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

Abstract

Let Ω be a smooth bounded domian in $\mathbb{R}^{2}$, $H_{0}^{1}(\Omega)$ be the standard Sobolev space, and $\lambda_{f}(\Omega)$ be the first weighted eigenvalue of the Laplacian, namely,

$\lambda_{f}(\Omega)=\inf _{u \in H_{0}^{1}(\Omega), \int_{\Omega} u^{2} \mathrm{~d} x=1} \int_{\Omega}|\nabla u|^{2} f \mathrm{~d} x$

where f is a smooth positive function on Ω. In this paper, using blow-up analysis, we prove

$\sup _{u \in H_{0}^{1}(\Omega), \int_{\Omega}|\nabla u|^{2} f \mathrm{~d} x \leq 1} \int_{\Omega} e^{4 \pi f u^{2}\left(1+\alpha\|u\|_{2}^{2}\right)} \mathrm{d} x<+\infty$

for any 0≤α<λ_f(Ω). Furthermore, extremal functions for the above inequality exist when α>0 is chosen sufficiently small.

Keywords

Trudinger-Moser inequality / extremal functions / blow-up analysis