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

Abstract

The main goal of this paper is to study the asymptotic expansion near the boundary of the large solutions of the equation -Delta u + lambda u(m) = f in Omega, where lambda > 0, m > 1, f is an element of c(Omega), f >= 0, and Omega is an open bounded set of R-N, 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 in 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 partial derivative Omega, 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, partial derivative Omega). For high explosive sources the preliminary coefficients, relative to the non-geometrical part, are independent on Omega and the diffusion. Finally, the geometrical part does not exist for very high explosive sources consists in two eventual geometrical and non-geometrical parts, separated by a term independent on the geometry of $\partial\Omega$∂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 ${\rm dist}(x,\partial\Omega)$dist(x,∂Ω). For high explosive sources the preliminary coefficients, relative to the non-geometrical part, are independent on $\Omega$Ω and the diffusion. Finally, the geometrical part does not exist for very high explosive sources.