A characterization related to the dirichlet problem for an elliptic equation

Abstract

Let Ω be a bounded domain in RN with smooth boundary. Let f: [0+∞[→[0,+ ∞[, with f (0) = 0, be a continuous function such that, for some a > 0, the function (Formula presented) is non increasing in ]0, a[. Finally, let (Formula presented) [ be a continuous function with α(x) > 0, for all x ∈ Ω. We establish a necessary and suffcient condition for the existence of solutions to the following problem -∆u = 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.