Elliptic bifurcation problems that are singular in the dependent and in the independent variables. Issue 9 (1st September 2020) - Godoy, T.; Guerin, A.
Abstract : We consider singular problems of the form − Δ u = k ., u + λ h ., u in Ω, u =0 on ∂ Ω, u >0 in Ω, where Ω is a bounded C 1, 1 domain in R n, n ≥ 2, h : Ω × 0, ∞ → 0, ∞ and k : Ω × 0, ∞ → 0, ∞ are Carathéodory functions such that h x, . is nondecreasing, and k x, . is nonincreasing and singular at the origin a . e . x ∈ Ω . Additionally, k ., s and h ., s are allowed to be singular at ∂ Ω for s > 0. Under suitable additional hypotheses on h and k, we prove that there is a positive λ ∗ such that, for any λ ∈ 0, λ ∗, a minimal positive weak solution u λ ∈ H 0 1 Ω ∩ C Ω ¯ exists. The monotonicity of λ → u λ, and the behaviour of u λ at ∂ Ω, is addressed. In addition, we prove that no weak solution exists if λ > λ ∗ .