A complete characterization of the existence of extremals for the Trudinger-Moser inequality on $\mathbb{R}^2$ under sharp $L^p$-perturbations
In this paper, we investigate the following critical Trudinger--Moser inequality on $\mathbb R^2$ under sharp $L^p$-perturbations: $$ S(\lambda,p) := \sup_{\substack{u\in H^{1}(\mathbb R^{2})\\ \int_{\mathbb R^2}(|\nabla u|^2+|u|^2)\,dx\le 1}} \int_{\mathbb R^2} \left(e^{4\pi u^2}-1-\lambda |u|^p\right)\,dx . $$ For $2 \lambda^{\ast}$. 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 $\mathbb R^2$.