Regularity of the lower positive branch for singular elliptic bifurcation problems
Date
2019-04-12
Authors
Godoy, Tomas
Guerin, Alfredo
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We consider the problem
-∆u = αu-α + ƒ(λ, ∙, u) in Ω,
u = 0 on ∂Ω,
u > 0 in Ω,
where Ω is a bounded domain in ℝn, λ ≥ 0, 0 ≤ α ∈ L∞(Ω), and 0 < α < 3. It is known that, under suitable assumptions on ƒ, there exists Λ > 0 such that this problem has at least one weak solution in H10(Ω) ∩ C(Ω̅) if and only if λ ∈ [0, Λ]; and that, for 0 < λ < Λ, at least two such solutions exist. Under additional hypothesis on α and ƒ, we prove regularity properties of the branch formed by the minimal weak solutions of the above problem. As a byproduct of the method used, we obtain the uniqueness of the positive solution when λ = Λ.
Description
Keywords
singular elliptic problems, positive solutions, bifurcation problems, implicit function theorem, sub and super solutions
Citation
Godoy, T., & Guerin, A. (2019). Regularity of the lower positive branch for singular elliptic bifurcation problems. <i>Electronic Journal of Differential Equations, 2019</i>(49), pp. 1-32.
Rights
Attribution 4.0 International