We consider the boundary value problem, g is a positive weight function and λ is a positive parameter. We derive an estimate for u λ which describes its exact behavior when the parameter λ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems. Here, as an application of this estimate, we obtain a uniqueness result for a nonlinear elliptic boundary value problem with a singular nonlinearity.