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On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball
Author
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2009, Issue 2009 (31 Dec. 2009), pp.1-11, 11 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2009-11-02
Country of Publication
Egypt
No. of Pages
11
Main Subjects
Abstract EN
We consider the following eigenvalue problem: −Δu+f(u)=λu, u=u(x), x∈B={x∈ℝ3:|x|<1}, u(0)=p>0, u||x|=1=0, where p is an arbitrary fixed parameter and f is an odd smooth function.
First, we prove that for each integer n≥0 there exists a radially symmetric eigenfunction un which possesses precisely n zeros being regarded as a function of r=|x|∈[0,1).
For p>0 sufficiently small, such an eigenfunction is unique for each n.
Then, we prove that if p>0 is sufficiently small, then an arbitrary sequence of radial eigenfunctions {un}n=0,1,2,…, where for each n the nth eigenfunction un possesses precisely n zeros in [0,1), is a basis in L2r(B) (L2r(B) is the subspace of L2(B) that consists of radial functions from L2(B).
In addition, in the latter case, the sequence {un/∥un∥L2(B)}n=0,1,2,… is a Bari basis in the same space.
American Psychological Association (APA)
Zhidkov, Peter. 2009. On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball. International Journal of Mathematics and Mathematical Sciences،Vol. 2009, no. 2009, pp.1-11.
https://search.emarefa.net/detail/BIM-988541
Modern Language Association (MLA)
Zhidkov, Peter. On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball. International Journal of Mathematics and Mathematical Sciences No. 2009 (2009), pp.1-11.
https://search.emarefa.net/detail/BIM-988541
American Medical Association (AMA)
Zhidkov, Peter. On the Existence, Uniqueness, and Basis Properties of Radial Eigenfunctions of a Semilinear Second-Order Elliptic Equation in a Ball. International Journal of Mathematics and Mathematical Sciences. 2009. Vol. 2009, no. 2009, pp.1-11.
https://search.emarefa.net/detail/BIM-988541
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-988541