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March/April 2015 A Meshkov-type construction for the borderline case
Blair Davey
Differential Integral Equations 28(3/4): 271-290 (March/April 2015).

Abstract

We construct functions $u: \mathbb R^2 \to \mathbb C$ that satisfy an elliptic eigenvalue equation of the form $-\Delta u + W \cdot \nabla u + V u = \lambda u$, where $\lambda \in \mathbb C$, and $V$ and $W$ satisfy $| V ({x} ) | \lesssim \langle {x} \rangle ^{-N}$, and $ | W ({x})| \lesssim \langle{x} \rangle ^{-P}$, with $\min\{N, P\} = 1/2$. For $|{x}|$ sufficiently large, these solutions satisfy $|{u(x)}| \lesssim \exp ({- c|{x}|})$. In the author's previous work, examples of solutions over $\mathbb R^2$ were constructed for all $N, P$ such that $\min\{N,P\} \in [0, 1/2)$. These solutions were shown to have the optimal rate of decay at infinity. A recent result of Lin and Wang shows that the constructions presented in this note for the borderline case of $\min\{N, P\} = 1/2$ also have the optimal rate of decay at infinity.

Citation

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Blair Davey. "A Meshkov-type construction for the borderline case." Differential Integral Equations 28 (3/4) 271 - 290, March/April 2015.

Information

Published: March/April 2015
First available in Project Euclid: 4 February 2015

zbMATH: 1363.35094
MathSciNet: MR3306563

Subjects:
Primary: 35B60, 35J10, 35J15

Rights: Copyright © 2015 Khayyam Publishing, Inc.

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Vol.28 • No. 3/4 • March/April 2015
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