Li, Hong-Ying; Pu, Yang; Liao, Jia-Feng Some results for a class of Kirchhoff-type problems with Hardy-Sobolev critical exponent. (English) Zbl 1421.35112 Mediterr. J. Math. 16, No. 3, Paper No. 63, 16 p. (2019). Summary: We study a class of Kirchhoff equations \[\begin{cases} -\left( a+b \int_{\Omega}|\nabla u|^2\mathrm{d}x\right) \Delta u= \frac{u^{3}}{|x|}+\lambda u^{q},\quad &\text{in } \Omega, \\ u=0, & \text{on }\partial \Omega, \end{cases}\] 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 \(\lambda >0\). MSC: 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs 35B09 Positive solutions to PDEs Keywords:Kirchhoff equations; Hardy-Sobolev critical exponent; positive solutions; variational method PDFBibTeX XMLCite \textit{H.-Y. Li} et al., Mediterr. J. Math. 16, No. 3, Paper No. 63, 16 p. (2019; Zbl 1421.35112) Full Text: DOI References: [1] Alves, C.O., Corrêa, F.J.S.A., Figueiredo, G.M.: On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2, 409-417 (2010) · Zbl 1198.35281 [2] Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85-93 (2005) · Zbl 1130.35045 · doi:10.1016/j.camwa.2005.01.008 [3] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. 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