Variational Analysis of Toda Systems?

Andrea MALCHIODI

(To Ha?m with admiration and gratitude)

1 Introduction

The Toda system consists of coupled Liouville equations of the type

whereis Laplace-Beltramioperator and A=(aij)ijis the Cartan matrix of SU(N+1),

This system arises in the study of self-dual non-abelian Chern-Simons models(see[22,44,45])(also for further details and an up-to-date set of references),and the right-hand side of the equations might contain singular sources corresponding to vortices,namely points where the wave function appearing in the physical model vanishes.The system also has an interest in geometry,as it describes to the Frenet frame of(possibly ramificated)holomorphic curves in CPn(see[8,10,16,26]).

The following non-homogeneous version with two components was extensively studied on compact,boundary-less Riemannian surfaces(Σ,g):

Here h1,h2are smooth positive functions onand ρ1,ρ2are real parameters.Flat tori might model for example periodic physical systems in the plane.

(1.2)has variational structure,and the corresponding Euler functionalhas the expression

where Q(u1,u2)is the positive-definite quadratic form as follows:

It is well-known that H1(Σ)embeds into any Lpspace,and that indeed the embedding can be pushed to exponential class via the Moser-Trudinger inequality.Concerning the functional Jρthe sharp inequality for the Toda system was found in[26].

Theorem 1.1(see[26])For ρ =(ρ1,ρ2)the functionalis bounded from below if and only if both ρ1and ρ2satisfy ρi≤ 4π.

By the latter theorem we have that when boththe functional Jρis coercive,and solutions can be found by minimization via the direct methods of the calculus of variations(see also[28]for the case whenwhen the energy is bounded below but non-coercive).When one of the ρi’s exceeds 4π the energy becomes unbounded from below,and solutions have to be found as saddle points.One result in this direction is as follows.

Theorem 1.2(see[36])Suppose that m is a positive integer,and letbe smooth positive functions.Then for ρ1∈ (4πm,4π(m+1))and for ρ2<4π,(1.2)is solvable.

Remark 1.1The case m=1 in Theorem 1.2 was proved in[25]for surfaces with positive genus.The assumption ρ14πN is due to compactness reasons(see Section 2).

To describe the general ideas beyond the proof of Theorem 1.2,we recall first the mean field or Liouville equation:

whereandis smooth and positive.The interest in(1.5)arises from the abelian version of(1.2)and,in geometry,from the problem of conformally prescribing the Gaussian curvature of a compact surface(see[1]).

One approach to attack the existence problem for(1.5)relies on computing the Leray-Schauder degree of the equation(see[14,29])(more comments on this approach will be given in the next section).Another one exploits the variational structure of the problem,via the Euler functional

The counterpart of Theorem 1.1 in the scalar case is the classical Moser-Trudinger inequalitystands for the average of u onΣ)

which gives coercivity offorρ<4π(see[18,40]for the borderline case).The supercritical caseρ>4π can be treated through improvements of the latter inequality under suitable conditions on the function u(see[13]).Roughly,the improvement states that if the function euspreads into two separate regions ofΣ,then the constant in(1.7)can be nearly halved.In[19],(1.5)was studied forand the improvement in[13]was used to show that ifis large negative,then the probability measurehas to concentrate near a single point of Σ.This fact was used jointly with a variational scheme to prove existence of critical points of saddle point type whenΣhas positive genus(see[43]for a different argument on the flat torus,using the mountain-pass theorem).

This strategy was then pursued in[21](for the prescribed Q-curvature problem in four dimension)and in[20]to treat the caseAn extension of the argument in[13,19],with a more involved topological construction,allowed to show that for low energy the measureconcentrates near at most k points of the surface.This induces to consider the familyof formal sums

called the barycentric sets ofΣof order k.The above set of measures,which is naturally endowed with the weak topology of distributions,does not have a smooth structure for k≥2(whileΣ1is homeomorphic toΣ),that is,it is a stratified set,namely union of open manifolds of different dimensions(see[27]for further characterizations,especially of topological type).The basic property used in[20–21]is thatΣkis non-contractible,which allows to define proper min-max schemes to attack the existence problem.In[35],it was also used to deduce the degree-counting formula from[14]with a different approach.We also mention the role of these sets for the study of the Yamabe equation in Euclidean domains(see[2]).

