Remarks on the Moser-Trudinger inequality

We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincar\'e's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.

(Ω), Ω |∇u(x)| N dx≤1 Ω is the N − 1-dimensional measure of the unit sphere S N −1 . This work is focused on extensions of the inequality to infinite-measure domains. Clearly, in this case the integral in (1) is infinite, since the integrand is greater than 1; moreover one should remove the first terms of the power series expansion of e αN |u| Gi. Mancini and Sandeep [MS10] studied the problem on the unit disc M = B 1 (0) ⊂ R 2 endowed with a conformal metric g ρ = ρg e , and they found that the Moser-Trudinger inequality holds for (M, g ρ ) if and only if g ρ is bounded by the hyperbolic metric, namely ρ(x) ≤ C (1 − |x| 2 ) 2 ∀ x ∈ M for some C > 0.
A particular case is given, through the Riemann map, by Euclidean simply connected domains: in this case, they showed that the Moser-Trudinger inequality is actually equivalent to Poincaré's inequality (4).
In this paper we prove that the same equivalence holds for any Euclidean domain Ω ⊂ R N , even for N ≥ 3 (Theorem 1.2). We will also investigate whether the equivalence between the Moser-Trudinger and the Poincaré inequality even holds for conformal metrics on the unit ball B 1 (0) ⊂ R N , and we discover that this does not occur. We build counterexamples of conformal metrics such that Poincaré's inequality holds but Moser-Trudinger's does not.
In these examples we show that the highest exponent for exponential integrability on non-compact manifolds may be any number smaller than α N ; moreover, exponential integrability might not occur at all, and even summability of higher powers fails for suitable metrics (Theorems 2.5 and 2.9).
In the last part of this paper, we study the existence of extremal functions for the Moser-Trudinger inequality: Carleson-Chang [CC86], Flucher [Flu92] and K.-C. Lin [Lin96] solved this problem for any Euclidean bounded domain, but nothing had been done, up to our knowledge, for unbounded domains yet.
Here we give the first existence result for unbounded domains, precisely for the strip Ω = R × (−1, 1) ⊂ R 2 (Theorem 3.2). When passing from bounded to unbounded domains, the main difficulty is that P.-L. Lions' concentration-compactness principle [Lio85] is no longer true in its original form: non-compact sequences may "vanish" at infinity, besides "concentrating"; however, the symmetry of Ω with the respect to both axes allows to exclude both vanishing and concentration whereas the Riemann map between Ω and the unit disc allows to exclude concentration, as for bounded domains. Section one is about the Moser-Trudinger inequality on unbounded euclidean domains, section two concerns the Moser-Trudinger inequality on the unit ball endowed with a conformal metric, and finally in section three we study the existence of extremals on the strip.
2. Poincaré's inequality holds for Ω, that is We stress that removing the hypothesis of simple connectedness, the finiteness of inradius is weaker than the other two statements; a counterexemple satisfying r(Ω) < +∞ and λ 1 (Ω) = 0 is shown indeed in [MS10]. The equivalence between Moser-Trudinger and Poincaré's inequalities, instead, will be now extended to any Euclidean domain. An essential tool in the proof of Theorem 1.2 is the Schwarz symmetrization. The Schwarz symmetrization of a non-negative u ∈ W 1,N 0 (Ω) is a non-negative radially nonincreasing u * ∈ W 1,N 0 R N that is equidistributed with u, namely |{u > t}| = |{u * > t}| for any t ≥ 0, hence (see for instance [Kes06]) it holds and in particular for f (u) = |u| p and f = Φ, so for any function u ∈ W 1,N 0 (Ω), the Moser-Trudinger functional has the same value in u and |u| * . Moreover, due to Pólya-Szegő inequality the condition on the Dirichlet integral in (3) also holds for |u| * if it does for u; so, the Moser-Trudinger inequality can be proved without loss of generality just on Schwarz-symmetrized functions. For the definition of the Schwarz symmetrization and a detailed list of its properties see for instance [Bae94]. Remark 1.3. Any admissible u for the supremum in Theorem 1.2 can be seen as a function in however, if one extends the set of admissible functions to any u ∈ W 1,N 0 R N satisfying (5), the supremum is infinite for any C > 0, even if one considers just radially nonincreasing functions: taking a sequence of Moser-like functions On the other hand, replacing α N with any smaller exponent gives uniform boundedness of the Moser-Trudinger functional even in this case, as proved by Adachi and Tanaka [AT00].
The proof of Theorem 1.2 will need some lemmas, the first of which extends the "radial lemma" from Berestycki and P.-L. Lions [BL83]: R N is a radially nonincreasing function, then |x| .
Now we have all the tools necessary to prove the main result of this section.

