The Hardy–Schrödinger 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 $\gamma 0$ 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 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 $s=0$, $\gamma \leq 0$.