A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS ANDREA MALCHIODI(1)

Abstract. We consider a class of Liouville equations motivated by uniformization problems in a singular setting, as well as from models in Mathematical Physics. We will study existence from a variational point of view, using suitable improvements of the Moser-Trudinger inequality. These improvements allow to study the concentration properties of conformal volume, and to reduce the problem to studying explicit finite-dimensional objects.

1. Introduction This paper is meant to describe in more detail the content of a plenary talk at the XX Congresso dell’Unione Matematica Italiana, held in Siena during September 2015. Some further perspectives are also presented. The author is very grateful to the organizers for having given him the opportunity to speak at this important meeting. A classical problem in Riemannian geometry is to endow a given manifold with a canonical structure. Apart from the general interest in such a question, having a canonical structure is doubtfully useful in classifying objects. When dealing with two-dimensional surfaces, a natural choice of canonical structure is a metric with constant Gaussian curvature, and having to find it is the classical Uniformization problem. A possible way to deform a metric is to choose a so-called conformal representative, namely a pointwise scaling by a suitable positive function on the surface. Given a compact surface (Σ, g) with Gaussian curvature Kg , consider the conformal change of metric g 7→ g˜ = e2w g. Here w is a smooth function on Σ, and e2w a convenient way to write a general conformal factor. It is known that under conformal changes the Gaussian curvature transforms according to the law (1)

−∆g w + Kg = Kg˜ e2w .

Therefore, if one wants to look for constant curvature, it is sufficient to find a solution of (U )

−∆g w + Kg = Ke2w ,

where K is a real constant. Clearly, by the Gauss-Bonnet formula, the sign of K has to be the same as that of the Euler characteristic of Σ. We are interested here in a singular version of (U ). Singular spaces drew a lot of attention over the past decades, as they arise in many different situations such as limits of Einstein metrics ([1], [9], [56]), K¨ ahler-Einstein metrics ([22]), as well as in more physical situations as the study of interfaces or in general relativity. Here we will consider some of the simplest singular objects, namely compact surfaces with finitelymany conical points. The model is a standard cone, which can be realized from an isometry to a circular sector of the plane. Since isometries preserve the Gaussian curvature, a cone is flat at all its regular points however, viewing the curvature as a measure, it is non-zero at the vertex. More precisely, if the opening angle θ of the cone is written as θ = 2π(1 + α), α > −1, the curvature at the vertex is a Dirac mass with magnitude −2πα. Notice that for the round angle θ = 2π there is no singular part either. We will also consider angles greater than 2π which, as we will explain, have particular interest for the physical motivations of the equation. 2000 Mathematics Subject Classification. 35B33, 35J35, 53A30, 53C21. Key words and phrases. Geometric PDEs, Singular Liouville equation, Variational Methods, Min-max Schemes. (1) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy. email: [email protected]. 1

ANDREA MALCHIODI(1)

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Having this model in mind, for p1 , . . . , pm points in Σ and α1 , . . . , αm > −1, we will consider the following problem on a compact, closed surface (Σ, g) of total volume 1 −∆g u + Kg = ρ e2u − 2π

(2)

m X

ρ ∈ R.

αj δpj ;

j=1

Equation (2) is a singular version of (U ), and a solution will endow Σ with a constant-curvature metric on Σ \ ∪m i=1 {pi }, and conical angles θi = 2π(1 + αi ) at each point pi . All the singular structure is encoded in the divisor m X αi pi . α := i=1

Still by the Gauss-Bonnet formula (assuming w.l.o.g. that V olg (Σ) = 1), the constant ρ should satisfy the geometric constraint m X (3) ρ = 2πχ(Σ) + 2π αi . i=1

The above formula can be obtained, for example, rounding-off the conical points and applying the usual Gauss-Bonnet theorem. To study existence for (2), it is useful to desingularize it. This can be done considering the Green’s function of −∆g on Σ with pole p, namely the solution to ˆ (4) −∆g Gp (x) = δp − 1 on Σ, with Gp (x) dVg = 0. Σ 1 log dg (x, p) near the singularity, where dg (·, ·) stands for Notice that Gp has the asymptotics Gp ' − 2π the distance induced by g. Making the following change of variables m X (5) u 7→ u + 2π αj Gpj (x), j=1

problem (2) becomes   2u ˜ −∆g u = ρ h(x)e −a ˜(x)

(6)

on Σ,

´ ˜ ˜(x) is a smooth function on Σ such that Σ a where h(x) = e−2π j=1 αj Gpj (x) . Here a ˜(x)dVg = 1, while ˜ by the asymptotics of the Green’s function h satisfies ˜ > 0 on Σ \ ∪j {pj }; ˜ (7) h h(x) ' γj dg (x, pj )2αj near pj Pm

for some constant γj > 0. Problem (6) is the Euler-Lagrange equation of the energy ˆ ˆ ˆ 2u ˜ |∇g u|2 dVg + 2ρ a ˜(x) u dVg − ρ log h(x)e dVg ; (8) Iρ,α (u) = Σ

Σ

u ∈ H 1 (Σ).

Σ

Recall that in two dimensions H 1 (Σ) embeds into every Lp space: however the embedding can be extended up to exponential class. Indeed one has the well-known Moser-Trudinger inequality, giving a quantitative estimate on exponential integrals ˆ ˆ 1 (9) log e2(u−u) dVg ≤ |∇g u|2 dVg + CΣ,g , 4π Σ Σ ffl where u = Σ u dVg stands for the average of u on Σ. In the singular case the Moser-Trudinger inequality on Σ has a different best constant, as was proven by Chen and Troyanov in [17] and [58] (see also [15]). ˜ : Σ → R be as in (7). Then one has the Proposition 1.1. ([17], [58]) Let αj > −1 for all j, and let h inequality ˆ ˆ 1 2(u−u) ˜ (10) log h(x)e dVg ≤ |∇u|2 dVg + Ch,g ˜ 4π min{1, 1 + minj αj } Σ Σ for all u ∈ H 1 (Σ).

