# American Institute of Mathematical Sciences

September  2009, 25(3): 963-979. doi: 10.3934/dcds.2009.25.963

## Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension

 1 Department of Mathematics, Wayne State University, Detroit, MI 48202, United States 2 Department of Mathematics, Information School, Renmin University of China, Beijing 100872, China

Received  March 2008 Revised  February 2009 Published  August 2009

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain, and $H_0^1(\Omega)$ be the standard Sobolev space. Define for any $p>1$,

$\lambda_p(\Omega)=$i n f$_u\in H_0^1(\Omega),$u≠0(for some u)$\|\|\nabla u\|\|_2^2/\|\|u\|\|_p^2,$

where $|\|\cdot\||_p$ denotes $L^p$ norm. We derive in this paper a sharp form of the following improved Moser-Trudinger inequality involving the $L^p$-norm using the method of blow-up analysis:

$s u p_{u\in H_0^1(\Omega),\|\|\nabla u\|\|_2=1}\int_{\Omega} e^{4\pi (1+\alpha\|\|u\|\|_p^2)u^2}dx<+\infty$

for $0\leq \alpha <\lambda_p(\Omega)$, and the supremum is infinity for all $\alpha\geq \lambda_p(\Omega)$. We also prove the existence of the extremal functions for this inequality when $\alpha$ is sufficiently small.

Citation: Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963
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