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

Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ ($n\geq 3$) such that $0\in\partial \Omega$. In this memoir, we consider issues of non-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 high energy-- solutions of corresponding subcritical problems to establish, for example, 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, then the above equation has an infinite number of (possibly sign-changing) solutions in ${H_{1,0}^2(\Omega)}$. This complements results of the first and third authors, who had previously shown that if $\gamma\leq \frac{n^2}{4}-\frac{1}{4}$ and the mean curvature of $\partial\Omega$ at $0$ is negative, then the equation has a positive solution. On the other hand, the sharp 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 a surprising stability under $C^1$-perturbations of the potential $h$ of those regimes where no variational positive solutions exist. In particular, and in sharp contrast with the non-singular case (i.e., when $\gamma=s=0$), we show non-existence of such solutions for (E) in any dimension, whenever $\Omega$ is star-shaped and $h$ is close to $0$, which include situations not covered by the classical Pohozaev obstruction.