The Moser–Trudinger inequality and its extremals on a disk via energy estimates

Abstract

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.

A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc

“…Similar to (5) ( [11], Lemma 4), there holds w ǫ → w 0 in C 1 loc (R 2 ), where w 0 is given as in (6). Then repeating the argument of the proof of ( [12], Propositions 12 and 14), we conclude…”

“…for two real numbers γ ǫ and α ǫ . Adapting the arguments of Malchiodi-Martinazzi [11] and Mancini-Martinazzi [12], we shall prove the following:…”

“…For the proof of Theorem 1, we shall modify the arguments in [11,12]. Let (u ǫ ) ǫ>0 be a sequence of decreasing radially symmetric solutions to (10) for some γ ǫ > 0 and α ǫ → α < λ 1…”

Sharp threshold nonlinearity for maximizing the Trudinger-Moser inequalities

“…This means that the existence of maximizer is more stable for the original Trudinger-Moser (1.1) on the disk D than (1.2) on the whole plane R 2 with the critical nonlinearity u −2 e u 2 , which makes the latter problem more delicate. It is worth noting, however, that vanishing of the second order term was also observed in the asymptotic expansion by Mancini and Martinazzi [10] of ∇u 2 L 2 (D) with respect to u L ∞ for concentrating sequences of critical points for (1.1). It does not seem clear if there is any relation to the vanishing observed in this paper on R 2 .…”

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

“…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).…”