A Characterization Related to the Dirichlet Problem for an Elliptic Equation

Abstract

Let Ω be a bounded domain in R^N with smooth boundary. Let f: [0, + ∞[ → [0,+∞[, with f(0) = 0, be a continuous function such that, for some a > 0, the function ξ∈]0, +∞[ → ξ⁻² · ∫₀^ξ f(t)dt is non increasing in ]0,a[. Finally, let α: Ω → [0,+∞[ be a continuous function with α(x) > 0, for all x ∈ Ω. We establish a necessary and sufficient condition for the existence of solutions to the following problem −Δu = λα(x)f(u) in Ω, u > 0 in Ω, u = 0 on ∂Ω, where λ is a positive parameter. Our result extends to higher dimension a similar characterization very recently established by Ricceri in the one dimensional case.