Bifurcation analysis of a singular nonlinear Sturm–Liouville equation

Abstract

In this paper we study existence of positive solutions to the following singular nonlinear Sturm–Liouville equation [Formula: see text] where α > 0, p > 1 and λ are real constants. We prove that when 0 < α ≤ ½ and p > 1 or when ½ < α < 1 and [Formula: see text], there exists a branch of continuous positive solutions bifurcating to the left of the first eigenvalue of the operator ℒαu = -(x2αu′)′ under the boundary condition limx→0x2αu′(x) = 0. The projection of this branch onto its λ component is unbounded in two cases: when 0 < α ≤ ½ and p > 1, and when ½ < α < 1 and [Formula: see text]. On the other hand, when ½ < α < 1 and [Formula: see text], the projection of the branch has a positive lower bound below which no positive solution exists. When 0 < α < ½ and p > 1, we show that a second branch of continuous positive solution can be found to the left of the first eigenvalue of the operator ℒαunder the boundary condition limx→0u(x) = 0. Finally, when α ≥ 1, the operator ℒαhas no eigenvalues under its canonical boundary condition at the origin, and we prove that in fact there are no positive solutions to the equation, regardless of λ ∈ ℝ and p > 1.