Abstract
We find a second-order approximation of the boundary blowup solution of the equation, with, in a bounded smooth domain. Furthermore, we consider the equation. In both cases, we underline the effect of the geometry of the domain in the asymptotic expansion of the solutions near the boundary.
Highlights
Let Ω ⊂ RN be a bounded smooth domain
We look for a super-solution of the form w(x) = Φ(δ) + β−1Hδ Φ(δ) 1−β + αδ Φ(δ) 1−2β, (2.15)
We look for a subsolution of the form v(x) = Φ(δ) + β−1Hδ Φ(δ) 1−β − αδ Φ(δ) 1−2β, (2.38)
Summary
Let Ω ⊂ RN be a bounded smooth domain. In 1916, Bieberbach [10] has investigated the problemΔu = eu in Ω, u(x) −→ ∞ as x −→ ∂Ω, (1.1)and has proved the existence of a classical solution called a boundary blowup (explosive, large) solution. K(x) is the mean curvature of the surface {x ∈ Ω : δ(x) = constant}, and O(1) denotes a bounded quantity. The effect of the geometry of the domain in the behaviour of boundary blowup solutions for special equations has been observed in various papers, see for example, [2, 7, 9, 11]. In what follows we denote with O(1) a bounded quantity.
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