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Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) suchthat $0\in\partial \Omega$. In this memoir, we consider issues ofnon-existence, existence, and multiplicity of variational solutions in$H_{1,0}^2(\Omega)$ for the borderline Dirichlet problem, $-\Delta u-\gamma\frac{u}{|x|^2}- h(x) u = \frac{|u|^{{2^\star(s)}-2}u}{|x|^s}$ in $\Omega$,where $0<s<2$, ${{2^\star(s)}}:=\frac{2(n-s)}{n-2}$, $\gamma\in\mathbb{R}$ and$h\in C^0(\overline{\Omega})$. We use sharp blow-up analysis on --possibly highenergy-- solutions of corresponding subcritical problems to establish, forexample, that if $\gamma<\frac{n^2}{4}-1$ and the principal curvatures of$\partial\Omega$ at $0$ are non-positive but not all of them vanishing, thenthe above equation has an infinite number of (possibly sign-changing) solutionsin ${H_{1,0}^2(\Omega)}$. This complements results of the first and thirdauthors, who had previously shown that if $\gamma\leq\frac{n^2}{4}-\frac{1}{4}$ and the mean curvature of $\partial\Omega$ at $0$ isnegative, then the equation has a positive solution. On the other hand, thesharp blow-up analysis also allows us to prove that if the mean curvature at$0$ is non-zero and if the mass (when defined) does not vanish, then there is asurprising stability under $C^1$-perturbations of the potential $h$ of thoseregimes where no variational positive solutions exist. In particular, and insharp contrast with the non-singular case (i.e., when $\gamma=s=0$), we shownon-existence of such solutions for (E) in any dimension, whenever $\Omega$ isstar-shaped and $h$ is close to $0$, which include situations not covered bythe classical Pohozaev obstruction.

Multiplicity and stability of the Pohozaev obstruction for Hardy-Schr\"odinger equations with boundary singularity