \(\frac{1}{|x|^{2}}\) is called the Hardy potential and \(g: {\mathbb {R}} \rightarrow {\mathbb {R}} \) is a continuous function that satisfies the Berestycki–Lion type condition. Using variational methods, we establish two existence results for problem (A) under different conditions for g. Furthermore, if \(\mu < 0\), we prove that the mountain pass level in \(H^{1}({\mathbb {R}}^{3})\) can not be achieved.

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Acknowledgements

This paper is supported by Natural Science Foundation of Fujian Province (No.2020J01708 & No.2022J01339) and National Foundation Training Program of Jimei University (ZP2020057).

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  1. School of Sciences, Jimei University, Xiamen, 361021, People’s Republic of China

    MengYun Zhou & YongYi Lan

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MZ,YL wrote the main manuscript text. All authors reviewed the manuscript.

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Zhou, M., Lan, Y. Existence of Ground State Solutions for Kirchhoff Problems with Hardy Potential. Qual. Theory Dyn. Syst. 22, 141 (2023). https://doi.org/10.1007/s12346-023-00841-9

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