A singular Moser-Trudinger embedding and its applications

Abstract.

Let Ω be a bounded domain in ${\mathbb{R}}^{n}$ , we prove the singular Moser-Trudinger embedding: $\mathop {\sup\limits_{\parallel u\parallel \leqslant 1\Omega } \int {\frac{{e^{\alpha |u|^{\frac{n} {{n - 1}}} } }}{{|x|^\beta }}} } < \infty$ if and only if $\frac{\alpha}{{\alpha _n }} + \frac{\beta}{n} \leqslant 1$ where $\alpha > 0,\beta \in [0,n),u \in W_0^{1,n} (\Omega )$ and $\parallel u\parallel = \left({\int\limits_\Omega {|\nabla u|^n } } \right)^\frac{1}{n}$ . We will also study the corresponding critical exponent problem.