  We consider the boundary value problem $-\Delta_p u_\lambda -\Delta_q
u_\lambda =\lambda g(x) u_\lambda^{-\beta}$ in $\Omega$ , $u_\lambda=0$ on
$\partial \Omega$ with $u_\lambda>0$ in $\Omega.$ We assume $\Omega$ is a
bounded open set in $\mathbb{R}^N$ with smooth boundary, $1<p<q<\infty$,
$\beta\in [0,1),$ $g$ is a positive weight function and $\lambda$ is a positive
parameter. We derive an estimate for $u_\lambda$ which describes its exact
behavior when the parameter $\lambda$ 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.