A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS

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Notice that the multiplicative constant appearing in the last formula is determined by the most singular ˜ which becomes unbounded at the points pi with negative weights αi . The behaviour of the function h, constant instead coincides with the one in (9) when all weights αi are positive. Depending then on the value of ρ, we distinguish three geometric cases. Subcritical case: ρ < 4π min{1, 1 + minj αj }. In this case the latter term in Iρ,α can be absorbed into the first one, giving coercivity of the energy. As a consequence, one always finds solutions using the direct methods of the Calculus of Variations, i.e. taking weak limits of minimizing sequences. See for example [41], [55], [57]. In the regular case (U ), this situation corresponds to the negative or zero curvature case. Critical case: ρ = 4π min{1, 1 + minj αj }. This time the energy Iρ,α is bounded below but coercivity is lost, so it is unclear whether minimizing sequences would converge. If compactness fails, a typical behaviour of solutions (described in more detail later) leads to indefinite concentration of conformal volume at a finite number of points. For example, in the positive curvature case of (U ) (i.e. on the sphere), the loss of compactness is caused by the action on Sobolev functions of the (non-compact) M¨ obius group, which might cause all conformal volume to concentrate to a single point, but leaving the Euler-Lagrange energy for (U ) invariant. A careful blow-up analysis of the minimizing sequences might still lead to existence results: we will not discuss the details here, referring the reader to [27], [50] (and to [26] specifically for the uniformization case). Supercritical case: ρ > 4π min{1, 1 + minj αj }. This is the most delicate case, and has no regular counterpart in (U ). The fact that ρ exceeds the Chen-Troyanov constant causes unboundedness from below of the energy, so it is hopeless to try to find global minima as before. Worse than that, there are situations in which solutions do not exist: one well-known example is the tear-drop, namely a spherical surface with only one singularity. It is known that there is indeed no constant curvature metric on such an object (and more examples will be discussed later on). This is the case we will mostly be interested in, and we will show that a variational approach might still give conclusions in the search of critical points of saddle type. In the last section another approach to the problem, which relies on degree-theoretical arguments, will be discussed and compared to ours. We want to describe here a general strategy for the supercritical cases. This relies on finding saddle points for Iρ,α , using min-max methods that combine topological and variational arguments. In order to apply them, as for the Direct minimization methods, one fundamental condition is compactness. Concerning problem (2), an alternative was proved in [6] (after previous results in [10], [39] for the regular case): either a sequence (un )n of solutions to (Eρn ) (with ρn → ρ ∈ R) stays uniformly bounded, or it develops a finite number of spheres at regular points and/or american footballs at singular points, see Theorem 3.8 for a precise statement. An american football is obtained from a sphere (possibly covered multiple times) by cutting two meridians and by gluing the remaining edges. This results in a constant-curvature singular surface having two equal conical angles θ = 2π(1 + α): by the modified Gauss-Bonnet formula (1) the total curvature of this object must be 4π(1 + α). In the blow-up alternative, is it possible to prove that all the curvature is exhausted in this way, and ˜ 2un , must converge to a number in this therefore ρn , the total curvatures of the conformal metric he discrete set ( ) X (11) S = ρ | ρ = 4πn + 4π (1 + αi ), n ∈ N, I ⊆ {1, . . . , m} . i∈I

On the other hand, if ρ does not belong to this set, solutions have to stay compact and variational methods can be applied. Recall that in the super-critical regime the Euler-Lagrange energy is unbounded from below. However there is a way to describe how the lower bounds fail, in terms of concentration of conformal volume. More precisely, it turns out that the multiplicative constant in (10) improves if the conformal volume spreads over Σ, see Lemma 2.1. Having a better constant implies more chances to bound the energy from below, and therefore a low energy forbids too much spreading of the volume. Roughly, localizing (via cut-off functions) (10) near a regular or a singular points, in the denominator one finds respectively the value 4π or 4π(1 + α) (indeed, points with positive α’s need a special care, as explained in Subsection

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ANDREA MALCHIODI(1)

2.2). This suggests to introduce a weighted cardinality χ on points of Σ as follows: set ( χ(q) = 4π if q ∈ Σ \ {p1 , . . . , pm }; (12) χ(pi ) = 4π(1 + αi ) for all i = 1, . . . , m. Define also P(Σ) = {µ : µ is a probability measure on Σ} . As the total curvature we have at hand is ρ, this counting suggest that the limit measures for small energy are the following (13)

Σρ,α = {µ ∈ P(Σ) : 4πχ(supp(σ)) < ρ} ;

σ ∈ S.

We endow this set, naturally, with the weak topology of distributions: the latter is in fact induced by a metric d on measures, see (22). Our main result reads as follows. Theorem 1.2. ([14]) Suppose that ρ 6∈ S and that Σρ,α is not contractible. Then (6) has a solution. We notice that some assumption has to be imposed on Σ and the αi ’s in order to have existence, as the tear-drop example shows. In this case Σρ,α is homeomorphic to either a point or to S 2 with a point removed: in either case contractible. Liouville equations arise in Mathematical Physics as well, to describe mean field vorticity in steady flows (see [11], [16]), Chern-Simons vortices in superconductivity or Electroweak theory (see [54], [60]). For these problems ρ represents a positive physical parameter, and is not assumed to satisfy (3). The points pj are called vortices, and describe either points where vorticity is imposed by external forces, or vortex points, namely zeroes of the Higgs field with vanishing order αi (a positive integer, in the latter case). The case of the torus is particularly meaningful, as it represents a periodic configuration in the plane. We discuss next the plan of the paper. In Section 2 we will analyse the variational structure of the problem, and in particular the role of the Moser-Trudinger inequality and some of its improvements. We will see how this is helpful to characterize the concentration properties of e2u when the Euler-Lagrange energy is low enough. More precisely, in the regular case we will see that for ρ ∈ (4kπ, 4(k + 1)π) if u has low energy then e2u may concentrate near at most k points of Σ. The singular case will be then discussed, with particular attention to the case of positive weight. For these, a scaling-invariant improved inequality is needed: this is derived in terms of angular moments of the volume measure around such singular points. Section 3 will be devoted to the existence part. For the regular case, we will introduce a family of test functions with low energy for which the exponentials are concentrated near k arbitrary points, giving a sort of reverse of the previous characterization. Out of this construction we build a min-max scheme proving existence for a generic choice of the parameter ρ. We turn then to the singular case, and describe for simplicity only the cases of negative weights as well as a model problem on the unit ball of R2 , containing all the features of the positive-weight case. In Section 4 instead we will collect some non-existence results, compare our results to others in the literature and discuss some perspectives. Acknowledgements The author has been supported by the PRIN project Variational and perturbative aspects of nonlinear differential problems and by the project Geometric Variational Problems by Scuola Normale Superiore. The author is a member of the group G.N.A.M.P.A., as a part of INdAM. 2. Improved Moser-Trudinger inequalities In this section we describe how improved Moser-Trudinger inequalities can be used to deduce information on functions whose Euler-Lagrange energy is small enough. We first discuss the regular case, passing then to the singular one. 2.1. The regular case. Here we are interested in the regular case of the Liouville equation, so we will assume that ˜ h(x) is a smooth positive function, and we set Iρ := Iρ,0 . We let ρ be any positive parameter, not satisfying in general the geometric constrain (3). As we explained, this extra freedom allows to treat as well cases with physical motivations.

A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS

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We are mainly interested here in the case ρ > 4π, and would like to describe some conditions on u in order to obtain lower bounds on Iρ even beyond the coercivity threshold. It turns out that the spreading of the function e2u (in a sense to be described below) gives sufficient conditions to ensure this lower bound, which is deduced via some improved versions of the Moser-Trudinger inequality. Loosely speaking, the more e2u is spread over Σ, the better is the constant one can put in front of the Dirichlet norm in (9). Two well-known examples were due to J.Moser and T.Aubin, see [2], [48]. Moser proved 1 1 that one can replace 4π by 8π on the standard sphere (S 2 , gS 2 ) provided u is antipodally symmetric. 1 provided u is balanced, Aubin showed instead that on (S 2 , gS 2 ) one can take any constant larger than 8π which means that ˆ xi e2u dVS 2 = 0;

i = 1, 2, 3.

S2

Here xi stand for the Euclidean i-th coordinate function, so the balancing condition means having zero center of mass in R3 for the conformal volume. W.Chen and C.Li, [18], extended this argument to arbitrary surfaces, showing that if e2u has integral 1 in (9) can be basically divided bounded from below into two separate subsets of Σ, then the constant 4π by two. The next lemma extends this result in [18] to an arbitrary number of spreading regions with minor modifications. Lemma 2.1. For a fixed integer `, let Ω1 , . . . , Ω`+1 of Σ satisfying dg (Ωi , Ωj ) ≥ δ0 for i 6= j,  be subsets  1 where δ0 is a positive real number, and let γ0 ∈ 0, `+1 . Then, for any ε˜ > 0 there exists a constant C = C(`, ε˜, δ0 , γ0 ) such that

ˆ e2(u−u) dVg ≤ C +

log Σ

1 4(` + 1)π − ε˜

for all the functions u ∈ H 1 (Σ) satisfying ´ e2u dVg ´Ωi (14) ≥ γ0 , e2u dVg Σ

ˆ |∇g u|2 dVg Σ

∀ i ∈ {1, . . . , ` + 1}.

