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- Title
Bifurcation analysis of a singular nonlinear Sturm-Liouville equation.
- Authors
Castro, Hernán
- Abstract
In this paper we study existence of positive solutions to the following singular nonlinear Sturm-Liouville equation 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 , 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 x→0 x2αu′(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 . On the other hand, when ½ < α < 1 and , 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 ℒα under the boundary condition x→0 u(x) = 0. Finally, when α ≥ 1, the operator ℒα 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 λ ∈ ℝ and p > 1.
- Subjects
NUMERICAL solutions to Sturm-Liouville equations; MATHEMATICAL constants; EXISTENCE theorems; NONLINEAR equations; BIFURCATION theory; EIGENVALUES; BOUNDARY value problems
- Publication
Communications in Contemporary Mathematics, 2014, Vol 16, Issue 5, p-1
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199714500126