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

Abstract

We consider the Moore–Nehari equation, $u'' + h(x, \lambda) |u|^{p-1} u = 0$ in $(-1, 1)$ with $u(-1) = u(1) = 0$, where $p > 1$, $h(x, \lambda) = 0$ for $|x| < \lambda$, $h(x, \lambda) = 1$ for $\lambda \leq |x| \leq 1$ and $\lambda \in (0, 1)$ 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 $n$.