A remark on the existence of large solutions via sub and supersolutions

Abstract

We study the boundary blow-up elliptic problem ∆u = α(x) f(u) in a smooth bounded domain Ω ⊂ ℝN, with u|∂Ω = +∞. Under suitable growth assumptions on α near ∂Ω and on f both at zero and at infinity, we prove the existence of at least a positive solution. Our proof is based on the method of sub and supersolutions, which permits on the one hand oscillatory behaviour of f(u) at infinity and on the other hand positive weights α(x) which are unbounded and/or oscillatory near the boundary.