Poincar\'e-Sobolev equations with the critical exponent and a potential in the hyperbolic space
Abstract
On the hyperbolic space, we study a semilinear equation with non-autonomous nonlinearity having a critical Sobolev exponent. The Poincar\'e-Sobolev equation on the hyperbolic space explored by Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] resembles our equation. As seen from the profile decomposition of the energy functional associated with the problem, the concentration happens along two distinct profiles: localised Aubin-Talenti bubbles and hyperbolic bubbles. Standard variational arguments cannot obtain solutions because of nontrivial potential and concentration phenomena. As a result, a deformation-type argument based on the critical points at infinity of the associated variational problem has been carried out to obtain solution for Conformal change of metric is used for proofs, enabling us to convert the original equation into a singular equation in a ball in and perform a fine blow-up analysis.
Keywords
Cite
@article{arxiv.2410.03164,
title = {Poincar\'e-Sobolev equations with the critical exponent and a potential in the hyperbolic space},
author = {Mousomi Bhakta and Debdip Ganguly and Diksha Gupta and Alok Kumar Sahoo},
journal= {arXiv preprint arXiv:2410.03164},
year = {2024}
}