We classify the solutions to the equation (− ) m u = (2m − 1)!e 2mu on R 2m giving rise to a metric g = e 2u g R 2m with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of u at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e 2u g R 2m at infinity, and we observe that the pull-back of this metric to S 2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.
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...
We study the Dirichlet energy of non-negative radially symmetric critical points u µ of the Moser-Trudinger inequality on the unit disc in R 2 , and prove that it expands as 4π + 4πwhere µ = u µ (0) is the maximum of u µ . As a consequence, we obtain a new proof of the Moser-Trudinger inequality, of the Carleson-Chang result about the existence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicity result in the supercritical regime (only in the case of the unit disk). Our results are stable under sufficiently weak perturbations of the Moser-Trudinger functional. We explicitly identify the critical level of perturbation for which, although the perturbed Moser-Trudinger inequality still holds, the energy of its critical points converges to 4π from below. We expect, in some of these cases, that the existence of extremals does not hold, nor the existence of critical points in the supercritical regime.
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