We study a class of Kirchhoff equations -a+b∫Ω|∇u|2dxΔu=u3|x|+λuq,inΩ,u=0,on∂Ω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( a+b\displaystyle \int _{\Omega }|\nabla u|^2\mathrm{d}x\right) \Delta u=\displaystyle \frac{u^{3}}{|x|}+\lambda u^{q},&{}\hbox {in } \Omega , \\ u=0, &{}\hbox {on } \partial \Omega , \end{array}\right. } \end{aligned}$$\end{document}where Ω⊂R3\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^{3}$$\end{document} is a bounded domain with smooth boundary and 0∈Ω\documentclass[12pt]{minimal}
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\begin{document}$$0\in \Omega $$\end{document}, a,b,λ>0,0<q<1.\documentclass[12pt]{minimal}
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\begin{document}$$a,b,\lambda >0,0<q<1.$$\end{document} By the variational method, two positive solutions are obtained. Moreover, when b>1A12\documentclass[12pt]{minimal}
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\begin{document}$$b>\frac{1}{A_{1}^{2}}$$\end{document} (A1>0\documentclass[12pt]{minimal}
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\begin{document}$$A_{1}>0$$\end{document} is the best Sobolev–Hardy constant), using the critical point theorem, infinitely many pairs of distinct solutions are obtained for any λ>0.\documentclass[12pt]{minimal}
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\begin{document}$$\lambda >0.$$\end{document}
