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: −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.