Bifurcation of symmetric solutions for the sublinear Moore-Nehari differential equation

Abstract

We study the bifurcation of symmetric nodal solutions for the equation u″+h(x,λ)|u|p−1u=0 in (−1,1) with u(−1)=u(1)=0, where 0<p<1, h(x,λ)=0 for |x|<λ and h(x,λ)=1 for λ≤|x|≤1 and λ∈(0,1) is a bifurcation parameter. For a non-negative integer n, we call a solution u(x)n-nodal if it has exactly n zeros in (−1,1). We call a solution u symmetric if it is even or odd. For each n, the equation has a unique n-nodal symmetric solution un(x,λ), which is a continuous curve of λ∈(0,1). We prove that when n is even, this curve does not bifurcate and when n is odd, it bifurcates.