A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials

Yongsheng Jiang; Huan-Song Zhou

doi:10.3934/cpaa.2010.9.1723

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2010, Volume 9, Issue 6: 1723-1730. Doi: 10.3934/cpaa.2010.9.1723

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A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials

Yongsheng Jiang^1, and
Huan-Song Zhou^1,

1.
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071

Received: July 31, 2009

Revised: November 30, 2009

Published: October 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35J10, 35B40.

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