TY - JOUR
T1 - A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two
AU - Zhao , Juan
AU - Yu , Pengxiu
JO - Journal of Partial Differential Equations
VL - 4
SP - 402
EP - 416
PY - 2024
DA - 2024/12
SN - 37
DO - http://doi.org/10.4208/jpde.v37.n4.3
UR - https://global-sci.org/intro/article_detail/jpde/23688.html
KW - Trudinger-Moser inequality, extremal functions, blow-up analysis.
AB - <p style="text-align: justify;">Let $Ω$ be a smooth bounded domian in $\mathbb{R}^2$ , $H^1_0
(Ω)$ be the standard Sobolev
space, and $λ_f
(Ω)$ be the first weighted eigenvalue of the Laplacian, namely, $$\lambda_f(\Omega)=\inf\limits_{u\in H^1_0(\Omega),\int_{\Omega}u^2{\rm dx}=1}\int_{\Omega}|\nabla u|^2f{\rm dx},$$where $f$ is a smooth positive function on $Ω.$ In this paper, using blow-up analysis, we
prove$$\sup\limits_{u\in H^1_0(\Omega),\int_{\Omega}|\nabla u|^2f{\rm dx}\le 1}\int_{\Omega}e^{4\pi fu^2(1+\alpha||u||^2_2)}{\rm dx}&lt;+\infty$$for any $0≤α&lt;λ_f
(Ω).$ Furthermore, extremal functions for the above inequality exist
when $α&gt;0$ is chosen sufficiently small.</p>