Proof. Assuming without loss of generality that u = 0, one can find ` + 1 functions g1 , . . . , g`+1 such that  gi (x) ∈ [0, 1] for every x ∈ Σ;    gi (x) = 1, for every x ∈ Ωi , i = 1, . . . , ` + 1; (15) g (x) = 0, if dg (x, Ωi ) ≥ δ40 ;    i kgi kC 4 (Σ) ≤ Cδ0 , where Cδ0 is a positive constant depending only on δ0 . By interpolation, for any ε > 0 there exists Cε,δ0 (depending only on ε and δ0 ) such that, for any v ∈ H 1 (Σ) and for any i ∈ {1, . . . , ` + 1} we have ˆ ˆ ˆ ˆ 2 2 2 2 (16) |∇g (gi v)| dVg ≤ gi |∇g v| dVg + ε |∇g v| dVg + Cε,δ0 v 2 dVg . Σ

We next notice that

Σ

ˆ

1 e dVg ≤ γ 0 Σ

Σ

ˆ

Σ

ˆ 2u

2u

e2gi u dV.

e dVg ≤ Ωi

Σ

Using the standard Moser-Trudinger inequality one finds ˆ l+1 ˆ l+1 X 1 1 X log e2u dVg ≤ log + |∇g (gi v)|2 dVg + gi u + Cl,Σ,g . γ0 4π i=1 Σ Σ i=1 By (16) we then deduce ˆ ˆ l+1 ˆ l+1 X 1 1+εX 2u 2 + |∇g u| dVg + gi u + Cl,Σ,g Cε,δ0 v 2 dVg . log e dVg ≤ log γ0 4π i=1 Σ Σ Σ i=1 Since we are assuming the average of u to be zero, the average terms in the last formula are bounded by a constant times the Dirichlet norm of u by Poincar´e’s inequality. Therefore, using the elementary

ANDREA MALCHIODI(1)

6

1 inequality t ≤ εt2 + 4ε we find that ˆ ˆ l+1 ˆ 1 1+εX log e2u dVg ≤ log + |∇g u|2 dVg + Cl,Σ,g,ε + Cε,δ0 v 2 dVg . γ0 4π i=1 Σ Σ Σ

It can be shown, for example using truncations (in height or in Fourier modes) that the last term is lower order. By Lemma 2.1, spreading of e2u into ` + 1 regions improves the Moser-Trudinger constant nearly by 1 a factor `+1 . This immediately implies a lower bound on Iρ for ρ < 4π(` + 1). As a consequence, if ρ < 4π(` + 1) and if Iρ (u) is sufficiently low, the function e2u is not allowed to spread into ` + 1 regions, and should instead concentrate near at most ` points of Σ. We will state next two lemmas making this reasoning rigorous via a covering procedure. Lemma 2.2. Let ` ∈ N, and fix two positive number ε and r. Suppose for a non-negative function f ∈ L1 (Σ) with kf kL1 (Σ) = 1 the following condition holds ˆ f dVg < 1 − ε for every `-tuple q1 , . . . , q` ∈ Σ. ∪`i=1 Br (qi )

Then there exist ε > 0 and r > 0, depending only on ε, r, ` and Σ (but not on f ), and ` + 1 points q 1 , . . . , q `+1 ∈ Σ (which depend on f ) satisfying ˆ ˆ f dVg ≥ ε, . . . , f dVg ≥ ε; B2r (q i ) ∩ B2r (q j ) = ∅ for i 6= j. Br (q 1 )

Br (q `+1 )

Proof. Arguing by contradiction, assume that for every ε, r > 0 there exists f as in the statement and such that for every (` + 1)-tuple of points q1 , . . . , q`+1 in Σ we have ˆ (17) f dVg ≥ ε ∀j = 1, . . . , ` + 1 ⇒ B2r (qi ) ∩ B2r (qj ) 6= ∅ for some i 6= j. Br (qj ) r 8,

Let r = where r is given in the statement. We can find h ∈ N and h points x1 , . . . , xh ∈ Σ such that ε ∪hi=1 Br (xi ) covers Σ. For ε as in the statement of the Lemma, we also define ε = 2h . We remark that the choice of r and ε depends on r, ε, ` and Σ only, as required.´ Let {˜ x1 , . . . , x ˜j } ⊆ {x1 , . . . , xh } denote the points for which Br (˜xi ) f dVg ≥ ε. Define x ˜ j1 = x ˜1 , and let A1 denote the set xi ) : B2r (˜ xi ) ∩ B2r (˜ xj1 ) 6= ∅} ⊆ B4r (˜ xj1 ). A1 = {∪i Br (˜ If there exists x ˜j2 with B2r (˜ xj2 ) ∩ B2r (˜ xj1 ) = ∅, we set A2 = {∪i Br (˜ xi ) : B2r (˜ xi ) ∩ B2r (˜ xj2 ) 6= ∅} ⊆ B4r (˜ xj2 ). Proceeding in this way, we choose recursively points x ˜j3 , x ˜ j4 , . . . , x ˜js such that xjs ) ∩ B2r (˜ xja ) = ∅ B2r (˜

∀ 1 ≤ a < s,

and introduce sets A3 , . . . , As by As = {∪i Br (˜ xi ) : B2r (˜ xi ) ∩ B2r (˜ xjs ) 6= ∅} ⊆ B4r (˜ xjs ). Because of (17), the process cannot go further than x ˜j` , and hence s ≤ `. Using the definition of r we obtain (18)

∪ji=1 Br (˜ xi ) ⊆ ∪si=1 Ai ⊆ ∪si=1 B4r (˜ xji ) ⊆ ∪si=1 Br (˜ xji ).

x1 , . . . , x ˜j } and by (18) one has Then by our choice of h, ε, {˜ ˆ ˆ ˆ f dVg ≤ f dVg ≤ f dVg Σ\∪si=1 Br (˜ xji ) Σ\∪ji=1 Br (˜ xi ) (∪hi=1 Br (xi ))\(∪ji=1 Br (˜xi )) ε < (h − j)ε ≤ . 2 Finally, if we chose qi = x ˜ji for i = 1, . . . , s and qi = x ˜js for i = s + 1, . . . , `, we get a contradiction to the assumptions of the lemma.

