A remark on Ricceriʹs conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem ( P λ f ˜ ) : − Δ u = λ f ( x , u ) in Ω, u = 0 on ∂Ω, where Ω is a bounded smooth domain in R N . Denote by C L ˜ the set of all Carathéodory functions f : Ω × R → R such that for a.e. x ∈ Ω , f ( x , ⋅ ) is Lipschitzian with Lipschitz constant L, f ( x , 0 ) = 0 and sup ξ ∈ R ∫ 0 ξ f ( x , t ) d t = 0 , and denote by Λ f ˜ (resp. Λ f ˜ w ) the set of λ > 0 such that ( P λ f ˜ ) has at least one nonzero classical (resp. weak) solution. Let λ 1 be the first eigenvalue for the Laplacian–Dirichlet problem. We prove that inf f ∈ C L ˜ inf Λ f ˜ = inf f ∈ C L ˜ inf Λ f ˜ w = 3 λ 1 L and { inf Λ f ˜ | f ∈ C L ˜ \ { 0 } } = { inf Λ f ˜ | f ∈ C L ˜ \ { 0 } } = [ 3 λ 1 L , + ∞ ] . Our result is a positive answer to Ricceriʹs conjecture if use f ( x , u ) instead of f ( u ) in the conjecture.