Expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation

The main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation −Δu+λum=finΩ, where λ>0,m>1,f∈C(Ω),f≥0λ>0,m>1,f∈C(Ω),f≥0, and ΩΩ is an open bounded set of RN, N>1, with boundary smooth enough. Roughly speaking, we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite, but it goes to infinity as mm goes to 1. We prove that the expansion consists in two eventual geometrical and non-geometrical parts separated by a term independent on the geometry of ∂Ω∂Ω, but dependent on the diffusion. For low explosive sources the non-geometrical part does not exist; all coefficients depend on the diffusion and the geometry of the domain by means of well-known properties of the distance function dist(x,∂Ω). For high explosive sources the preliminary coefficients, relative to the non-geometrical part, are independent on ΩΩ and the diffusion. Finally, the geometrical part does not exist for very high explosive sources.