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

{-(x(2 alpha)u')' = lambda(u) + u(p) in (0, 1), u(1) = 0,

where alpha > 0, p > 1 and lambda are real constants. We prove that when 0 < alpha <= 1/2 and p > 1 or when 1/2 < alpha < 1 and 1 < p <= 3-2 alpha/2 alpha-1, there exists a branch of continuous positive solutions bifurcating to the left of the first eigenvalue of the operator L(alpha)u = -(x(2 alpha)u')' under the boundary condition lim(x -> 0) x(2 alpha)u'(x) = 0. The projection of this branch onto its lambda component is unbounded in two cases: when 0 < alpha <= 1/2 and p > 1, and when 1/2 < alpha < 1 and p < 3-2 alpha/2 alpha-1. On the other hand, when 1/2 < alpha < 1 and p = 3-2 alpha/2 alpha-1, the projection of the branch has a positive lower bound below which no positive solution exists. When 0 < alpha < 1/2 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 L alpha under the boundary condition lim(x -> 0) u(x) = 0. Finally, when alpha >= 1, the operator L alpha has 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 lambda is an element of R and p > 1.