Supposing instead
|∇u(x)| N dx < 1, one can estimate the Moser-Trudinger functional on B 1 (0) c through Lemma 1.4. In fact one has hence, using the estimate for any x ≥ 0 , 2 ≤ N ∈ N one gets To estimate the integral over B 1 (0), it suffices to consider the function where u(1) indicates, with a little abuse of notation, the trace of u on the boundary of B 1 (0); the elementary inequality for A ≥ 0, B ≥ −A and p ∈ [0, 2] and a weighted Young inequality give, for any ε > 0, hence we can apply the Moser-Trudinger inequality on B 1 (0) to obtain Moreover, from Lemma 1.6, and the conclusion follows from (7) and (8).
With few modifications, the proof of Theorem 1.2 can be extended to a Moser-like functional with a different number of term removed, that is Even in this case, if the first power of the functional is controlled by the L N norm of the gradient, then all the functional is, precisely

The Moser-Trudinger inequality for conformal metrics
A natural question to ask is whether Theorems 1.1 and 1.2 can be extended to other mainfolds besides Euclidean domains, such as the unit ball in R N endowed with a conformal metric. A characterization of the metrics where the Moser-Trudinger inequality holds was given by Gi. Mancini and Sandeep in the 2-dimensional case [MS10] and it was later extended by themselves and Tintarev [MST] to any dimension.
Let g ρ = ρ(x)g e be a conformal metric on B = B 1 (0) ⊂ R N . Then the following conditions are equivalent: 1. g ρ is bounded by the hyperbolic metric, that is 2. The Moser-Trudinger inequality holds for the metric g, that is In view of Theorem 2.1, the question can be rewritten as: are the conformal metrics on B such that Poincaré's inequality holds all and only the ones which are bounded by the hyperbolic metric g h ? As usual, Poincaré's inequality is implied by the Moser-Trudinger inequality, and therefore by the boundedness of the conformal factor with respect to the hyperbolic metric. Moreover, a partial converse can be shown easily: is the conformal factor with respect to the hyperbolic metric. Then, λ 1 (B, g ρ ) = 0.
Remark 2.3. Condition (9) is actually stronger than the unboundedness of ζ, that can be expressed as lim sup To prove Proposition 2.2, and also later on, one requires some conformal diffeomorphisms, called Möbius maps, that extend in higher dimension the well-known biholomorphisms of the complex unit disc ϕ a (z) = z + a 1 + az .
The following lemma, whose proof is a simple calculation, lists the main properties of the Möbius maps (see [Rat06] for more details): For any a ∈ B and x ∈ R N \ − a |a| 2 , the map ϕ a (x) = 1 − |a| 2 x + |x| 2 + 2 a, x + 1 a |a| 2 |x| 2 + 2 a, x + 1 has the following properties: can be built in this way: given a function 0 whereas, to work with the L N norm, one observes that for for every M > 0 the set {ζ ≥ M } is a neighborhood of x, hence, due to the properties of ϕ x k , it contains ϕ x k B 1 2 (0) for k sufficiently large. Thus, being the ϕ x k 's hyperbolic isometries, However, despite this result, in general Poincaré's inequality does not imply exponential integrability up to the critical exponent, that is Theorems 1.1 and 1.2 cannot be extended in this context. Moreover, not only exponential but also polynomial integrability is not ensured; actually, for any couple of nonlinearities satisfying some basic properties one can build a metric, which attains higher and higher values on small balls accumulating on ∂B, such that the former nonlinearity is uniformly summable and the latter is not. Precisely, the result we obtained is the following: Let f 1 , f 2 ∈ C(R, R) be even, positive, nondecreasing on [0, +∞) and satisfying Then, there exists a conformal metric g ρ = ρ(x)g e such that Remark 2.6. Notice that if we choose the functions f i 's such that f 1 (u) ≥ C|u| N and f 2 (u) ≤ CΦ(u) we get that Poincaré's inequality holds whereas the Moser-Trudinger inequality does not, thus showing that Theorems 1.1 and 1.2 are no longer true in this setting.
To prove Theorem 2.5, we will use a radial estimate which plays a similar role of Lemma 1.4; the following Lemma is by Tintarev [Tin11], who used it to give another proof of the hyperbolic Moser-Trudinger inequality.
Lemma 2.7. Let u ∈ W 1,N 0 (B, g ρ ) be a radially nonincreasing function. Then Proof of Theorem 2.5. The metric will be built as where the ϕ x k 's are as in Lemma 2.4, the x k are taken such that spt(η k ) are pairwise disjoint, for instance Moser-Trudinger inequality on the hyperbolic disc, Lemma 2.7 and the properties of On the other hand, for f 2 , one takes a sequence of Moser functions whereas the fact that ϕ x k preserves hyperbolic measure and the construction of η k allow to write Taking the quantity (11) becomes whereas the addend in the sum (10) becomes The last condition in the statement of Theorem 2.5 seems tricky but it is actually satisfied by most elementary functions which satisfy the limitation on the growth.
The following result is a simple calculus exercise: Proposition 2.8.
, the following functions satisfy the third condition of Theorem 2.5:

