A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two (Journal of Partial Differential Equations, 2024)
Let $Ω$ be a smooth bounded domian in $\mathbb{R}^2$ , $H^1_0(Ω)$ be the standard Sobolevspace, and $λ_f(Ω)$ be the first weighted eigenvalue of the Laplacian, namely, $$λ_f(Ω)=\inf\limits_{u\in H^1_0(Ω),\int_{Ω}u^2{\rm dx}=1}\int_{Ω}| abla u|^2f{\rm dx},$$where $f$ is a smooth positive function on $Ω.$ In this paper, using blow-up analysis, weprove$$\sup\limits_{u\in H^1_0(Ω),\int_{Ω}| abla u|^2f{\rm dx}\le 1}\int_{Ω}e^{4π fu^2(1+α||u||^2_2)}{\rm dx}
Keywords: Trudinger-Moser inequality; extremal functions; blow-up analysis.