A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS

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Using Lemmas 2.1 and 2.2 we will now be able to analyse functions u ∈ H 1 (Σ) for which Iρ is large negative, when ρ < 4(k + 1)π, k ∈ N. Lemma 2.3. Suppose ρ < 4(k + 1)π, with k ∈ N. Then for any ε > 0 and any r > 0 there exists a large L = L(ε, r) such that for every u ∈ H 1 (Σ) with Iρ (u) ≤ −L there exist k points q1,u , . . . , qk,u ∈ Σ with ˆ 1 ´ e2u dVg < ε. (19) 2u dV k e g B (q ) Σ\∪ r i,u Σ i=1 Proof. Suppose again by contradiction that the statement is not true, namely there exist ε, r > 0´ and (un )n ⊆ H 1 (Σ) with Iρ (un ) → −∞ and such that for every k-tuple q1 , . . . , qk in Σ one has eun dVg < 1 − ε. Noting that Iρ is invariant under adding constants, we can assume that for ∪k i=1 Br (qi ) ´ every n we have the normalization Σ e2un dVg = 1. Then we can apply Lemma 2.2 with ` = k, f = e2un , and afterwards Lemma 2.1 with δ0 = 2r, Ω1 = Br (q 1 ), . . . , Ωk+1 = Br (q k+1 ) and γ0 = ε, where ε, r and (q i )i are given by Lemma 2.2. This implies that for any ε˜ > 0 there exists C > 0 depending only on ε, ε˜ and r such that ˆ ˆ Iρ (un ) ≥ |∇g un |2 dVg + 2ρ a ˜ un dVg − Cρ Σ Σ ˆ ρ |∇g un |2 dVg − ρ un , − 4(k + 1)π − ε˜ Σ ρ with C independent of n. Since ρ < 4(k + 1)π, we can choose ε˜ > 0 so small that 1 − 4(k+1)π−˜ ε := δ > 0. Hence using also the Poincar´e inequality we find ˆ ˆ 2 Iρ (un ) ≥ δ |∇g un | dVg + 2ρ a ˜(un − un )dVg − Cρ Σ

Σ

ˆ (20)

≥

ˆ

2

|∇g un | dVg − C

δ

2

|∇g un | dVg

Σ

 21 − Cρ ≥ −C.

Σ

This goes against our contradiction assumption, and allows us to conclude the proof. By the previous lemma it follows that if the Euler-Lagrange energy is low enough then the function e2u , normalized in L1 , is localized near at most k points of Σ. Distributionally it behaves then like a sum of Dirac masses, with at most k summands. Recalling that P(Σ) stands for the space of probability measures on Σ, it is therefore natural to introduce the set of formal barycenters of Σ   k k  X X tj = 1, xj ∈ Σ . tj δxj : (21) Σk = {µ ∈ P(Σ) : cardg (supp(µ)) ≤ k} =   j=1

j=1

Σk is naturally endowed with the weak topology of distributions: since it is a finite-dimensional object, this topology is realized by the following distance, called Kantorovich-Rubinstein distance ˆ ˆ (22) d(µ, ν) = sup f dν ; µ, ν ∈ M(Σ). f dµ − kf kLip(Σ) ≤1

Σ

Σ

When k = 1 Σk is clearly homeomorphic to Σ, which is smooth: however, when k ≥ 2 Σk becomes a stratified set, union of open manifolds of different dimensions. This is due to the fact that the representation of the elements in the second bracket of the above formula is not a good parametrization for all points (as the coefficients ti might vanish and the points xi might merge), and hence singularities appear. Notice also that for ρ ∈ (4kπ, 4(k + 1)π), Σk coincides with Σρ,0 . A further characterization of this set will be given in the next section, devoted to existence of solutions. As a consequence of Lemma 2.3 we have, using the last notations, that when ρ < 4(k + 1)π then d(e2u , Σk ) → 0 as Iρ (u) → −∞. In fact, with some extra work, one can show the following result, see [30], [29]. Proposition 2.4. For ρ < 4(k + 1)π there exist L and a continuous map Ψk : {Iρ ≤ −L} → Σk such that, if e2un * σ ∈ Σk , then Ψk (un ) * σ. The latter result allows to assign in a natural way an element of Σk to any low-energy function. We will see in the next section how to construct a sort of inverse map, which the min-max scheme will rely on.

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ANDREA MALCHIODI(1)

2.2. The singular case. We discuss next the counterpart of the previous subsection in presence of singularities. Recall first of all the inequality in Proposition 1.1. Again, we wish to derive some improved inequalities in terms of spreading of the function 2u ˜ h(x)e . f˜u := ´ 2u dV ˜ h(x)e Σ

Similarly to Lemma 2.1, (10) can also be localized. If some portion of f˜u is localized near a regular point, the corresponding gain in the constant will still be 4π. If instead f˜u is localized near a singular point pi with negative weight αi , we will gain locally a quantity of size 4π(1 + αi ). We get therefore the following result. Lemma 2.5. Let n ∈ N, let I ⊆ {1, . . . , m} with n + card (I) > 0, and let αi < 0 for all i ∈ I. Assume there exist r > 0, δ0 > 0 and pairwise distinct points {q1 , . . . , qn } ⊆ Σ \ {p1 , . . . , pm } such that: • for any couple {a, b} ⊆ {q1 , . . . , qn ∪ (∪i∈I pi )} with a 6= b one has distg (Br (a) , Br (b)) ≥ 4δ0 ; • for any a ∈ {q1 , . . . , qm } one has dg (pi , Br (a)) ≥ 4δ0 for any i ∈ {1, . . . , m} \ I;   1 . and consider any γ0 ∈ 0, n+card(I) Then, for any e  > 0 there exists a constant C := C(Σ, g, n, I, r, δ0 , γ0 , e ) such that ˆ ˆ 1 e
|∇g u|2 dVg + C P (23) log he2(u−u) dVg ≤ 4π n + i∈I (1 + αi ) − e  Σ Σ for all functions u ∈ H 1 (Σ) satisfying ´ e he2u dVg Br (a) ≥ γ0 , ´ 2u dV e he g Σ

∀ a ∈ {q1 , . . . , qn ∪ (∪i∈I pi )} .

Similarly to Lemma 2.3, we obtain a concentration behaviour for the function f˜u provided the EulerLagrange energy is low enough. The following result holds, and takes into account the different characters of the accumulation points, regular or singular with negative weights. To introduce it, let us consider the subset of indices i corresponding to pi ’s with negative weights α and define the negative divisor X αi pi , α− = pi ∈I−

and the corresponding weighted barycentric set Σρ,α− = {µ ∈ P(Σ) : χ− (supp(µ)) < ρ} . Here χ− stands for the weighted cardinality defined as ( χ− (pi ) = 4π(1 + αi ) for all i = 1, . . . , m with αi < 0; (24) χ− (q) = 4π otherwise. As a counterpart of Proposition 2.4 we obtain the following result, in which we set S− as the counterpart of (11) with α replaced by α− . n o Proposition 2.6. For ρ 6∈ S− , there exist L and a continuous (natural) map Ψρ,α− : Iρ,α− ≤ −L → Σρ,α− such that, if e2un * σ ∈ Σρ,α− , then Ψρ,α− (un ) * σ. As we noticed, there is a difference here between points with positive and negative weights, as in (10) ˜ vanishes near singular point only negative α’s count. Notice also that for positive weights the function h with some power rate, so one could hope to improve the constant if u concentrates enough near such a singularity. This is in fact a delicate issue, and to obtain rigorous estimates it is necessary to understand the behaviour of f˜u near singular points in more detail. To fix the ideas, we discuss next a toy model, in which the surface is replaced by the unit ball B of R2 , and we work in the space H01 (B), so that Dirichlet boundary conditions are imposed. We discuss the simple case in which a singularity of weight α > 0 is at the center of the ball.

A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS

9

An improvement of inequality (10) was derived in [31] for radial functions: it was shown that for any α > 0 there exists Cα > 0 such that ˆ ˆ 1 2α 2(u−u) 1 dx ≤ (25) log |x| e |∇g u|2 dx + Cα ; u ∈ H0,r (B). 4(1 + α)π B B 1 This improves the original Troyanov constant 4π . We will show that there is still a way to improve the constant, but imposing only finitely-many constraints on u, more or less in the same spirit of Aubin’s result compared to Moser’s (see Subsection 2.1). To explain this condition in more detail, let f˜u,B denote the probability measure on B

|x|2α e2u . f˜u,B = ´ |x|2α e2u dx B

(26)

Define then the probability measure on the circle ˆ (27) µu (A) = f˜u,B dx;

A ⊆ S1,

˜ A

A˜ = ∪t∈(0,1] tA,

and then introduce the k-moments of the above measure ˆ ˆ ˆ z dµu , z 2 dµu , . . . , (28) Fk (µu ) = S1

S1

k



z dµu .

