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

We study the bifurcation of symmetric nodal solutions for the equation u '' + h(x, lambda)|u|(p-1)u = 0 in (-1, 1) with u(-1) = u(1) = 0, where 0 < p < 1, h(x, lambda) = 0 for vertical bar x vertical bar lambda and h(x, lambda) = 1 for lambda <= |x| <= 1 and lambda is an element of (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, lambda), which is a continuous curve of lambda is an element of (0, 1). We prove that when n is even, this curve does not bifurcate and when n is odd, it bifurcates. (C) 2022 Elsevier Inc. All rights reserved.