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|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) (𝐿𝑟2(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.