Symmetric and asymmetric nodal solutions for the Moore–Nehari differential equation

Abstract

We consider the Moore–Nehari equation, 𝑢” + ℎ(𝑥, 𝜆) |𝑢|^{𝑝 −1} 𝑢 = 0 in (−1, 1) with 𝑢(−1) = 𝑢(1) = 0, where 𝑝 > 1, ℎ(𝑥, 𝜆) = 0 for |𝑥| < 𝜆, ℎ(𝑥, 𝜆) = 1 for 𝜆 ≤ |𝑥| ≤ 1 and 𝜆 ∈ (0, 1) is a parameter. We prove the existence of a solution which has exactly 𝑚 zeros in the interval (−1, 0) and exactly 𝑛 zeros in (0, 1) for given nonnegative integers 𝑚 and 𝑛.