International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 243048, 11 pages
Research Article

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

Bogoliubov 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.


We consider the following eigenvalue problem: Δu+f(u)=λu, u=u(x), xB={x3:|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 n0 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/unL2(B)}n=0,1,2, is a Bari basis in the same space.