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arXiv · 2410.03164

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

Mousomi Bhakta·Debdip Ganguly·Diksha Gupta·Alok Kumar Sahoo
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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.

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Mousomi Bhakta, Debdip Ganguly, Diksha Gupta, Alok Kumar Sahoo. 2024-10-04. Poincar\'e-Sobolev equations with the critical exponent and a potential in the hyperbolic space. https://arxiv.org/abs/2410.03164

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