The Hardy-Schrödinger operator with interior singularity: mass and blow-up analysis
We consider the remaining unsettled cases in the problem of existence of positive 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-Schrödinger operator \(L_\gamma:=-\Delta -\frac{\gamma}{|x|^2}\). The higher dimensional case (i.e., when \(\gamma \leq \frac{(n-2)^2}{4}-1\)) has been settled sometime ago. In this paper we deal with the case when \( \frac{(n-2)^2}{4}-1<\gamma< \frac{(n-2)^2}{4}\). If either \(s>0\) or \(s=0\) and \(\gamma > 0\), we show that 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, the classical positive mass theorem is needed for when \(s=0\), \(\gamma \leq 0\) and \(n=3\), which in this case is the critical dimension. This is joint work with Nassif Ghoussoub (UBC, Vancouver).