DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

BOUNDARY ESTIMATES FOR BLOW-UP SOLUTIONS
OF ELLIPTIC EQUATIONS WITH EXPONENTIAL GROWTH
C. ANEDDA, A. BUTTU, AND G. PORRU
We investigate blow-up solutions of the equation Δu = e u + g(u) in a bounded smooth
domain Ω.Ifg(t) satisfies a suitable growth condition (compared with the growth of e t )
as t goes to infinity, we find second-order asymptotic estimates of the solution u(x) in
terms of the distance of x from the boundary ∂Ω.
Copyright © 2006 C. Anedda et al. This is an open access article distributed under the
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and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let Ω ⊂ R N be a bounded smooth domain. It is known since 1916 [10] that the problem
Δu = e u in Ω, u(x) −→ ∞ as x −→ ∂Ω, (1.1)
has a classical solution called a boundary blow-up (explosive, large) solution. Moreover,
if δ = δ(x) denotes the distance from x to ∂Ω,wehave[10]
2
u(x) − log
δ 2 −→ 0 asx −→ ∂Ω. (1.2)
(x)
Recently, Bandle [3] has improved the previous estimate, finding the expansion
2
u(x) = log
δ 2 (x) +(N − 1)K(x)δ(x)+o( δ(x) ) , (1.3)
where K(x) denotes the mean curvature of ∂Ω at the point x nearest to x,ando(δ)hasthe
usual meaning. Boundary estimates for more general nonlinearities have been discussed
in several papers, see [4–6, 8, 12, 14–17].
In Section 2 of the present paper we investigate the problem
Δu = e u + g(u) inΩ, u(x) −→ ∞ as x −→ ∂Ω, (1.4)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 47–55