Symmetry-breaking bifurcation for the Moore–Nehari differential equation

We study the bifurcation problem of positive solutions for the Moore-Nehari differential equation, u^{′ ′}+ h(x, λ) u^p= 0 , u> 0 in (- 1 , 1) with u(- 1) = u(1) = 0 , where 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. We shall show that the problem has a unique even positive solution U(x, λ) for each λ∈ (0 , 1). We shall prove that there exists a unique λ_∗∈ (0 , 1) such that a non-even positive solution bifurcates at λ_∗ from the curve (λ, U(x, λ)) , where λ_∗ is explicitly represented as a function of p.