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

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 that consists of radial functions from . In addition, in the latter case, the sequence is a Bari basis in the same space.