Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

On the Hardy-Schrödinger operator with a boundary singularity (1410.1913v2)

Published 7 Oct 2014 in math.AP

Abstract: We investigate the Hardy-Schr\"odinger operator $L_\gamma=-\Delta -\frac{\gamma}{|x|2}$ on domains $\Omega\subset\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in the interior of $\Omega$. For one, if $0\in\Omega$, then $L_\gamma$ is positive if and only if $\gamma<\frac{(n-2)2}{4}$, while if $0\in\partial\Omega$ the operator $L_{\gamma}$ could be positive for larger value of $\gamma$, potentially reaching the maximal constant $\frac{n2}{4}$ on convex domains. We prove optimal regularity and a Hopf-type Lemma for variational solutions of corresponding linear Dirichlet boundary value problems of the form $L_{\gamma} u=a(x)u$, but also for non-linear equations including $L_{\gamma} u=\frac{|u|{\crits-2}u}{|x|s}$, where $\gamma <\frac{n2}{4}$, $s\in [0,2)$ and $\crits:=\frac{2(n-s)}{n-2}$ is the critical Hardy-Sobolev exponent. We also provide a Harnack inequality and a complete description of the profile of all positive solutions --variational or not-- of the corresponding linear equation on the punctured domain. The value $\gamma=\frac{n2-1}{4}$ turned out to be another critical threshold for the operator $L\gamma$, and our analysis yields a corresponding notion of "Hardy singular boundary-mass" $m_\gamma(\Omega)$ of a domain $\Omega$ having $0\in \partial \Omega$, which could be defined whenever $\frac{n2-1}{4}<\gamma<\frac{n2}{4}$.

Summary

We haven't generated a summary for this paper yet.