Poincar\'e-Sobolev equations with the critical exponent and a potential in the hyperbolic space

TL;DR

This paper investigates a critical exponent semilinear PDE on hyperbolic space, revealing concentration phenomena and employing advanced variational methods to establish solutions for dimensions greater than six.

Contribution

It introduces a novel deformation approach using critical points at infinity and conformal transformations to solve the Poincaré-Sobolev equation with potential on hyperbolic space.

Findings

Concentration occurs along Aubin-Talenti and hyperbolic bubbles.

Solutions exist for dimensions N > 6.

A new variational method handles nontrivial potential and concentration phenomena.

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 $N>6.$ Conformal change of metric is used for proofs, enabling us to convert the original equation into a singular equation in a ball in $\mathbb{R}^N$ and perform a fine blow-up analysis.