Large solutions to the p-Laplacian for large p

Abstract

In this work we consider the behaviour for large values of p of the unique positive weak solution u p to pu = uq in , u = +∞ on ∂, where q > p − 1. We take q = q(p) and analyze the limit of u p as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of max{− ∞u, −|∇u| + uQ} = 0 with u = +∞ on ∂. If Q = 1 then solutions go to ∞ in the whole and when Q = ∞ solutions converge to 1 uniformly in compact subsets of , hence the boundary blow-up is lost in the limit.