S1

We have then the following result. Lemma 2.7. Let α > 0, k ∈ N, and let Fk denote the map in (28). Then for every ε > 0 there exists a constant Cε,α,k , depending only on ε, k and α, such that ˆ ˆ 1+ε 2α 2u log |x| e dx ≤ |∇u|2 dx + Cε,α,k , 4π min{1 + k, 1 + α} B B whenever Fk (µu ) = 0. Let us comment on the above result. Qualitatively, it is saying that the more angular moments of µu 1 vanish, the more the constant improves. However, it never goes below 4π(1+α) , as this is the sharp one among radial functions (for which all moments vanish). The main feature of the inequality is that it is scaling-invariant, differently from the one in Lemma 2.1, and it allows concentration of f˜u,B at any pace. Macroscopically, the latter measure might look like a Dirac mass, and still one could gain in the constant. We will not report here the proof of the latter inequality, which is quite technical, and limit ourselves to describe the main ideas involved. While the unit ball is compact, to retain the geometric invariance of the problem by dilation we regard it as an infinite half-cylinder blowing-up the metric near the singularity as g˜ = |x|1 2 (dx)2 . We then measure the spreading of f˜u,B on this cylinder looking at the following quantity, for ρ ∈ (4kπ, 4(k + 1)π), and a given small δ > 0 ˆ ˜ (29) Jk,δ (fu ) = sup f˜u dx. x1 ,...,xk 6=0

∪k i=1 Bδ|xi | (xi )

Two alternatives then may occur: when Jk,δ (f˜u ) is close to 1 and when it is not. Pk When Jk,δ (f ) is close to 1 then µu ' i=1 ti δθi (in the distributional sense) for some ti ≥ 0 and some θi ∈ S 1 . In our notation, this corresponds to a barycentric set over S 1 (see (13), replacing Σ by S 1 ). The k-barycenters of S 1 , (S 1 )k , are known to be homotopically equivalent to S 2k−1 , see Theorem 1.1 and Corollary 1.5 in [37]. It is however possible to understand this set in more detail, proving that it is indeed homeomorphic to a (piecewise smooth) immersed sphere Sk in Ck , see Section 3 in [10], where the following result was proved. Lemma 2.8. The map Fk realizes a homeomorphism between (S 1 )k and a topological sphere Sk in Ck which bounds a neighbourhood Uk of 0 ∈ Ck . The other case is when Jk,δ (f ) is not close to 1: then, similarly in spirit to Lemma 2.2 one is able to find k + 1 separate regions on the cylinder all containing a finite portion of the mass of f˜u,B . Once this is achieved, it is possible to use a more accurate version of Lemma 2.1 that uses cut-off functions not to zero, but to suitable values that take into account the locations of this regions: the choice of these values is made to optimize the the cost of the cut-offs. Notice that when Fk (µu ) = 0 we must be in the second

ANDREA MALCHIODI(1)

10

alternative otherwise, by Proposition 2.8, the image of Fk would be close to Sk = Fk ((S 1 )k ), which is bounded away from zero. As an application of the above proposition we consider the following problem in B  ˜ 2u he  in B; −∆u = ρ ´ ˜ 2u dx (30) he B  u=0 on ∂B, ˜ satisfying with h ˜ ' |x|2α h

near the origin.

This problem is variational, and has Euler-Lagrange energy ˆ ˆ B 2 2u ˜ Iρ,α = |∇u| dx − ρ log h(x)e . B

B

We have then the following result. Proposition 2.9. Suppose ρ < 4π(1 + α), and let k be the smallest  B integer greater1 or equal to α. Then there exists a large constant LB and a continuous map ψk : Iρ,α ≤ −LB → (S )k such that, if µun * σ ∈ (S 1 )k , then ψk (un ) * σ. B The result is obtained in the following way. If Iρ,α (u) is low enough, then the improved MoserTrudinger constant is less than 4π(1 + α) ≤ 4π(1 + k). Therefore, the assumptions of Lemma 2.7 are not satisfied and Fk (µu ) 6= 0. By the previous lemma, Fk (µu ) belongs to a pointed ball that can be retracted to its boundary, which is Fk ((S 1 )k ). Applying then Fk−1 , we obtain the desired map.

3. Existence of solutions Here we will discuss existence of solutions, first for the regular case and then for the singular one. 3.1. Existence for the regular case. Proposition 2.4 gave us a description of low-energy levels in terms of the barycentric sets Σk . The next proposition shows that the characterization is somehow optimal. Pk Pk Given σ ∈ Σk , σ = i=1 ti δxi ( i=1 ti = 1) and λ > 0, define ϕλ,σ : Σ → R by  2 k X λ 1 1 ti − log π, (31) ϕλ,σ (y) = log 2 d2 (y) 2 1 + λ 2 i i=1 where xi , y ∈ Σ.

di (y) = dg (y, xi ), We have the following result.

Proposition 3.1. Let ρ > 4kπ and let ϕλ,σ be defined as in (31). Then, as λ → +∞ the following properties hold true (i) e2ϕλ,σ * σ weakly in the sense of distributions; (ii) Iρ (ϕλ,σ ) → −∞ uniformly in σ ∈ Σk ; (iii) if Ψk is given by Proposition 2.4 and if ϕλ,σ is as in (31), then for λ sufficiently large the map σ 7→ Ψk (ϕλ,σ ) is homotopic to the identity on Σk . Proof. To prove (i) we first consider  ϕ˜λ,x (y) =

λ 1 + λ2 d2g (x, y)

2 ;

y ∈ Σ,

for x a fixed point of Σ. It is easy to show that ϕ˜λ,x → πδx as λ → +∞. Then (i) follows immediately from the explicit expression of ϕλ,σ , since its exponential is a linear combinations of ϕ˜λ,xi .

A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS

To show (ii), we prove the following estimates ˆ (32) ρ a ˜(x)ϕλ,σ dVg = −(ρ + oλ (1)) log λ

11

(oλ (1) → 0 as λ → +∞) ;

Σ

ˆ (33)

˜ e2ϕλ,σ dVg = O(1) h

log

as λ → +∞;

Σ

ˆ |∇g u|2 dVg ≤ 8kπ(1 + oλ (1)) log λ

(34)

as λ → +∞.

Σ

Once we have these, (ii) follows immediately. Proof of (32). Fixing any δ 0 small we have that log

π π λ λ − log ≤ ϕλ,σ (y) ≤ log − log , 1 + λ2 diam(Σ)2 2 1 + λ2 δ 2 2

y ∈ Σ \ ∪ki=1 B2δ (xi ),

and (35)

λ 1 π π 1 log − log ≤ ϕλ,σ (y) ≤ log λ − log , 2 1 + 4λ2 diam(Σ)2 2 2 2

for y ∈ ∪ki=1 B2δ (xi ).

