Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition

Abstract

Abstract. We consider asymptotic behavior of the following fourth order equationwhere Ω is a smooth oriented bounded domain in R 4 . Assuming that 0 < ρ ≤ C, we completely characterize the asymptotic behavior of the unbounded solutions.

Asymptotic behavior of SU(3) Toda system in a bounded domain

“…Instead, we shall use the Pohozaev identity and precise information on blow-ups to exclude boundary bubbles. This idea was introduced first by Robert-Wei [21] in studying the fourth order mean field equation with Dirichlet boundary conditions.…”

Topological degree for solutions of fourth order mean field equations

“…Then using Lemma 2.4 and Green's representation, along the line of [24], we can prove the following claims step by step. The proof is omitted here.…”

“…In general, G 4 is not positive. We collect the property of G 4 in the following lemma which has been proved in [24].…”

“…In [24], Robert and Wei have given a complete proof of (P1)-(P3) when h(x) ≡ 1 for problem (1.9). Now let h(x) be a positive C 2,β function.…”

“…There is now an extensive literature about this problem, we refer to Adimurthi et al [1], Baraket et al [3], Clapp et al [9], Druet and Robert [10], Hebey and Robert [11], Hebey et al [12], Malchiodi [20], Robert and Wei [24] and the references therein. In particular, we mention the two papers [3,9], where they constructed m-point blowing-up solutions for…”