SHARP CONSTANT AND EXTREMAL FUNCTION FOR THE IMPROVED MOSER-TRUDINGER INEQUALITY INVOLVING Lp NORM IN TWO DIMENSION
SHARP CONSTANT AND EXTREMAL FUNCTION FOR THE IMPROVED MOSER-TRUDINGER INEQUALITY INVOLVING Lp NORM IN TWO DIMENSION
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DOI：
10.3934/dcds.2009.25.963
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发表时间：
2009-11-01
影响因子：
1.1
通讯作者：
Yang, Yunyan
中科院分区：
文献类型：
--
作者：
Lu, Guozhen;Yang, Yunyan
Let Omega subset of 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)= inf(u is an element of H01(Omega), u not equivalent to 0) parallel to del u parallel to(2)(2)/parallel to u parallel to(2)(p),where parallel to center dot parallel to(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:sup(u is an element of H01(Omega),parallel to del u parallel to 2=1)integral(Omega) (e4 pi(1+alpha parallel to u parallel to p2)u2) dx < +infinityor for 0 = lambda(p)(Omega). We also prove the existence of the extremal functions for this inequality when alpha is sufficiently small.
