Turkish Journal of Mathematics
Abstract
Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain and $W_0^{1,2}(\Omega)$ be the usual Sobolev space. Let $\beta$, $0\leq\beta1$, $$\lambda_{p,\beta}(\Omega)=\inf_{u\in W_0^{1,2}(\Omega),\,u\not\equiv 0}{\ \nabla u\ _2^2}/{\ u\ _{p,\beta}^2},$$ 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
DOI
10.3906/mat-1501-63
Keywords
Trudinger-Moser inequality, singular Trudinger-Moser inequality
First Page
874
Last Page
883
Recommended Citation
YUAN, A, & HUANG, Z (2016). An improved singular Trudinger-Moser inequality in dimension two. Turkish Journal of Mathematics 40 (4): 874-883. https://doi.org/10.3906/mat-1501-63