The first of these two estimates can be written as     diam(Σ)2 δ2 π π (36) − log λ − log 1 + − log ≤ ϕλ,σ ≤ − log λ − log 1 + 2 − log λ2 2 λ 2 in Σ \ ∪ki=1 Bδ (xi ). From (35), (36) and the fact that a ˜ integrates to 1 on Σ we deduce that ˆ   ρ ϕλ,σ dVg = ρ (− log λ)(1 + O(δ 2 )) + O(1) + O(δ 2 )(| log λ| + | log δ|) , Σ

as λ → +∞. By the arbitrarity of δ, (32) follows. ˜ is bounded above and below by positive Proof of (33). It follows easily from (i) and the fact that h 2ϕλ,σ constants, since we can integrate the function e over Σ, to obtain a quantity with uniformly bounded logarithm. Proof of (34). We will show the following two pointwise estimates on the gradient of ϕλ,σ (37)

|∇ϕλ,σ (y)| ≤ Cλ;

where C is a constant independent of σ and λ, and 2 where (38) |∇ϕλ,σ (y)| ≤ dmin (y)

for every y ∈ Σ,

dmin (y) = min {dg (y, xi )}. i=1,...,k

To check (37) we notice that (39)

λ2 dg (y, xi ) ≤ Cλ, 1 + λ2 d2g (y, xi )

i = 1, . . . , k,

where C is a fixed constant (independent of λ and xi ). Moreover P ti (1 + λ2 d2i (y))−3 ∇y (d2i (y)) . (40) ∇ϕλ,σ (y) = −λ2 i P 2 2 −2 j tj (1 + λ dj (y)) Using |∇y (d2i (y))| ≤ 2di (y) and inserting (39) into (40) we obtain immediately (37). Similarly we get P P 2 2 −2 di (y) i ti (1 + λ di (y)) ti (1 + λ2 d2i (y))−3 di (y) λ2 d2i (y) 2 2 iP P |∇ϕλ,σ (y)| ≤ 2λ ≤ 2λ 2 2 2 −2 2 −2 j tj (1 + λ dj (y)) j tj (1 + λ dj (y)) P 1 2 2 −2 2 i ti (1 + λ di (y)) dmin (y) P ≤ 2 ≤ , 2 d2 (y))−2 t (1 + λ d (y) j min j j which is (38).

ANDREA MALCHIODI(1)

12

(37) then implies

ˆ |∇g u|2 dVg ≤ Ck

(41) ∪k i=1 B 1 (xi ) λ

for some fixed C depending only on Σ. Introducing the sets   k Ai = y ∈ Σ : dg (y, xi ) = min{dg (y, xj )} . j=1

Then we have ˆ |∇g u|2 dVg

≤

Σ\∪k i=1 B 1 (xi )

k ˆ X i=1

λ

≤

4

|∇g u|2 dVg

Ai \B 1 (xi )

k ˆ X i=1

λ

Ai \B 1 (xi )

1 dVg ≤ 8π(1 + oλ (1)) log λ dg (y, xi )2

λ

as λ → +∞. From (41) and the last formula we finally deduce (34). To show (iii) it is sufficient to use (i) and the last statement of Proposition 2.4. We will need next arguments more topological in nature. We begin with the following property on Σk . Lemma 3.2. For any k ≥ 1, Σk is non-contractible. Proof. If k = 1 the statement is obvious as Σ1 ' Σ, so we turn to the case k ≥ 2. The set Σk \ Σk−1 is an open manifold of dimension 3k − 1 (in fact, the cell of maximal dimension in Σk ). Σk−1 is a retraction of some neighbourhood with smooth boundary (see [42]). Therefore Σk has an orientation (mod 2) with respect to Σk−1 , namely the relative homology class H3k−1 (Σk , Σk−1 ; Z2 ) is non-trivial. Consider next the exact homology sequence of the pair (Σk , Σk−1 ) · · · → H3k−1 (Σk−1 ; Z2 ) → H3k−1 (Σk ; Z2 ) → H3k−1 (Σk , Σk−1 ; Z2 ) → → H3k−2 (Σk−1 ; Z2 ) → · · · . Since the dimension of Σk−1 is less or equal to 3(k − 1) − 1 < 3k − 2, both the homology groups H3k−1 (Σk−1 ; Z2 ) and H3k−2 (Σk−1 ; Z2 ) vanish, and hence H3k−1 (Σk ; Z2 ) ' H3k−1 (Σk , Σk−1 ; Z2 ) 6= 0. ˆ k be the cone We next introduce a variational scheme for obtaining existence of solutions for (6). Let Σ ˆ over Σk , which can be represented as Σk = Σk × [0, 1] with Σk × {0} identified to a single point. Let first L be so large that Proposition 2.4 applies with L4 , and then let λ be so large that Iρ (ϕλ,σ ) ≤ −L uniformly for σ ∈ Σk (see Proposition 3.1 (ii)). Fixing this value of λ, we define the family of maps n o ˆ k → H 1 (Σ) : $ is continuous and $(· × {1}) = ϕ on Σk . (42) Πλ = $ : Σ λ,· Lemma 3.3. Πλ is non-empty and moreover, letting Πλ = inf

$∈Πλ

sup Iρ ($(m)),

one has

ˆk m∈Σ

L Πλ > − . 2

Proof. To show that Πλ 6= ∅, it suffices to consider the map (43)

$(z, t) = tϕλ,z ,

ˆ k. (z, t) ∈ Σ

which Arguing by contradiction, suppose that Πλ ≤ − L2 . Then there would exist a map $ ∈ Πλ with supm∈Σˆ k Iρ ($(m)) ≤ − 38 L. Since by our choice of L Proposition 2.4 applies with L4 , writing m = (z, t), with z ∈ Σk , the map t 7→ Ψ ◦ $(·, t) realizes a homotopy in Σk between Ψ ◦ ϕλ,z and a constant map. However, this cannot be, as Σk is non-contractible (see Lemma 3.2) and since Ψ ◦ ϕλ,z is homotopic to the identity on Σk , by Proposition 3.1 (ii). Hence we deduce Πλ > − L2 .

A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS

13

By the statement of Lemma 3.3 and standard variational arguments, one can find a Palais-Smale sequence (un )n for Iρ at level Πλ , namely a sequence for which Iρ0 (un ) → 0.

Iρ (un ) → Πλ ;

Unfortunately it is not known whether Palais-Smale sequences converge. To show this property, from the fact that u 7→ e2u is compact from H 1 (Σ) to L1 (Σ), it would be sufficient to show that a Palais-Smale sequence is bounded. This is in fact proven indirectly, following an argument in [52], by slightly modifying the value of the parameter ρ. Fixing ρ ∈ (4kπ, 4(k + 1)π), we choose a small ρ0 > 0, and allow ρ to vary in the interval [ρ − ρ0 , ρ + ρ0 ]. We consider then the functional Iρ for these values of ρ. If ρ0 is sufficiently small, the interval [ρ − ρ0 , ρ + ρ0 ] will be compactly contained in (4kπ, 4(k + 1)π). Following the previous estimates with minor changes, one easily checks that the min-max scheme applies uniformly for ρ ∈ [ρ − ρ0 , ρ + ρ0 ] and for λ sufficiently large. Precisely, given any large L > 0, there exist ρ0 sufficiently small and λ so large that for ρ ∈ [ρ − ρ0 , ρ + ρ0 ] sup Iρ ($(m)) < −2L;

(44)

L sup Iρ ($(m)) > − , 2 ˆk m∈Σ

Πρ := inf

$∈Πλ

ˆk m∈∂ Σ

where Πλ is defined in (42). Moreover, using for example the test map (43), one shows that for ρ0 sufficiently small there exists a large constant L such that Πρ ≤ L

(45)

for every ρ ∈ [ρ − ρ0 , ρ + ρ0 ].

