Uniqueness of bubbling solutions of mean field equations

Abstract

We prove the uniqueness of blow up solutions of the mean field equation as ρ n → 8πm, m ∈ N. If u n,1 and u n,2 are two sequences of bubbling solutions with the same ρ n and the same (non degenerate) blow up set, then u n,1 = u n,2 for sufficiently large n. The proof of the uniqueness requires a careful use of some sharp estimates for bubbling solutions of mean field equations [24] and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in [51] in the context of the Chern-Simons-Higgs equations. Moreover, motivated by the Onsager statistical description of two dimensional turbulence, we are bound to obtain a refined version of an estimate about ρ n − 8πm in case the first order evaluated in [24] vanishes.

Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions

“…On the other hand, the blow-up set (p 1 , p 2 ) is a degenerate critical point of f 2 defined in (1.9) due to the translation invariance. The proof of the uniqueness result follows the one in [6] by taking advantage of the evenly symmetric property to bypass the non-degeneracy assumption. More precisely, assuming by contradiction the existence of two distinct blow-up solutions u (i) n of (1.4) we consider their normalized difference…”

“…Together with the non-degeneracy assumption on the critical point p, Bartolucci et al [6] concluded that…”

“…In particular, we recall l( p) = 32πe G * 1 (p1) = 0. We would like to remark that a more refined estimate involving D( q) on the total mass has been derived in [6,Theorem 1.3] which is crucial in the case where l( p) vanishes.…”

Local uniqueness and non-degeneracy of blow up solutions of mean field equations with singular data

“…For what concerns regular bubbling solutions, for q = (q 1 , · · · , q m ) ∈ Ω × · · · × Ω, we let G * j (x) = 8πR(x, q j ) + 8π ∑ For (x 1 , · · · , x m ) ∈ Ω × · · · Ω, we also define the m-vortex Hamiltonian, G(x l , x j ). (1.1) Then, by assuming suitable non-degeneracy conditions the authors in [8,9] proved that regular m-bubbling solutions are unique and non-degenerate (see also [10] for an analogous result for the Gelfand equation).…”

“…Theorem A ( [8,9]). Let u (1) n and u (2) n be two regular m-bubbling solutions of (P ρ n ), with ρ (1) n = ρ n = ρ (2) n , blowing up at the points q j / ∈ {p 1 , · · · , p N }, j = 1, · · · , m, where q = (q 1 , · · · , q m ) is a critical point of H m .…”