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Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

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    * Corresponding author

The author was partly supported by grant from the NNSF of China (No.11371056)

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  • We present the singular Hardy-Trudinger-Moser inequality and the existence of their extremal functions on the unit disc $ B $ in $ \mathbb{R}^2 $. As our first main result, we show that for any $ 0<t<2 $ and $ u \in C_0^\infty({B}) $ satisfying

    $ \int_{{B}}|\nabla u|^2 dx- \int_{{B}}\frac{u^2}{(1-|x|^2)^2}dx\leq1, $

    there exists a constant $ C_{0}>0 $ such that the following inequality holds

    $ \int_{{B}}\frac{e^{4\pi(1-t/2)u^2}}{|x|^t} dx\leq C_{0}. $

    Furthermore, by the method of blow-up analysis, we establish the existence of extremal functions in a suitable function space. Our results extend those in Wang and Ye [36] from the non-singular case $ t = 0 $ to the singular case for $ 0<t<2 $.

    Mathematics Subject Classification: Primary: 35J50; Secondary: 46E30, 46E35.


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