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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