Proof.
Notice that the third hypothesis of Theorem 2.5 only depends on the behavior of f 1 around infinity.
One can easily see that both R N f 1 (R) and hence the lim inf can be computed as a limit using l'Hôpital's rule and a change of variable: .
The existence and positiveness of the last limit is a simple calculation.

We have
if and only if r ∈ [N, p).
Notice that for α ≥ α N Theorem 2.1 implies that, if g ρ is as in the proof of Theorem 2.5, moreover, one cannot take f 1 (u) = |u| p for p < N because one does not have |u| p ≤ CΦ α (u) when |u| is small; actually, it holds |u(x)| p dV g h (x) = +∞ hence these powers are not even summable on the disc endowed with the metric g ρ .

Proof.
All but the last point follow from applying Theorem 2.5 with suitable f 1 , f 2 , that can be chosen as in Proposition 2.8: 1. It suffices to take f 1 (u) = |u| N and f 2 (u) = max e − 1 u , |u| N log |u| since |u| q ≥ C(q)f 2 (u) for any q > N .

It suffices to take
6. It suffices to take 7. For the last point, it suffices to show a metric g ρ such that is uniformly integrable on W 1,N 0 (B, g ρ ); however, f 1 does not satisfy all the hypotheses of Theorem 2.5, since it has the same asymptotic behavior of and, being However, one can argue as in the proof of Theorem 2.5, building the metric in the same way and choosing later a k and R k : that converges choosing for instance a k = k and R k = e −k 3 .

