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

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

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