The Brezis-Nirenberg problem for the Laplacian with a singular drift in $\mathbb{R}^n$ and $\mathbb{S}^n.$

TL;DR

This paper analyzes the Brezis-Nirenberg problem for the Laplacian with a singular, symmetric drift in balls within Euclidean and spherical spaces, precisely characterizing the parameter regions for existence and non-existence of positive solutions.

Contribution

It provides an exact characterization of the parameter space where positive solutions exist or do not exist for the problem with a singular drift in both Euclidean and spherical settings.

Findings

Identifies the precise parameter regions for solution existence.

Determines the critical thresholds for the parameters involved.

Extends classical results to include singular drift scenarios.

Abstract

We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le n \le 5$. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ($\delta$ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space is characterized by the (geodesic) radius of the ball, $\delta$, and $\lambda$, the coupling constant of the linear term of the Brezis-Nirenberg problem.