Let $Ω$ be a bounded domain of class $C^2$ in $R^n$. If $u$ is a positive solution of $-∆u + u^q = 0$ in $Ω$ with $q > 1$ it admits a boundary trace $ν = Tr_{∂Ω}(u)$ in the class $B^+_{reg}(∂Ω)$ of outer regular Borel measures on $∂Ω$, not necessarily locally bounded. It is known that the correspondence $u ↔ Tr_{∂Ω}(u)$ is one to one if $1 < q < q_c = (N +1)/(N −1)$, which is no longer the case if $q ≥ q_c$. One of the key problems raised by Dynkin is to prove that, for any compact set $K ⊂ ∂Ω$, the maximal solution $U_K$ of this equation which vanishes on $K^c$ is $σ$-moderate, that is an increasing limit of solutions with boundary data belonging to the space of positive Radon measures. By means of boundary Bessel capacity estimates we describe the precise asymptotic behavior of $U_K$ at points $σ ∈ K$. The Dynkin conjecture follows from these estimates.