In[36],it wasshown that this latter approach can be extended to study(1.2)in the situation described in Theorem 1.2.When ρ1∈ (4kπ,4(k+1)π),k ∈ N,and ρ2<4π the Euler-Lagrange energy Jρis virtually(even though not literally)coercive in the second component u2,and the setΣkagain appears in the distributional description of the function eu1when Jρis low enough.Using the non-contractibility ofΣk,one can then prove Theorem 1.2 using variational techniques.

We are interested here in the situation when both the ρi’s exceed the threshold coercivity value 4π.Using improved inequalities in the spirit of[13],it is possible to prove that ifρ1<4(k+1)π,ρ2<4(l+1)π,k,l∈ N,and if Jρ(u1,u2)is sufficiently low,then either eu1is close toΣkor eu2is close toΣlin the distributional sense.This(non-mutually exclusive)alternative can be expressed in term of the topological join ofΣkandΣl.Recall that,given two topological spaces A and B,their join A?B is defined as the family of elements of the form(see[23])

where E is an equivalence relation given by

This construction allows to map low sublevels of JρintoΣk?Σl,with the join parameter s expressing whether distributionally eu1is closer toΣkor whether eu2is closer toΣl.

However,as for the scalar problem(1.5)the above description is somehow optimal,it is no more the case for(1.2).The optimality of the description of low-energy levels ofmeans of theΣk’s has to be understood in the following sense.Whenfor any measure σ ∈ Σkthere exists a test function φλ,σ,depending on a large parameterλ >0 with the following properties:

with the second property holding uniformly in the choice ofσ∈Σk.This means that copies ofΣkcan naturally be embedded into arbitrary low sublevels ofIn this way,using also the previous statements,one finds a way to go back and forth from large-negative sublevels ofto the set of measuresΣk.

Turning to(1.2),things get more complicated,due to the interaction form Q(u1,u2).Looking at its structure,it clearly appears that when the gradients of u1and u2point in the same direction Q gets larger.This means that if u1and u2are peaked near the same point a higher energy isexpected,which putsextra constraintsin thechoiceof thetest functions.In some cases this problem can be overcome:By restricting the location of the peaks of the two components u1,u2to disjoint curvesγ1,γ2? Σ,the following result was obtained in[7].

Theorem 1.3(see[7])Suppose thatfor both i=1,2 and that Σ has positive genus.Then(1.2)is solvable.

The assumption on the genus ofΣis used to construct global maps from Σ intoγi,and then their push-forwardsfrom Σk(resp. Σl)into(γ1)k(resp.(γ2)l)(the barycentric sets of the γi’s),and then reason only in terms of the topological join of(γ1)kand(γ2)l.Some counterpart of this argument for the singular Liouville equation can be found in[3].

In general,one would need to understand in more depth the interaction between the two components when they concentrate near the same point.We will see that one has to take into account not only of the location of the concentration points,but also of the scale of concentration.One can indeed show for example that when both ρi’s belong to(4π,8π),Jρ(u1,u2)low enough implies that either eu1,eu2are concentrated at different points,or that if they are concentrated at the same point but with different scales of concentration(see[37]).

This argument relies on improved Moser-Trudinger inequalities which are different in spirit from those in[13],and which have the feature of being scaling-invariant.They can be applied to functions that are arbitrarily concentrated and that,at a macroscopic level,look just like Dirac masses.A different improved inequality,but with the same scaling-invariant feature,was proved in[24].Werefer to Section 3 for details,just remarking herethat it appliesto caseswhen one component is much more concentrated than the other.This has some relation to improved inequalities for the Singular Liouville equation:A version of(1.5)with singular sources(in the form of Dirac masses)on the right-hand side,representing either conical points on surfaces,or vortex points from the physical point of view.The relation to this latter problem can be seen from the right-hand side of(1.2):A highly concentrated function appears asa Dirac delta when looking at the scale of the less concentrated component.These improved inequalities have the effect of removing suitable subsets in the topological join Σk?Σl(see Sections 3–4 for details).One can then prove the following result via min-max theory.

