Maximal solutions for −Δu+uq=0 in open and finely open sets

AbstractWe derive sharp estimates for the maximal solution U of (∗) −Δu+uq=0 in an arbitrary open set D⊂RN. The estimates involve the Bessel capacity C2,q′, for q in the supercritical range q⩾qc:=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 C2,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