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Positive solutions of nonpositone sublinear elliptic problems

Godoy Tomas

Opuscula Mathematica

2024

Vol. 44, no. 6

827--851

Consider the problem [formula].

Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions

Godoy Tomas

Opuscula Mathematica

2023

Vol. 43, no. 1

19--46

Let Ω be a C2 bounded domain in Rn such that ∂Ω = Γ1 ∪ Γ2, where Γ1 and Γ2 are disjoint closed subsets of ∂Ω, and consider the problem −Δu = g(·, u) in Ω, u = τ on Γ1, ∂u ∂ν = η on Γ2, where 0 ≤ τ ∈ W 1 2 ,2(Γ1), η ∈ (H1 0, Γ1(Ω))′, and g : Ω×(0,∞) → R is a nonnegative Carathéodory function. Under suitable assumptions on g and η we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow g to be singular at s = 0 and also at x ∈ S for some suitable subsets S ⊂ Ω. The Dirichlet problem −Δu = g(·, u) in Ω, u = σ on ∂Ω is also studied in the case when 0 ≤ σ ∈ W 1 2 ,2(Ω).