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In this article, we investigate necessary and sufficient conditions on the perturbation ρ for the existence of positive least energy solutions of the critical singular semilinear elliptic equation 
$ -\Delta u = \frac{|u|^{2^{*}(s)-2}}{|x|^s}u + \rho(u) $ with Dirichlet boundary condition in a bounded smooth domain in 
$\mathbb R^n$ containing the origin, where 
$2^*(s)=\frac{2(n-s)}{n-2}$, 
$0\leq s \lt 2 \lt n$. We show that the almost necessary and sufficient condition obtained for the case s = 0 in [1] differs conceptually when 
$0 \lt s \lt 2$.
In this paper, we study the following Kirchhoff-type equation:
where a, b are positive constants and N = 1, 2, 3. Under appropriate assumptions on V, K and g, we obtain a ground-state solution by using the approach developed by Szulkin and Weth in 2010.
