A note on the log-perturbed Brézis-Nirenberg problem on the hyperbolic space
Monideep Ghosh, Anumol Joseph, Debabrata Karmakar
TL;DR
The paper analyzes the log-perturbed Brézis–Nirenberg problem on hyperbolic space, identifying a threshold in the logarithmic perturbation parameter that governs existence versus nonexistence of positive solutions. It develops a constrained variational framework on a radial log-integrable space and proves existence of a positive ground state for θ>0, both in subcritical and critical regimes, by a limiting argument from subcritical problems. For θ<0, the authors establish nonexistence by deriving sharp lower decay estimates for potential solutions, demonstrating that no positive energy solution can exist in the natural function spaces. The results extend BN-type theory to the hyperbolic setting with a sign-changing perturbation and illuminate how the logarithmic term affects compactness and nonexistence mechanisms. The work combines barrier constructions, decay estimates, and careful variational analysis to deliver an almost sharp threshold and rigorous nonexistence conclusions with potential physical relevance to logarithmic Schrödinger dynamics on curved spaces.
Abstract
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.