On the Brezis-Nirenberg problem on S3, and a conjecture of Bandle-Benguria

Abstract

We consider the following Brezis-Nirenberg problem on S3 -ΔS3 u = λu + u5 in D, u > 0 in D and u = 0 on ∂ D, where D is a geodesic ball on S3 with geodesic radius θ1, and ΔS3 is the Laplace-Beltrami operator on S3. We prove that for any λ < π with π - θ1 sufficiently small (depending on λ), there exists bubbling solution to the above problem. This solves a conjecture raised by Bandle and Benguria [J. Differential Equations 178 (2002) 264-279] and Brezis and Peletier [C. R. Acad. Sci. Paris, Ser. I 339 (2004) 291-394]. © 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.