Abstract
Given a Riemannian manifold (M,g) and a group G of isometries of (M,g), we investigate the existence G-invariant positive solutions u:M→R to the nonlinear equation Δgu+au=u2⋆(k,s)−1dg(x,Gx0)s+huq−1 where Δg=−divg(∇). The singularity and the nonlinearity are such that the problem is critical for the Hardy-Sobolev embeddings. We prove existence by using the Aubin minimization and the Mountain-Pass lemma of Ambrosetti-Rabinowitz. As a by product of our analysis, we find the value of the best-constant in the associated Hardy-Sobolev inequality.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
