$\boldsymbol{\rm R}^2$上带精确$L^p$扰动的Trudinger-Moser不等式极值函数存在性的完整刻画
A complete characterization of the existence of extremals for the Trudinger-Moser inequality on $\mathbb{R}^2$ under sharp $L^p$-perturbations
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中文总结 AI 辅助
本文完整刻画了$\boldsymbol{\rm R}^2$上带精确$L^p$扰动的临界Trudinger-Moser不等式极值函数的存在性，揭示了不同$p$取值下的阈值结构及与有界域情形的本质差异。
中文摘要 AI 辅助
本文研究$\boldsymbol{\rm R}^2$上带精确$L^p$扰动的临界Trudinger–Moser不等式：$$ S(λ,p) := \sup_{\substack{u\in H^{1}(\mathbb R^{2})\\\\\\\\ \int_{\mathbb R^2}(|\nabla u|^2+|u|^2)\\\\,dx\le 1}} \int_{\mathbb R^2} \left(e^{4πu^2}-1-λ|u|^p\right)\\\\,dx . $$ 对于$2<p\le 4$的情形，我们证明存在临界值$λ^{\ast}\in(0,+\infty)$，使得当$λ<λ^{\ast}$时$S(λ,p)$可达，当$λ>λ^{\ast}$时$S(λ,p)$不可达。此外，我们证明该参数范围内的不可达性是由消失现象导致的。\n 对于$p=2$的情形，我们将自身分析与文献[Chenluzhu]中得到的带$L^2$扰动的Trudinger–Moser不等式的不存在性结果相结合，证明存在两个有限阈值$λ_{\ast}>-\infty$和$λ^{\ast}<+\infty$，使得当$λ_{\ast}<λ<λ^{\ast}$时$S(λ,2)$可达，当$λ<λ_{\ast}$或$λ>λ^{\ast}$时$S(λ,2)$不可达。\n 与之相对，对于$p>4$的情形，我们证明对所有可容许的$λ$值，$S(λ,p)$均可达。\n 我们的结果表明，在全空间设定下，$L^p$扰动项通过集中现象或消失现象影响极值函数的存在性与不存在性，这与有界区域情形存在本质区别——有界区域中极值的存在与否仅由集中现象决定。\n 这些结果完整刻画了精确$L^p$扰动如何决定全空间$\boldsymbol{\rm R}^2$上临界Trudinger–Moser不等式极值函数的存在性与不存在性。由此得到的存在与不存在理论呈现出关于$L^p$扰动的阈值结构，让人联想到全空间$\boldsymbol{\rm R}^2$中的经典Brezis–Nirenberg现象。
英文摘要
In this paper, we investigate the following critical Trudinger--Moser inequality on $\mathbb R^2$ under sharp $L^p$-perturbations: $$ S(λ,p) := \sup_{\substack{u\in H^{1}(\mathbb R^{2})\\ \int_{\mathbb R^2}(|\nabla u|^2+|u|^2)\,dx\le 1}} \int_{\mathbb R^2} \left(e^{4πu^2}-1-λ|u|^p\right)\,dx . $$ For $2<p\le 4$, we prove the existence of a critical value $λ^{\ast}\in(0,+\infty)$ such that $S(λ,p)$ is attained when $λ<λ^{\ast}$ and is not attained when $λ>λ^{\ast}$. 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 $λ_{\ast}>-\infty$ and $λ^{\ast}<+\infty$ such that $S(λ,2)$ is attained when $λ_{\ast}<λ<λ^{\ast}$, and is not attained when $λ<λ_{\ast}$ or $λ>λ^{\ast}$. In contrast, for $p>4$, we prove that $S(λ,p)$ is attained for all admissible values of $λ$. 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 $\mathbb R^2$.
