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November  2010, 9(6): 1723-1730. doi: 10.3934/cpaa.2010.9.1723

A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China, China

Received  August 2009 Revised  December 2009 Published  August 2010

The aim of this paper is to establish a sharp decay estimate for radially symmetric solutions of the following type of nonlinear Schrödinger equations:

$-\Delta u + V(|x|)u =Q(|x|)|u|^{p-2}u, x\in R^N, $

$u(x)\rightarrow 0$ as $|x|\rightarrow+\infty,$

where $N\geq 3$, $p\in(2,+\infty)$, $V(x)$ and $Q(x)$ are continuous functions which vanishes at infinity and may change sign. As a special case, our result shows that the solutions obtained by Su-Wang-Willem in [11, Theorem 3] must decay precisely like $|x|^{-(N-2)}$ as $|x|\rightarrow+\infty$ if $V(|x|)$ decays faster than $|x|^{-2}$ at infinity.

Citation: Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723
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