Theorem 1.4(see[24,37])Let h1,h2be two positive smooth functions,and letΣbe any compact surface.Suppose thatρ1∈ (4kπ,4(k+1)π),k ∈ N and ρ2∈ (4π,8π).Then(1.2)has a solution.

The existence problem for the case of general parameters and genus is still open.We hope that the topological join construction might still play a role.An interesting variant of(1.2)regards the presence of singular sources on the right-hand sides of the equations.For this case the progress is still limited(see[5,7,11]in the scalar case)for some particular situations.

The plan of the paper is the following.In Section 2,some compactness results are presented,showing the role of the multiples of 4π:Applications to min-max constructions are also discussed.In Section 3,some improved inequalities are shown,both at a macroscopic and at a scaling-invariant level.Section 4 treats the weighted barycentric sets and the topological join construction.Finally,in Section 5,some test functions are constructed,which allow to define suitable min-max schemes to prove existence of solutions.

Some NotationsGiven points x,y∈Σ,d(x,y)will stand for the metric distance between x and y on Σ.Similarly,for any,we set

The symbol Bs(p)stands for the open metric ball of radius s and center p,and the complement of a set?inΣwill be denoted by?c.

Given a function u∈L1(Σ)and??Σ,the average of u on? is denoted by the symbol

We denote bythe average of u in Σ.Since we are assuming|Σ|=1,we have

The sub-levels of the functional Jρwill be indicated as

Throughout this paper,the letter C will stand for large constants which are allowed to vary among different formulas or even within the same lines.We denote M(Σ)the set of all Radon measures onΣ,and introduce a norm by using duality versus Lipschitz functions,that is,we set

We denote by d the corresponding distance,which is known as the Kantorovich-Rubinstein distance.

2 Analytic Asp ects of the Problem

In thissection,wecollect someuseful compactnessresultsthat mainly arisefrom a concentration-compactness alternative.We discuss first the scalar case,then the vector case,and finally we turn to applications to min-max theory.

2.1 Compactness p rop erties of(1.5)

In this subsection,we describe the analytic aspects of the problem,especially for what concernsthecompactness properties.The first result in this direction was proved in[9],concerning the scalar Liouville equation in a bounded domain?of the Euclidean plane.For a smooth positive function h(x)and a real-valued sequence of positive numbersρnwe consider the following problem:

Define then the blow-up set S as

Theorem 2.1(see[9])Suppose that??R2is a bounded domain,and consider a sequence of solutions of(2.1).Then,up to a subsequence,one of the following three possibilities holds true:

(i)unis bounded in

(ii)un→?∞ on every compact set of?.

(iii)The blow-up set S of(un)is finite,on the compact sets of?S and moreover

with βi≥2π for every xi∈S.

This result was proved using potential theory jointly with Jensen’s inequality.Later,it was specialized by Li and Shafrir in the following sense.

Theorem 2.2(see[30])If the case(iii)occurs in Theorem 2.1,thenβiis a positive multiple of 4π for every xi∈S.

One of the main extra ingredients in the proof of the latter result is the blow-up analysis of solutions.The un’s are rescaled near its local maxima.Using the dilation invariance of the limit equation

it is proved that in the limit a scaled unsolves(2.2)and has a uniformly bounded global maximum at zero.Solutions of(2.2)with such property were classified in[13].It was proved that,up to a dilation,the limit profile has the form

Corollary 2.1Consider the problem(1.5).Then for any compact set K of R4πN there exists a positive constant CKsuch that all solutions of(1.5)are bounded in C2,α(Σ)by CKwheneverρ∈K.