Extremals for the Moser-Trudinger inequality on strips
The last section is devoted to the problem of extremal functions for the Moser-Trudinger inequality. As mentioned before, the existence of extremals has been already proved for any bounded set Ω ⊂ R N : Then, there exists a function u ∈ W 1,N 0 (Ω) with This has been first proved by Carleson and Chang [CC86], when Ω = B R ( x) is a ball, then by Flucher [Flu92] for any planar domain, and finally by K.-C. Lin [Lin96] in the general case. Here, a first existence result is given for an unbounded domain, precisely Ω = R × (−1, 1): Theorem 3.2.
A basic tool for the proof of Theorem 3.1 is the concentration-compactness principle by P.-L. Lions [Lio85]; it states that a bounded sequence in W 1,N 0 (Ω) can either be compact for the Moser-Trudinger functional or concentrate at some point: Then, up to a subsequence, one of the following alternatives holds true: 1. As k → +∞, u k ⇀ k→+∞ 0 and |∇u k | N dx ⇀ k→+∞ δ x for some x ∈ Ω, and The sequence {u k } will be called "concentrating" at the point x in the first case, and "compact" in the latter.
In view of this, the proof of Theorem 3.1 will follow by showing that maximizing sequences for the Moser-Trudinger inequality cannot concentrate, and this is what was actually done by [CC86,Flu92,Lin96]. When dealing with general domains, one can use a slight modification of the previous result: Then, up to a subsequence, one of the following alternatives holds true:

Proof.
We first notice that, in view of Theorem K j = Ω and apply Theorem 3.4 to each of these sets, starting from K 1 . Suppose that, applying the Concentration-Compactness Theorem on K 1 , one finds that u k concentrates in some x ∈ Ω; then, since µ Ω ≤ 1, the same alternative will occur when the Concentration-Compactness principle will be applied on the other sets K j , hence one gets the first alternative in Theorem 3.5.
On the other hand, if we get compactness on K 1 , we continue to apply 3.4 on the sets K j : if we have concentration in one of these sets, we argue as before getting the first alternative; otherwise, we will find compactness of the functional on any bounded set, that is the second alternative in 3.5.
Remark 3.6. Notice that in Theorem 3.5 the convergence in L 1 loc (Ω) actually means convergence in L 1 (K) for any bounded set K ⊂ Ω, even if K ⊂ Ω. This little abuse of notation will be also done in the rest of this paper; the same holds when convergence in L p loc (Ω) will be considered for some other p > 1. Intuitively, the main difference between the Theorems 3.4 and 3.5 is that the mass might 'disappear' at the infinity; taking for instance, on the aforementioned strip Ω = R × (0, 1), on every compact subset of Ω, u k ≡ 0 definitively, so This phenomenon is known as "vanishing". Moreover there is a fourth scenario to be considered, the so-called dichotomy, that is when just a part of the mass is vanishing; as an example, one may take v k = (1 − θ)u k + θw k with u k as before, w k compact for the Moser-Trudinger functional and θ ∈ (0, 1). The strategy to exclude vanishing and dichotomy is trying to restrict the set of admissible functions for the supremum to those which have some symmetries, hence satisfy some uniform decay estimates similar to Lemmas 1.4 and 2.7. The symmetry of the strip Ω = R × (−1, 1) ⊂ R 2 with respect to both axes allows to perform a trick similar to Schwarz symmetrization: it is possible to apply twice a one-dimensional symmetrization, with respect to each axes, to any u ∈ H 1 0 (Ω), and the symmetrized function is still defined on Ω. To be precise, for any fixed x 2 ∈ (−1, 1) one defines µ u,1 (t) = |{x 1 ∈ R : u(x 1 , x 2 ) > t}| and u * ,1 (x 1 , x 2 ) = inf{t ∈ R : µ u,1 (t) ≤ 2|x 1 |}; in the same way, one puts µ u,2 (t) = |{x 2 ∈ (−1, 1) : u(x 1 , x 2 ) > t}| and u * ,1 (x 1 , x 2 ) = inf{t ∈ R : µ u,1 (t) ≤ 2|x 2 |}, and finally sets u * ,Ω = (u * ,1 ) * ,2 ; since u * ,Ω is obtained by applying twice a Steiner symmetrization, some good properties of Schwarz symmetrization still hold: 1. u * ,Ω (x 1 , x 2 ) is even in both x 1 and x 2 , that is u * ,Ω (x 1 , x 2 ) = u * ,Ω (−x 1 , x 2 ) = u * ,Ω (x 1 , −x 2 ).
2. u * ,Ω (x 1 , x 2 ) is nonincreasing in both variables for nonnegative x 1 and x 2 , that is

