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Some Results for a Class of Kirchhoff-Type Problems with Hardy–Sobolev Critical Exponent

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Abstract

We study a class of Kirchhoff equations

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( a+b\displaystyle \int _{\Omega }|\nabla u|^2\mathrm{d}x\right) \Delta u=\displaystyle \frac{u^{3}}{|x|}+\lambda u^{q},&{}\hbox {in } \Omega , \\ u=0, &{}\hbox {on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^{3}\) is a bounded domain with smooth boundary and \(0\in \Omega \), \(a,b,\lambda >0,0<q<1.\) By the variational method, two positive solutions are obtained. Moreover, when \(b>\frac{1}{A_{1}^{2}}\) (\(A_{1}>0\) is the best Sobolev–Hardy constant), using the critical point theorem, infinitely many pairs of distinct solutions are obtained for any \(\lambda >0.\)

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Acknowledgements

The authors express their gratitude to the reviewer for careful reading and helpful suggestions which led to an improvement of the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Information, China West Normal University, Nanchong, 637002, Sichuan, People’s Republic of China

    Hong-Ying Li, Yang Pu & Jia-Feng Liao

  2. College of Mathematics Education, China West Normal University, Nanchong, 637002, Sichuan, People’s Republic of China

    Jia-Feng Liao

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  1. Hong-Ying Li
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Correspondence to Jia-Feng Liao.

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Supported by the Scientific Research Fund of Sichuan Provincial Education Department (18ZA0471), Fundamental Research Funds of China West Normal University (15D006, 17E089, 18B015, 18D052), Meritocracy Research Funds of China West Normal University (17YC383) and Innovative Research Team of China West Normal University (CXTD2018-8).

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Li, HY., Pu, Y. & Liao, JF. Some Results for a Class of Kirchhoff-Type Problems with Hardy–Sobolev Critical Exponent. Mediterr. J. Math. 16, 63 (2019). https://doi.org/10.1007/s00009-019-1349-3

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  • DOI: https://doi.org/10.1007/s00009-019-1349-3

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