A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space

2024/07/17 by Ghosh, Monideep, Joseph, Anumol, Karmakar, Debabrata
#35A01 #35A15 #35B09 #35B33 #35G30 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2407.12745

Abstract

We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space Δ_\mathbbB^Nu+λu +|u|^p-1u+θu ln u² =0, u ∈ H¹(\mathbbB^N), u gt; 0 in \mathbbB^N, and study the existence vs non-existence results. We show that whenever θ>0, there exists an H¹-solution, while for θ<0, there does not exist a positive solution in a reasonably general class. Since the perturbation u ln u² 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.

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