A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916563416776704 |
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| author | Ghosh, Monideep Joseph, Anumol Karmakar, Debabrata |
| author_facet | Ghosh, Monideep Joseph, Anumol Karmakar, Debabrata |
| contents | We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space
\begin{align*}
Δ_{\mathbb{B}^N}u+λu +|u|^{p-1}u+θu \ln u^2 =0, \ \ \ \ u \in H^1(\mathbb{B}^N), \ u > 0 \ \mbox{in} \ \mathbb{B}^N,
\end{align*}
and study the existence vs non-existence results. We show that whenever $θ>0,$ there exists an $H^1$-solution, while for $θ<0$, there does not exist a positive solution in a reasonably general class. Since the perturbation $ u \ln u^2$ changes sign, Pohozaev type identities do not yield any non-existence results. The main contribution of this article is obtaining an "almost" precise lower asymptotic decay estimate on the positive solutions for $θ<0,$ culminating in proving their non-existence assertion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space Ghosh, Monideep Joseph, Anumol Karmakar, Debabrata Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A01, 35A15, 35B09, 35B33, 35G30 We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space \begin{align*} Δ_{\mathbb{B}^N}u+λu +|u|^{p-1}u+θu \ln u^2 =0, \ \ \ \ u \in H^1(\mathbb{B}^N), \ u > 0 \ \mbox{in} \ \mathbb{B}^N, \end{align*} and study the existence vs non-existence results. We show that whenever $θ>0,$ there exists an $H^1$-solution, while for $θ<0$, there does not exist a positive solution in a reasonably general class. Since the perturbation $ u \ln u^2$ changes sign, Pohozaev type identities do not yield any non-existence results. The main contribution of this article is obtaining an "almost" precise lower asymptotic decay estimate on the positive solutions for $θ<0,$ culminating in proving their non-existence assertion. |
| title | A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35A01, 35A15, 35B09, 35B33, 35G30 |
| url | https://arxiv.org/abs/2407.12745 |