Critical points of the Moser-Trudinger functional on a disk

Abstract

On the unit disk B 1 ⊂ R 2 we study the Moser-Trudinger functionaland its restrictions E| MΛ , where M Λ := {u ∈ H 1 0 (B 1 ) : u 2 H 1 0 = Λ} for Λ > 0. We prove that if a sequence u k of positive critical points of E| MΛ k (for some Λ k > 0) blows up as k → ∞, then Λ k → 4π, and u k → 0 weakly in H 1 0 (B 1 ) and strongly in C 1 loc (B 1 \ {0}). Using this we also prove that when Λ is large enough, then E| MΛ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.Since then, a formidable amount of work has been devoted to the study of the functionaland in particular of its critical points. Clearly u ≡ 0 is the only global minimum of E, but because of (2) we cannot look for a global maximizer of E in H 1 0 . Instead one might hope to find a maximizer of E| M Λ , i.e. of E constrained to the manifoldfor Λ ∈ (0, 4π], or to find other kinds of critical points (local maxima or minima, saddle points, etc.) when Λ > 4π. As long as Λ < 4π the embedding (1) is in fact compact, so the existence of a maximizer is elementary, but when Λ ≥ 4π compactness is lost and also the Palais-Smale condition does not hold anymore, see [3].In spite of these difficulties Carleson and Chang [7] proved that when Ω = B 1 (0) (the unit disk in R 2 ) E| M 4π has a maximizer. This result was extended by Struwe [26] who proved the existence of a maximizer in M 4π when Ω is close to a ball, and finally by Flucher [13] for any bounded smooth Ω (see also [9] for a related result in higher dimension).The existence of critical points on M Λ in the supercritical regime, i.e. for Λ > 4π, is even more challenging, and to the fundamental question of the existence of critical points of E| M Λ for Λ large only few answers have been given. Monahan [22] gave numerical evidence that when Ω = B 1 (0) then for some Λ * > 4π the functional E| M Λ has a local maximum and a mountain pass critical point for every Λ ∈ (4π, Λ * ). Assuming that a local maximum of E| M 4π exists (which was later shown to be true for arbitrary domains by Flucher [13]) Struwe proved in [26] that for some Λ * = Λ * (Ω) > 4π and for a.e. Λ ∈ (4π, Λ * ) two critical points exists. This result was then extended in [16] to all values of Λ ∈ (4π, Λ * ) through the more precise information given by a parabolic flow, compared to the one given by the Palais-Smale condition.Further, using implicit function methods, Del Pino, Musso and Ruf [11] were able to characterize some of these critical points as one-peaked bubbling functions which blow-up as Λ ց 4π. In the same paper they showed that if Ω is not contractible, then for some Λ † > 8π the functional E| M Λ has a critical point of multi-peak type for Λ ∈ (8π, Λ † ). When Ω is a radially symmetric annulus they also proved for any 1 ≤ ℓ ∈ N the existence of some Λ * ℓ > 4πℓ such that E| M Λ has a critical point when Λ ∈ (4πℓ, Λ * ℓ ). We also refer to [27] and [16] for related results on domains with small holes, in the spirit of [8] (where the Yamabe equation was treated).T...

Glueing a peak to a non-zero limiting profile for a critical Moser–Trudinger equation

“…When considering the typical case (0.2), i.e. the Euler-Lagrange equation of the standard Moser-Trudinger inequality, existence results have been obtained using radial analysis [4,12] (see also [13]), variational [9,15], perturbative [5] or topological methods [7,11,16]. According to the previous discussion, contrary to those built in Theorem 0.1, the blow-up solutions obtained in these results always have a zero weak limit in H 1 0 .…”

Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in high dimension

“…In this spirit, pushing further the blow-up analysis of Theorem 1.1, case iv), we obtain sharp global estimates which relate the behaviour near the origin and the behaviour away from it. Moreover, using a linearization procedure partly inspired from [17], we are able to give a better asymptotic expansion of u k near the origin.…”

“…building upon [17]. A sequence (u k ) of positive (hence radial, by the moving-plane technique) solutions to (22) for some λ k > 0, with u k (0) → ∞ satisfies…”

“…By (9) we have u k ( 1 2 ) = (1 + o(1))β k ϕ( 1 2 ), and hence, from (59) we infer (18) follows at once. Then (17) follows from (59).…”

The nonlocal mean-field equation on an interval

“…Let us notice that if we replace the right-hand side of (1) with the nonlinearity e u 2 , nonlocal compactness problems have been studied in [14] and [17], but the techniques used there are different, for instance because of the lack of a Pohozaev-type identity. In fact a result analog to (8) is still unknown in the fractional case, although in dimension 2 it was recently proven by Druet-Thizy [10], see also [22].…”