A Characterization Related to the Dirichlet Problem for an Elliptic Equation

Let Omega be a bounded domain in R-N with smooth boundary. Let f : [0, +infinity[ -> [0, +infinity[, with f (0) = 0, be a continuous function such that, for some a> 0, the function xi is an element of]0, +infinity[->xi(-2).integral(xi)(0) f(t)dt is non increasing in]0, a[. Finally, let alpha : (Omega) over bar -> [0, +infinity[ be a continuous function with alpha(x) > 0, for all x is an element of Omega. We establish a necessary and sufficient condition for the existence of solutions to the following problem -Delta u = lambda alpha(x)f(u) in Omega, u > 0 in Omega, u= 0 on partial derivative Omega, where lambda is a positive parameter. Our result extends to higher dimension a similar characterization very recently established by Ricceri in the one dimensional case.