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: −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 Lr2 B L r 2 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.