Bifurcation of nodal solutions for the Moore–Nehari differential equation

Abstract

We study the bifurcation of nodal solutions for the Moore–Nehari differential 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 $|x|<\lambda$ and $h(x,\lambda )=1$ for $\lambda \le |x| \le 1$ and $\lambda \in (0,1)$ is a bifurcation parameter. For a non-negative integer n, we call a solution n-nodal if it has exactly n zeros in $(-1,1)$. We call a solution symmetric if it is even or odd. We prove that the equation has a unique n-nodal symmetric solution $(\lambda ,u_n(x,\lambda ))$, which is a continuous curve of $\lambda \in (0,1)$ in $C^1[-1,1]$. We show that when n is odd, this curve does not bifurcate and when n is even, the curve bifurcates and an n-nodal asymmetric solution emanates.