Second-order estimates for boundary blowup solutions of special elliptic equations

Abstract

We find a second-order approximation of the boundary blowup solution of the equation Δu = e u|u| β−1 , with β > 0, in a bounded smooth domain Ω ⊂ R N . Furthermore, we consider the equation Δu = e u+e u . In both cases, we underline the effect of the geometry of the domain in the asymptotic expansion of the solutions near the boundary ∂Ω.

Boundary behaviour for solutions of boundary blow-up problems in a borderline case

“…Putting t = φ(s) and observing that −φ (s) = (2F (φ(s))) 1 2 , estimate (31) follows. The lemma is proved.…”

“…Since condition (5) implies the Keller-Osserman condition, problem (2) with Ω = A(ρ, R) has a classical radial solution.…”

“…By using an argument of C. Bandle and M. Marcus [5], one can prove that condition (6) implies uniqueness of problem (2). In addition, we suppose there is C 0 finite such that for all θ ∈ (1/2, 2) and for t large we have…”

“…In the recent paper [2], with the aim to investigate the second order term of the expansion of the solution u(x) of problem (2), the following condition on f (t) is assumed…”

“…However, this special case has been studied in [3], and a more general situation has been discussed in [1]. Also the case p = 1 is a borderline case, and, as far as we know, it has not been investigated yet.…”