A complete characterization of the existence of extremals for the Trudinger-Moser inequality on ${\mathbb{R}}^2$ under sharp $L^p$-perturbations

Abstract

In this paper, we investigate the following critical Trudinger--Moser inequality on ${\mathbb{R}}^2$ under sharp $L^p$-perturbations: $$S(\lambda ,p):=\underset{\begin{array}{c}u\in H^1({\mathbb{R}}^2)\\ {\int }_{{\mathbb{R}}^2}(|\nabla u{|}^2+|u{|}^2)dx\le 1\end{array}}{\sup }{\int }_{{\mathbb{R}}^2}\left(e^{4\pi u^2}-1-\lambda |u{|}^p\right)dx.$$ For $2<p\le 4$, we prove the existence of a critical value ${\lambda }^{\ast }\in (0,+\infty )$ such that $S(\lambda ,p)$ is attained when $\lambda <{\lambda }^{\ast }$ and is not attained when $\lambda >{\lambda }^{\ast }$. Moreover, we show that the nonattainment in this range is caused by a vanishing phenomenon. For $p=2$, combining our analysis with the nonexistence results for $L^2$-perturbed Trudinger--Moser inequalities obtained in \cite{Chenluzhu}, we establish the existence of two finite thresholds ${\lambda }_{\ast }>-\infty$ and ${\lambda }^{\ast }<+\infty$ such that $S(\lambda ,2)$ is attained when ${\lambda }_{\ast }<\lambda <{\lambda }^{\ast }$, and is not attained when $\lambda <{\lambda }_{\ast }$ or $\lambda >{\lambda }^{\ast }$. In contrast, for $p>4$, we prove that $S(\lambda ,p)$ is attained for all admissible values of $\lambda$. Our results indicate that, in the whole-space setting, the $L^p$-perturbation term affects the existence and nonexistence of extremals through either concentration or vanishing phenomena, which is fundamentally different from the bounded-domain case, where existence or nonexistence is governed solely by concentration phenomena. These results provide a complete characterization of how sharp $L^p$ perturbations determine the existence and nonexistence of extremals for critical Trudinger--Moser inequalities on the entire ${\mathbb{R}}^2$. The resulting existence and nonexistence theory exhibits a threshold structure with respect to the $L^p$ pertubation reminiscent of the classical Brezis--Nirenberg phenomenon in the whole space ${\mathbb{R}}^2$.

43 pages