Remark 2.1(i)When h(x)is strictly positive on a compact manifold and is of class C1,the local accumulation of mass for(1.5)near a blow-up point is exactly 4π.This was proved in[29]using moving-plane arguments,and allowed the author to prove that the Leray-Schauder degree of the equation is well defined for ρ4πN.In[29],it was shown that the degree turns out to be always 1 when ρ <4π.Using a more refined blow-up analysis and a Lyapunov-Schmidt reduction in[14],the Leray-Schauder degree of(1.5)was computed for generalρ’s not belonging to 4πN(see also[35]for a different approach).

(ii)It was proved instead in[15]that in bounded planar domains the mass accumulation can be a multiple of 4π strictly larger than 4π.

2.2 Compactness prop erties of(1.2)

We next turn to(1.2).It turns out that in this case there are different types of blow-ups,but it is still possible to obtain compactness results under rather neat assumptions.We first consider a sequence of solutions to a counterpart of(1.2),where the coefficientsρnare allowed to vary,namely

Define the blow-up set

For a pointwe then define the local limit masses Ai(x)as

Then one has the following result(see also[32]).

Theorem 2.3(see[25])Suppose that x is a blow-up point for(2.4).Then only one among the following five possibilities may occur for(A1(x),A2(x)):

Thereason for the restriction to thesecouplesrelies on the different blow-up ratesof the two components.The first possibility(for the second one just exchange u1and u2)corresponds to the case when the first component u1,nblows-up while the second does not.Then the situation is quite similar to the scalar case described before,with a blow-up profile given by(2.3).

Solutionsof thissingular equation wereclassified in[41],and thesehavethefollowing expression:

whereξ>0 and whereθ0∈ [0,2π).We notice that,compared to(2.3),the solution is not unique,it is not radial and it depends on the angular parameterθ0.Indeed,if one considers the singular equation(2.5)but with a general weight?αin front of the Dirac mass,it turns out that solutions are always radial ifα is not a positive multiple of 4π while,as already noticed in[12],there always exist non-radial solutions in the complementary case.However,as it happens for U0in(2.3),it turns out thatindependently ofThis fact yields the third alternative in Theorem 2.3.

Finally,the fifth alternative occurs when both components blow-up with the same rate.After scaling,the profile of(u1,n,u2,n)is given by the vectorial solution of the entire system

Viewing(2.7)as a structure equation for holomorphic curves in CP2,such solutions were classified in[26].These depend indeed on eight parameters,however it happens that one always has the quantization property

This yields the fifth possibility for the mass accumulation values in Theorem 2.3.

Using Green’s representation formulas,in[6],it was proved that in case of blow-up at least onecomponent uimust accumulateat a finitenumber of points,and thereforethecorresponding limiting parameterρimust be quantized,according to Theorem 2.3.As a consequence one finds the following result.

Theorem 2.4(see[6,25])Consider the problem(1.2).Then for any compact set K of R4πN there exists a positive constant CKsuch that all solutions of(1.5)are bounded in C2,α(Σ)by CKwhenever ρi∈ K.

Remark 2.2In[17],it was proved that there exist blowing-up solutions of(1.2)for which only onecomponent concentratesnear finitely-many points,whiletheother doesnot.Therefore,compactness holds true provided the couple(ρ1,ρ2)stays bounded away from the grid

and not only from the squared lattice of points

As for(1.5),when(ρ1,ρ2)Λ,the Leray-Schauder degree of(1.2)is well defined for(ρ1,ρ2)Λ.Some degree-computations can be found in[31,39](and in[33]for other Liouville systems).

2.3 Ap p lications to min-max theory

We will now show how to apply the previous results to deduce existence of solutions.Let us fix(ρ1,ρ2)Λ,and letbe two compact sets.Letbe a continuous map,and define the following class of continuous maps:

Given this definition,one can introduce the corresponding min-max value

As the Palais-Smale condition is not known yet for the functional Jρ,one has to do some extra work in order to guarantee existence of critical points.We have the following proposition.

