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

  • Hong-Ying Li
  • Yang Pu
  • Jia-Feng LiaoEmail author
Article
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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 \(\lambda >0.\)

Keywords

Kirchhoff-type problems Hardy–Sobolev critical exponent positive solution variational method 

Mathematics Subject Classification

35A14 35B09 35B33 

Notes

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. 1.School of Mathematics and InformationChina West Normal UniversityNanchongPeople’s Republic of China
  2. 2.College of Mathematics EducationChina West Normal UniversityNanchongPeople’s Republic of China

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