Existence of large solutions for a semilinear elliptic problem via explosive sub- supersolutions

Abstract

We consider the boundary blow-up nonlinear elliptic problems Δu ± λ|∇u|q = k(x)g(u) in a bounded domain with boundary condition u|∂Ω = +∞, where q ∈ [0, 2] and λ ≥ 0. Under suitable growth assumptions on K near the boundary and on g both at zero and at infinity, we show the existence of at least one solution in C2(Ω). Our proof is based on the method of explosive sub-supersolutions, which permits positive weights k(x) which are unbounded and / or oscillatory near the boundary. Also, we show the global optimal asymptotic behaviour of the solution in some special cases.