Sur les Solutions D'un Opérateur Différentiel Singulier Semi-Linéaire

Abstract

This paper deals with the following Dirichlet problem Lu = 1A ( Au′ − qu = − f ( , u ) on ] 0, ω [ , u, ( 0 ) = 0, u ( ω ) = 0, where ω ɛ ] 0, + ∞ ], q ≤ 0 is continuous on [ 0, ω [ × ] 0, + ∞ [ → ] 0, + ∞ [ is continuous and A satisfies some appropriate conditions. The main result is the existence and the uniqueness of a strictly positive regular solution of the problem ( ✱ ). Moreover, we study the behaviour of this solution in a neighbourhood of ω. Our approach is based on the use of the Green's function of the homogeneous equation and Schauder's fixed point theorem.