On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a BallReportar como inadecuado




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International Journal of Mathematics and Mathematical SciencesVolume 2009 2009, Article ID 243048, 11 pages

Research ArticleBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

Received 29 July 2009; Accepted 21 October 2009

Academic Editor: Manfred H. Moller

Copyright © 2009 Peter Zhidkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the following eigenvalue problem: , , , , where is an arbitrary fixed parameter and is an odd smooth function. First, we prove that for each integer there exists a radially symmetric eigenfunction which possesses precisely zeros being regarded as a function of . For sufficiently small, such an eigenfunction is unique for each . Then, we prove that if is sufficiently small, then an arbitrary sequence of radial eigenfunctions , where for each the th eigenfunction possesses precisely zeros in , is a basis in is the subspace of thatconsists of radial functions from . In addition, in the latter case, the sequence is a Bari basis in the same space.





Autor: Peter Zhidkov

Fuente: https://www.hindawi.com/



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