The Hardy-Schr\"odinger operator with interior singularity: The remaining cases

Abstract

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 $s=0$, $\gamma \leq 0$.