Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus

TL;DR

This paper investigates the existence, multiplicity, and uniqueness of positive radial solutions to a super-critical Brezis-Nirenberg problem in an annulus, revealing conditions for multiple solutions and uniqueness in three dimensions.

Contribution

It establishes new uniqueness results in 3D and demonstrates the existence of multiple solutions depending on the annulus size and exponent range.

Findings

Unique positive radial solutions in 3D under certain conditions

Existence of at least three solutions for small inner radius

At least k solutions for exponents between critical and Joseph-Lundgren

Abstract

The super-critical Brezis-Nirenberg problem in an annulus is considered. The new uniqueness result of positive radial solutions is established for the three-dimensional case. It is also proved that the problem has at least three positive radial solutions when the inner radius of the annulus is sufficiently small and the outer radius of the annulus is in a certain range. Moreover, for each positive integer $k$, the problem has at least $k$ positive radial solutions when the exponent of the equation is greater than the critical Sobolev exponent and is less than the Joseph-Lundgren exponent.