Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

In this paper we prove uniqueness of positive solutions to logistic singular problems $-\Delta u=\lambda(x) u -a(x) u^{p}$, $u_{\vert\partial \Omega }=+\infty$, $p>1$, $a>0$ in $\Omega$, where the main feature is the fact that $a_{\vert\partial\Omega}=0$. More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near $\partial \Omega$. This expansion involves both the distance function $d(x)=\text{dist}(x,\partial \Omega)$ and the mean curvature $H$ of $\partial \Omega$.