We derive sharp estimates for the maximal solution U of ( ∗ ) − Δ u + u q = 0 in an arbitrary open set D ⊂ R N . The estimates involve the Bessel capacity C 2 , q ′ , for q in the supercritical range q ⩾ q c : = N / ( N − 2 ) . We provide a pointwise necessary and sufficient condition, via a Wiener type criterion, in order that U ( x ) → ∞ as x → y for given y ∈ ∂ D . This completes the study of such criterions carried out in [J.-S. Dhersin, J.-F. Le Gall, Wiener's test for super-Brownian motion and the Brownian snake, Probab. Theory Related Fields 108 (1997) 103–129] and [D.A. Labutin, Wiener regularity for large solutions of nonlinear equations, Ark. Mat. 41 (2003) 307–339]. Further, we extend the notion of solution to C 2 , q ′ finely open sets and show that, under very general conditions, a boundary value problem with blow-up on a specific subset of the boundary is well posed. This implies, in particular, uniqueness of large solutions.