  We are concerned with a semi-linear elliptic equation on a smooth bounded
domain $\Omega$ of $\mathbb{R}^n,\,n\geq 5,$ which involves a critical
nonlinearity and a linear term of the form $K(x)u^{(n+2)/(n-2)}$ and $\mu u,$
respectively. By using a test function procedure, we give an existence
criterion involving the parameter $\mu$ and the function $K(x).$ For a
particular case of $\Omega,\,K(x)$ and $n,$ we prove its optimality through a
Pohozaev type identity.
