Existence of positive radial solutions for a semipositone elliptic equation

Abstract

In this paper, we study the existence of positive radial solutions for a semipositone elliptic equation−Δu=λf(u)in Ω,u=0on ∂Ω, where Ω is a ball or an annulus in RN with N≥2, λ>0 is a parameter and f is a continuous function satisfying the semipositone condition f(0)<0. We give a weak and general sufficient condition on f for the existence of positive radial solutions for λ>0 large. We show that the linearized operator at each solution has the nonnegative first eigenvalue. These solutions are obtained as minimizers of a Lagrangian functional.