  We consider the remaining unsettled cases in the problem of existence of
energy minimizing solutions for the Dirichlet value problem $L_\gamma u-\lambda
u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in
$\mathbb{R}^n$ ($n\geq 3$) having the singularity $0$ in its interior. Here
$\gamma <\frac{(n-2)^2}{4}$, $0\leq s <2$, $2^*(s):=\frac{2(n-s)}{n-2}$ and
$0\leq \lambda <\lambda_1(L_\gamma)$, the latter being the first eigenvalue of
the Hardy-Schr\"odinger operator $L_\gamma:=-\Delta -\frac{\gamma}{|x|^2}$.
There is a threshold $\lambda^*(\gamma, \Omega) \geq 0$ beyond which the
minimal energy is achieved, but below which, it is not. It is well known that
$\lambda^*(\Omega) = 0$ in higher dimensions, for example if $0\leq \gamma \leq
\frac{(n-2)^2}{4}-1$. Our main objective in this paper is to show that this
threshold is strictly positive in "lower dimensions" such as when $
\frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}$, to identify the critical
dimensions (i.e., when the situation changes), and to characterize it in terms
of $\Omega$ and $\gamma$. If either $s>0$ or if $\gamma > 0$, i.e., in {\it the
truly singular case}, we show that in low dimensions, a solution is guaranteed
by the positivity of the "Hardy-singular internal mass" of $\Omega$, a notion
that we introduce herein. On the other hand, and just like the case wnen
$\gamma=s=0$ studied by Brezis-Nirenberg and completed by Druet, $n=3$ is the
critical dimension, and the classical positive mass theorem is sufficient for
the {\it merely singular case}, that is when $s=0$, $\gamma \leq 0$.