We have the following result, regarding the dependence in ρ of the min-max value Πρ , see [28]. Lemma 3.4. Let λ be so large and ρ0 be so small that (44) holds. Then the function ρ 7→

Πρ ρ

is non-increasing in [ρ − ρ0 , ρ + ρ0 ].

Proof. For ρ0 ≥ ρ, we have Iρ (u) Iρ0 (u) 1 − = ρ ρ0 2 which clearly implies

Πρ ρ

≥

Πρ0 ρ0



1 1 − ρ ρ0

ˆ

|∇g u|2 dVg ≥ 0,

Σ

. This concludes the proof.

From Lemma 3.4 we deduce that the function ρ 7→ the following corollary.

Πρ ρ

is differentiable almost everywhere, and we obtain

Corollary 3.5. Let λ and ρ0 be as in Lemma 3.4, and let Λ ⊂ [ρ − ρ0 , ρ + ρ0 ] be the (dense) set of Π

ρ for which the function ρρ is differentiable. Then for ρ ∈ Λ the functional Iρ possesses a bounded Palais-Smale sequence (ul )l at level Πρ , weakly converging to a solution of (Eρ,0 ). Proof. The existence of a Palais-Smale sequence (ul )l follows from Lemma 3.3, and the boundedness is proved exactly as in [28], Lemma 3.2. From the above result we obtained a sequence ρk → ρ such that (Eρk ,0 ) is solvable. We have next the following result. ˜ Theorem 3.6. (regular case [10], [39], [38]) Let Σ be a compact surface, and let ui solve (Eρi ,0 ) with h ´ 2ui ˜ smooth positive function on Σ, ρ = ρi , ρi → ρ. Suppose also ui satisfies Σ h(x)e dVg = 1 for all i. Then along a subsequence uik one of the following alternative holds: (i): uik is uniformly bounded on Σ;

ANDREA MALCHIODI(1)

14

(ii): maxΣ uik → +∞ and there exists a finite blow-up set S = {q1 , . . . , ql } ∈ Σ such that (a) for any s ∈ {1, . . . , l} there exist xsk → qs such that uik (xsk ) → +∞ and uik → −∞ uniformly on the compact sets of Σ \ S, 2ui Pl ˜ k * 4π s=1 δqs in the sense of measures. In particular one has that (b) ρik ´ ˜h(x)e2uik Σ

h(x)e

dVg

ρ = 4πn for some n ∈ N. We are now in position to state our main existence theorem concerning the regular case. Theorem 3.7. ([30], [29]) Suppose ρ 6= 4kπ for k = 1, 2, . . . . Then problem (6) is solvable for α = 0. Proof of Theorem 3.7 By Corollary 3.5 there exists a sequence ρk → ρ ∈ 6 4πN such that (Eρk ,0 ) admits a solution uk . By Theorem 3.6, since ρ is not a multiple of 4π, uk must then converge to a solution of (6) for α = 0. We also refer to [28], [53] for previous results on surfaces with positive genus. The above method can actually be used to find multiplicity results as well, see [24], [25].

3.2. Existence for the singular case. We consider in this section the singular case of (6). We recall that for α 6= 0 solutions may not exist, so one cannot expect a general existence result as in Theorem 3.7. For example, there is no constant curvature tear-drop, namely a sphere with one singular point. Recall next the definition of the singular values S defined in (11): for ρ 6∈ S, one has a compactness result similar to Theorem 3.6, see [5], [4], [6]. ˜ as in (7), ρ = ρi , ρi → ρ. Suppose Theorem 3.8. Let Σ be a compact surface, and let ui solve (6) with h ´ that Σ fui dVg ≤ C for some fixed C > 0. Then along a subsequence uik one of the following alternatives holds: (i): uik is uniformly bounded from above on Σ;  ´ (ii): maxΣ 2uik − log Σ fuik dVg → +∞ and there exists a finite blow-up set S = {q1 , . . . , ql } ∈ Σ such that (a) for any s ∈ {1, . . . , l} there exist xsk → qs such that uik (xsk ) → +∞ and uik → −∞ uniformly on the compact sets of Σ \ S, Pl (b) ρik f˜uik * s=1 βs δqs in the sense of measures, with βs = 4π for qs 6= {p1 , . . . , pm }, or βs = 4π(1 + αj ) if qs = pj for some j = {1, . . . , m}. In particular one has that X ρ = 4πn + 4π (1 + αj ), j∈J

for some n ∈ N ∪ 0 and J ⊆ {1, . . . , m} (possibly empty) satisfying n + card(J) > 0. Using this result and the general strategy of the previous section (with several non trivial adaptations though, due to the structure of Σρ,α ) one can then prove the following result. m

Theorem 3.9. ([13]) Suppose that α ∈ (−1, 0) and ρ ∈ R>0 \ S are such that the set Σρ,α is not contractible with respect to the topology induced by d(·, ·). Then problem (6) is solvable. We are extending the theorem (under the same assumptions) for general values of the α’s, namely when some (or all) of them are positive (partial progress in this direction has been obtained in [3], [10], [44]). The case with positive α’s is somehow more delicate, as there is no local improvement in the MoserTrudinger constant from Chen-Troyanov’s inequality. As an application of Lemma 2.7 we would like to illustrate the following theorem, for the model problem (30) in the unit ball B of R2 . Theorem 3.10. ([10], [44]) Suppose α > 0, and that ρ ∈ (4π, 4π(1 + α)) \ 4πN. Then problem (30) is solvable.

A VARIATIONAL APPROACH TO LIOUVILLE EQUATIONS

15

Proof. We let k be the unique integer for which ρ ∈ (4kπ, 4(k + 1)π), and we let Fk denote the map in (28), which realizes a homeomorphism between (S 1 )k and Sk , see Proposition 2.8. Choose a non negative cut-off function χ such that ( η ∈ Cc∞ (B); η(x) ≡ 1 in B 43 , Pk and for σ = i=1 ti δθi ∈ (S 1 )k , λ > 0, we define the test function !2 k X λ (46) ϕˆλ,σ (x) = η(x) log ti , xi = (cos θi , sin θi ). 2 1 + λ2 y − 1 xi i=1 2

Reasoning as in [29] (see also [42] for a simpler proof of this estimate) one can obtain the following result with minor modifications of the proof. Lemma 3.11. Let ϕˆλ,σ be defined as in (46). Then, recalling (26), as λ → +∞ one has (47)

d(f˜ϕλ,σ ,B , σ ˜ ) → 0,

σ ˜=

k X

ti δ 21 xi ,

i=1

and B Iρ,α (ϕˆλ,σ ) → −∞

uniformly for σ ∈ (S 1 )k . We next define the variational scheme which will allow us to find existence of solutions. Recalling that Uk denotes the interior of Sk in Ck , consider the family of continuous maps n o Kλ,ρ = h : Uk → H01 (B) : h(y) = ϕˆλ,F −1 (y) for every y ∈ Sk = ∂Uk . k

We define also the min-max value Kλ,ρ = inf

B sup Iρ,α (h(z)).

h∈Kλ,ρ z∈Uk

We have then the following result. Proposition 3.12. Under the assumptions of Theorem 3.10, if λ is sufficiently large then B Kλ,ρ > sup Iρ,α (ϕˆλ,F −1 (y) ). k

y∈Sk

B Moreover Kλ,ρ is a critical value of Iρ,α .