Proof.
Clearly, it suffices to provide estimates for nonnegative x 1 and x 2 ; since any function in H 1 (Ω) is absolutely continuous along almost every line, the decreasing character of u gives, for x 1 > 0, Moreover, for x 1 ∈ [0, 1] and x 2 > 0, This lemma implies that, away from the origin, the Moser-Trudinger functional is compact on H(Ω), except for the first term of its power series expansion: Corollary 3.9. Let u k be a sequence in H(Ω) such that u k → k→+∞ u almost everywhere in Ω. Then

Proof.
Using the estimate (6) and (12), one has hence Lebesgue's dominated convergence theorem gives the claim.
Another consequence of Lemma 3.8 is that when the limit is null, the only contributions in the integral come from neighborhoods of the origin and of infinity.
Corollary 3.10. Let u k be a sequence in H(Ω) such that u k → k→+∞ 0 almost everywhere in Ω.

Proof.
As before, one has hence the results follows applying again Lebesgue's dominated convergence theorem on the bounded set Ω R \B ε (0).
The proof of Theorem 3.2 will follow several steps: taking a maximizing sequence u k ∈ H 1 0 (Ω), one may suppose that, up to subsequences, one has u k → k→+∞ u weakly in H 1 0 (Ω), strongly in L 2 loc (Ω) and almost everywhere in Ω; moreover, up to replacing u k with |u k | * ,Ω ∇|u k | * ,Ω L 2 (Ω) , it is not restrictive to take u k ∈ H(Ω) and Denoting the supremum of the Moser-Trudinger functional as S := sup we will prove, in the order: 1. If u ≡ 0, then S is attained.
2. If u ≡ 0, set Then, by Corollary 3.10, it is independent on ε and smaller than 1 (No concentration).
For any u ∈ H 1 0 (Ω) it holds where g ψ = | det ∂ y ψ|g e = 16 π 2 1 ((y 1 + 1) 2 + y 2 2 ) ((y 1 − 1) 2 + y 2 2 ) g e is unbounded only around the two points (±1, 0), that are mapped by ψ at infinity. Moreover, the conformal invariance of the Dirichlet integral ensures that the sequence v k = u k • ψ is bounded in H 1 0 (B 1 (0), g ψ ), hence it must converge to v = u • ψ weakly, strongly in L 2 loc (B 1 (0)\{(±1, 0)}, g ψ ) and a.e. on B 1 (0). Following what was done by Moser on bounded domains [Mos71] and using the hyperbolic Moser-Trudinger inequality 2.1, we get: We apply the conformal diffeomorphism with the unit disc (13); since the metric g ψ generated by this map is bounded from above by the hyperbolic one, it suffices to verify Moreover, applying hyperbolic symmetrization, we can restrict our proof to nonnegative radially decreasing functions (see [MS10] for details), for which Lemma 2.7 holds; therefore, keeping in mind estimate (6) and the integrability of e αv 2 on any bounded Euclidean sets, we get that is what we claimed.
Some of the calculations used in this proof will be inspired by a work of Gi. Mancini and Sandeep [MS11], who studied existence of extremals for the Moser-Trudinger inequality on the hyperbolic disc.
Proof of step 1. The first case that will be considered is when the convergence is strong in L 2 (Ω); under this hypothesis, it will be shown that u itself attains the supremum, that is lim The modified concentration-compactness Theorem 3.5 gives, for any Ω R defined as in Corollary 3.10, whereas, near the infinity, one uses strong convergence for the quadratic term and Corollary 3.9 for the rest: If instead the convergence is just weak, a little more calculations are required; in this case, it holds 1 − σ := Ω |∇u(x)| 2 dx ∈ (0, 1), since σ = 1 would mean that the weak limit is null and σ = 0 would give strong convergence in H 1 0 (Ω) and so, by Poincaré's inequality, also in L 2 (Ω); moreover, weak convergence gives gives Passing to the limit for k → +∞, the last inequality becomes So, if we could pass to the limit inside the integral, we would get hence every