Sharp estimates for bubbling solutions of a fourth order mean field equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, p. 599-630
We consider a sequence of multi-bubble solutions u k of the following fourth order equation Δ 2 u k =ρ k h(x)e u k Ω he u k inΩ,u k =Δu k =0onΩ,(*) where h is a C 2,β positive function, Ω is a bounded and smooth domain in 4 , and ρ k is a constant such that ρ k C. We show that (after extracting a subsequence), lim k+ ρ k =32σ 3 m for some positive integer m1, where σ 3 is the area of the unit sphere in 4 . Furthermore, we obtain the following sharp estimates for ρ k : ρ k -32σ 3 m=c 0 j=1 m ϵ k,j 2 lj ΔG 4 (p j ,p l )+ΔR 4 (p j ,p j )+1 32σ 3 Δlogh(p j )+o j=1 m ϵ k,j 2 where c 0 >0, log64 ϵ k,j 4 =max xB δ (p j ) u k (x)-log( Ω he u k ) and u k 32σ 3 j=1 m G 4 (·,p j ) in C loc 4 (Ω{p 1 ,...,p m }). This yields a bound of solutions as ρ k converges to 32σ 3 m from below provided that j=1 m lj ΔG 4 (p j ,p l )+ΔR 4 (p j ,p j )+1 32σ 3 Δlogh(p j )>0. The analytic work of this paper is the first step toward computing the Leray-Schauder degree of solutions of equation (*).
Classification:  35B40,  35B45,  35J40
@article{ASNSP_2007_5_6_4_599_0,
     author = {Lin, Chang-Shou and Wei, Juncheng},
     title = {Sharp estimates for bubbling solutions of a fourth order mean field equation},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {4},
     year = {2007},
     pages = {599-630},
     zbl = {1185.35067},
     mrnumber = {2394412},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2007_5_6_4_599_0}
}
Lin, Chang-Shou; Wei, Juncheng. Sharp estimates for bubbling solutions of a fourth order mean field equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 599-630. http://www.numdam.org/item/ASNSP_2007_5_6_4_599_0/

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