Proof. If C := Cε,α,k is as in Corollary 2.7, we let L = 4C, and choose λ to be so large that B sup Iρ,α (ϕλ,F −1 (y) ) < −L,

y∈Sk

k

which is possible in view of Lemma 3.11. We will show that Kλ,ρ > − L2 . Indeed, assume by contradiction that there exists a continuous h0 such that 1 B (48) h0 ∈ Kλ,ρ and sup Iρ,α (h0 (z)) ≤ − L. 2 z∈Uk Then, by our choice of L, Proposition 2.9 would apply, yielding a continuous map Fλ,ρ : Uk → Sk defined as the composition Fλ,ρ = ψk ◦ h0 . Notice that, since h0 ∈ Kλ,ρ , h0 (·) coincides with ϕλ,F −1 (·) on Sk = ∂Uk , so by (47) we deduce that k

(49)

Fλ,ρ |Sk is homotopic to Id|Sk :

the homotopy is obtained by letting the parameter λ tend to +∞. Since Sk is homeomorphic to S 2k−1 , B it is non contractible, and we obtain a contradiction to (49). This proves Kλ,ρ > supy∈Sk Iρ,α (ϕλ,F −1 (y) ). k

16

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To check that Kλ,ρ is a critical level we proceed as in the regular case, using the monotonicity property from [52] and the compactness theorem 3.8, which can be easily adapted to the case of the Dirichlet problem (30). The conclusion of Theorem 3.10 follows from the previous proposition. Remark 3.13. Using an integration be parts, it is possible to show that problem (30) has no solutions ˜ when ρ ≥ 4π(1 + α) and when h(x) ≡ |x|2α , see [10]. Hence, the upper bound in Theorem 3.10 is sharp. 4. Other results and perspectives We collect here some other results and perspectives concerning non-existence results, non-compactness issues, degree formulas and Liouville systems. We mentioned in the introduction the tear-drop as a non-existence example for (6). In fact, it was shown in [23] that, also with two singularities on S 2 , the only case when one has existence is the american football, namely two antipodal singular points with equal weights. More recently, other non-existence results were obtained in [32], [33] and [47] on S 2 using monodromy methods (see also [41]). Notice that all these results hold true for the sphere. In fact, the introduction of the weighted barycentric sets Σρ,α , see (13), inspired the construction of new counterexamples when they are easily contractible. Recall that Theorem 3.7 requires topological non-triviality of Σρ,α , and hence we are working here in a complementary situation. The following result was proved recently. Theorem 4.1. ([12]) Let (Σ, g) be a compact surface of any genus, let p1 , . . . , pm be points on Σ with αj < 0 for all j, and let ρ 6∈ S. Let p0 ∈ Σ, α0 ∈ (−1, 0), and consider the singular problem (6) with one extra singularity at p0 with weight α0 . Then there exists α0∗ ∈ (−1, 0) such that if α0 < α0∗ the problem has no solutions. The result is proved by contradiction, analysing the behaviour of solutions corresponding to a sequence of weights α0,k → −1 as k → +∞. It is shown that necessarily concentration of volume occurs near p0 . Then a Pohozaev-type identity is used, similar to the argument discussed in Remark 3.13, but localized near the point p0 be carefully estimating the error terms from other regions. We wonder whether it might be possible to extend this latter non-existence result to more general cases, possibly choosing properly the points pi as well. Another approach for the study of (6) relies on computing the Leray-Schauder degree of the equation. In view of the compactness result in Theorem 3.8, the degree is well defined whenever ρ 6∈ S, see (11). For the regular case (when α = 0 S = 4πN) it was shown in [19], [20] that if ρ ∈ (4kπ, 4(k + 1)π) then the degree of (6) is given by the formula (1 − χ(Σ))(2 − χ(Σ)) · · · (k − χ(Σ)) . k! This result was proved via a refined blow-up analysis, computing the difference of degrees when ρ crosses multiples of 4π. The above formula yields existence for all positive genuses and k’s, but not for the sphere, except when k = 1. In [42] a different proof of the degree formula was given using the Poincar´e-Hopf theorem. Indeed it turns out that in the regular case the degree is given by 1 − χ(Σ). For a singular surface as in the introduction, define g(x) := (1 + x + x2 + x3 + · · · )−χ(Σ)+m

m Y

(1 − x1+αj ).

j=1

Expanding the products in powers of x (noticing that (1 + x + x2 + x3 + · · · )−χ(Σ)+m = (1 − x)χ(Σ)−m if −χ(Σ) + m < 0), write g(x) as g(x) = 1 + b1 xn1 + b2 xn2 + · · · + bk xnk + · · · . Pk It was shown in [21] that for ρ ∈ (4πnk , 4πnk+1 ) one has that the degree is given by the formula j=0 bj . It would be interesting to prove whether there is a relation of this formula to the weighted barycentric sets Σρ,α . A formula similar to the one for the regular case has been verified in special cases.

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17

It would be also interesting to understand cases in which ρ belongs to the discrete set S, in which case Theorem 3.8 does not apply. Very little is known in general for this case: we mention the interesting result in [40] that holds for the flat torus when m = 1 and α1 = 1. For this case the authors were able to give necessary and sufficient conditions for concentration, depending on the torus periodicity. We also mention a natural system of Liouville equations, the Toda system, motivated by the study of non-abelian Chern-Simons vortices, as well as from the study of holomorphic curves. The following non homogeneous version with two components has been extensively studied on compact, boundary-less Riemannian surfaces (Σ, g)      u2  −∆u1 = 2ρ1 ´ h1 euu1 ´ h2 eu − 1 − ρ − 1 , 2 1 2 h e dV h e dV   Σ 2 u g   Σ 1 u g (50) 1 2  −∆u2 = 2ρ2 ´ h2 eu − 1 − ρ1 ´ hh11eeu1 dVg − 1 . h2 e 2 dVg Σ

Σ

Here h1 , h2 are smooth positive functions on Σ and ρ1 , ρ2 are real parameters. Problem (50) has variational structure, and the corresponding Euler functional Jρ : H 1 (Σ) × H 1 (Σ) → R has the expression  ˆ ˆ ˆ 2 X ui (51) Jρ (u1 , u2 ) = Q(u1 , u2 ) dVg + ui dVg − log hi e dVg ; ρ = (ρ1 , ρ2 ), ρi Σ

i=1

Σ

Σ

where Q(u1 , u2 ) is the positive-definite quadratic form
1 |∇u1 |2 + |∇u2 |2 + ∇u1 · ∇u2 . 3 About Jρ , a sharp Moser-Trudinger inequality has been found in [36], stating that the energy is bounded from below if and only if both ρ1 , ρ2 are less than 4π. The above system is related to Liouville equations in two different aspects. When one of the ρi ’s is less than 4π, the system behaves like the regular version of (6), and similar existence results can be found, see [35], [43] (see also [51]). There is indeed some relation to the singular case as well, when one component is much more concentrated than the other one: this aspect is reflected in the blow-up analysis result in [35]. Moreover, the structure of the quadratic form Q penalizes vectorial functions for which both gradients point in the same direction: in fact, some new improved inequalities were obtained in [7], [45], [34]. It is now known that solutions always exist when none of the ρi ’s is a multiple of 4π and if Σ has positive genus, or for arbitrary genus if ρ1 is not a multiple of 4π and ρ2 ∈ (4π, 8π). The singular case of (50) (motivated by the study of vortices or ramification points) is also quite interesting, but for the moment almost entirely open, at least from the existence point of view. Some partial results are available in [8]. (52)

Q(u1 , u2 ) =

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