Poincaré-Sobolev equations with the critical exponent and a potential in the hyperbolic space
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910632505245696 |
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| author | Bhakta, Mousomi Ganguly, Debdip Gupta, Diksha Sahoo, Alok Kumar |
| author_facet | Bhakta, Mousomi Ganguly, Debdip Gupta, Diksha Sahoo, Alok Kumar |
| contents | On the hyperbolic space, we study a semilinear equation with non-autonomous nonlinearity having a critical Sobolev exponent. The Poincaré-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03164 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Poincaré-Sobolev equations with the critical exponent and a potential in the hyperbolic space Bhakta, Mousomi Ganguly, Debdip Gupta, Diksha Sahoo, Alok Kumar Analysis of PDEs Functional Analysis On the hyperbolic space, we study a semilinear equation with non-autonomous nonlinearity having a critical Sobolev exponent. The Poincaré-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. |
| title | Poincaré-Sobolev equations with the critical exponent and a potential in the hyperbolic space |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2410.03164 |