inequality would have to be actually an equality, and thus u √ 1 − σ would be an extremal function. Now we prove that one can actually take the limit inside the integral, studying separately what happens inside and outside Ω R . In the rectangles, we find boundedness in L 2 thanks to Lemma 3.11, hence we can pass to the limit by applying Vitali's convergence theorem: which is uniformly bounded if one takes On the other hand, the integral on Ω\Ω R is small for large R: the estimate (6) and the uniform boundedness of 8πu(u k − u) + 4πu 2 , which follows from estimate (12), give which goes to 0 as R goes to +∞; therefore, given ε > 0, one can find R such that (15) is smaller than ε and, using convergence in L 1 (Ω R ), we get which is, being ε arbitrary, (14).
Passing to the limit for ε → 0, ψ −1 (B ε (0)) shrinks around 0, so Since ψ is a (nonlinear) conformal diffeomorphism, | det ∂ y ψ| is (strictly) subharmonic, hence one can apply the (strict) mean value inequality; thus, if is a radial extremal for the Moser-Trudinger inequality on the unit disc, which is a contradiction.
The proof of step 2 closely follows the estimates in [Flu92] for the concentration level in the case of planar simply connected domain; moreover, we did not use any of the symmetry properties of Ω, which are instead crucial in most of the rest of the proof of Theorem 3.2: actually, the proof of this step can be reproduced for any domain which is conformally equivalent to the ball and where the Moser-Trudinger inequality holds. This can be intuitively explained by the fact that concentration is a local property, hence it does not depend on the shape of the domain.
Proof of step 5. An explicit calculation, by separation of variables, of the first eigenvalues and eigenfunctions of −∆ on Ω k shows that λ 1 (Ω) = π 2 4 , so it suffices to show that the functional can assume values strictly larger than 16 π . The first eigenfunctions on Ω k are respectively is a vanishing sequence and the evaluation of the Moser-Trudinger functional on u k tends to the critical value 16 π , as can be seen by the proof of step 5; we will now show that this value is reached from above. It holds Ω ϕ k (x 1 , x 2 ) 2 dx 1 dx 2 = that goes to 16 π from above; therefore, the vanishing value is exceeded by eigenfunctions of large rectangles.
Some of the results obtained can be extended to other simply connected domains which satisfy some hypotheses; the double symmetrization can be made in any domain Ω which coincides with its double symmetrized, for instance if Ω f = {(x 1 , x 2 ) ∈ R 2 : |x 2 | < f (|x 1 |)} with f ∈ C 1 ((0, +∞), (0, +∞)) nonincreasing (and actually the strip we considered in Theorem 3.2 is a domain of this kind with f ≡ 1). Probably something similar can be done if some other kind of symmetry holds, but it is not known if this implies some estimates like the ones in Lemma 3.8, which were essential in most of the proof. Another difficulty is that the precise value of λ 1 (Ω), that was needed to find functions above the vanishing level, is generally not known except in very special cases; this problem might be bypassed if a first eigenfunction ϕ exists for the laplacian, since in this case However, this condition is difficult to be verified, because the embedding of L 2 (Ω) in H 1 0 (Ω) is generally not compact, even if Ω has finite measure (see for instance [Ada75]). Finally, extenstions to strips in higher dimension seem difficult as well, because one has to deal with the N -laplacian ∆ N u = div(|∇u| N −2 ∇u) and much less is known about its spectral properties than for the usual Laplace operator; moreover, the domains which are conformally equivalent to the ball are much less than in the planar case.