A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two

Abstract

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$$\sup\limits_{u\in H^1_0(\Omega),\int_{\Omega}|\nabla u|^2f{\rm dx}\le 1}\int_{\Omega}e^{4\pi fu^2(1+\alpha||u||^2_2)}{\rm dx}<+\infty$$for any $0≤α<λ_f (Ω).$ Furthermore, extremal functions for the above inequality exist when $α>0$ is chosen sufficiently small.