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

Abstract

We consider the log-perturbed Brézis-Nirenberg problem on the hyperbolic space${\Delta }_{B^N}u+\lambda u+|u{|}^{p-1}u+\theta u\ln u^2=0,u\in H^1(B^N),u>0\text{in}{B}^N,$ and study the existence vs non-existence results. We show that whenever $\theta >0$, there exists an $H^1$-solution, while for $\theta <0$, 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 $\theta <0$, culminating in proving their non-existence assertion.