An improved singular Trudinger-Moser inequality in dimension two

Authors: ANFENG YUAN, ZHIYONG HUANG

Abstract: Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain and $W_0^{1,2}(\Omega)$ be the usual Sobolev space. Let $\beta$, $0\leq\beta<2$, be fixed. Define for any real number $p>1$, $$\lambda_{p,\beta}(\Omega)=\inf_{u\in W_0^{1,2}(\Omega),\,u\not\equiv 0}{\|\nabla u\|_2^2}/{\|u\|_{p,\beta}^2},$$ where $\|\cdot\|_2$ denotes the standard $L^2$-norm in $\Omega$ and $\|u\|_{p,\beta}=({\int_{\Omega}|x|^{-\beta}|u|^pdx})^{1/p}$. Suppose that $\gamma$ satisfies $\f{\gamma}{4\pi}+\f{\beta}{2}=1$. Using a rearrangement argument, the author proves that $$\sup_{u\in W_0^{1,2}(\Omega), \|\nabla u\|_2\leq 1}\int_{\Omega} |x|^{-\beta}e^{\gamma u^2 \le(1+\alpha\|u\|_{p,\beta}^2\ri) }dx$$ is finite for any $\alpha$, $0\leq\alpha<\lambda_{p,\beta}(\mathbb{B}_R)$, where $\mathbb{B}_R$ stands for the disc centered at the origin with radius $R$ verifying that $\pi R^2$ is equal to the area of $\Omega$. Moreover, when $\Omega=\mathbb{B}_R$, the above supremum is infinity if $\alpha\geq \lambda_{p,\beta}(\mathbb{B}_R)$. This extends earlier results of Adimurthi and Druet, Y. Yang, Adimurthi and Sandeep, Adimurthi and Yang, Lu and Yang, and J. Zhu in dimension two.

Keywords: Trudinger-Moser inequality, singular Trudinger-Moser inequality

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