International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 243048, 11 pages
doi:10.1155/2009/243048

On the existence, uniqueness, and basis properties of radial eigenfunctions of a semilinear second-order elliptic equation in a ball

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.