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In this paper, we study a class of critical elliptic problems of Kirchhoff type:    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$ \biggl[a+b \biggl( \int_{\mathbb{R}^{3}}\vert \nabla u\vert ^{2}-\mu \frac{u^{2}}{\vert x\vert ^{2}}\,dx \biggr)^{\frac{2-\alpha }{2}} \biggr]\biggl(-\Delta u- \mu \frac{u}{\vert x\vert ^{2}}\biggr) = \frac{\vert u\vert ^{2^{*}(\alpha )-2}u }{\vert x\vert ^{\alpha }}+\lambda \frac{f(x)\vert u\vert ^{q-2}u }{\vert x\vert ^{\beta }}, $$\end{document}    [  a  +  b    (   ∫   R  3    |  ∇  u   |  2   −  μ    u  2    |  x   |  2d  x  )2  −  α   2    ]    (  −  Δ  u  −  μ   u   |  x   |  2)   =    |  u   |    2  ∗   (  α  )  −  2    u    |  x   |  α+  λ    f  (  x  )  |  u   |   q  −  2    u    |  x   |  β,where    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$a,b>0$\end{document}   a  ,  b  >  0,    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\mu \in [0,1/4)$\end{document}   μ  ∈  [  0  ,  1  /  4  ),    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\alpha , \beta \in [0,2)$\end{document}   α  ,  β  ∈  [  0  ,  2  ), and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$q\in (1,2)$\end{document}   q  ∈  (  1  ,  2  )are constants and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$2^{*}(\alpha )=6-2\alpha $\end{document}    2  ∗   (  α  )  =  6  −  2  αis the Hardy–Sobolev exponent in    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\mathbb{R}^{3}$\end{document}    R  3. For a suitable function    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$f(x)$\end{document}   f  (  x  ), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$b>0$\end{document}   b  >  0as a parameter to obtain the convergence property of solutions for the given problem as    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$b\searrow 0^{+}$\end{document}   b  ↘   0  +by the mountain pass theorem and Ekeland’s variational principle.

Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy–Sobolev exponent and singular nonlinearity