Proposition 2.1Suppose that there exists t0>0 and β0>0 such thatand such that

Then Jρa(bǔ)dmits a critical point at level cρ.

The proof of this result relies on both a monotonicity method introduced by Struwe in[42](see also[34]for an alternative approach)and on the previous compactness results.First,one notices that since for t′≥ t

so we clearly have that

This implies that the functionis almost-everywhere differentiable.As in[19],one can prove that for the values of t for whichdifferentiable Jtρhas a bounded Palais-Smale sequence at level αtρ.It can be shown that this sequence then converges to a critical point of Jtρ.We will show it for the functionalcorresponding to the scalar equation(1.5)(withand t close to 1).The vectorial counterpart requires only minor changes.

Consider a Palais-Smale sequence ulforbounded in H1(Σ).The existence of a weak limitfollows from Theorem 2.2,asfor t close to 1.Let us show that u0satisfiesFor any function v∈H1(Σ)there holds

by(1.7),the boundedness ofand the fact thatweakly in

By the above reasoning,we proved that there existsfor which the system

is solvable.Then using Theorem 2.4,which applies when both ρ1,ρ2are not multiples of 4π,one obtains convergence of solutions and arrives to the desired conclusion.

3 Improved Inequalities

In this section,we collect some improved Moser-Trudinger type inequalities.These hold for functions satisfying certain requirements on the distribution of their exponentials.We will divide the discussion between cases in which the functions are macroscopically spread,and others that have scaling-invariant features.

3.1 Improved inequalities via macroscopic spreading

We present now the first improved inequality.Basically,if the mass of both h1eu1andis spread respectively on at least k+1 and l+1 different sets,then the values of the ρi’s for which one has coercivity increase by a factor(k+1)and(l+1)respectively.

We have first a couple of technical lemmas(see[7,Section 4]for details)that are useful for localizing the Moser-Trudinger inequality in Theorem 1.1.

Lemma 3.1Letδ>0 andbe such thatThen,for anyε>0 there exists C=C(ε,δ)such that for any

Lemma 3.2Letδ>0,θ>0,k,l∈N withbe non-negative functions withfor i=1,2 andsuch that

and

Then,there existindependent of fi,andsuch that

and

We then have the following result:It says qualitatively that the more the components(u1,u2)of thesystem are spread overΣ,the more effectively Q(u1,u2)controls theexponential integrals.

Proposition 3.1(see[7])Letδ>0,θ>0,k,l∈ N andbe such that

Then,for any ε >0,there exists C=C(ε,δ,θ,k,l,Σ),such that any×H1(Σ)satisfying

verifies

ProofIn the proof,we assume thatAfter relabelling the indexes,we can suppose k≥l and apply Lemma 3.2 withto getwith

and

Notice that

The average oncan be estimated by Poincaré’s inequality as follows:

For j ∈ {l+1,···,k},we have

The exponential term on the second component can be estimated by using Jensen’s inequality as follows:

Putting together(3.3)and(3.4),we have

Summing over all j ∈ {0,···,k}and taking into account(3.2)and(3.5),we obtain the result,renamingεappropriately.

3.2 Scaling-invariant improved inequalities

We next introduce two new improved inequalities that have the feature of being scalinginvariant.As we already remarked,this means that can be applied to functions which might be indefinitely concentrated near a single point,differently from the previous proposition.

We begin by considering the family of functions

First we have the following result about functions in L1that are sufficiently concentrated near a single point.

Proposition 3.2Fix R>1.Then there existsδ=δ(R)>0 and a continuous map

satisfying the following property:Given f∈A there exists p∈Σsuch that

(a)d(p,β)≤ C′σ for C′=max{3R+1,δ?1diam(Σ)},

(b)there existsτ>0 depending only on R andΣ such that

ProofWe only sketch the main arguments,referring to[38]for full details.Take R0=3R,and letσ:Σ×A→(0,+∞)(well defined and continuous)be such that

We notice thatσsatisfies

In fact,if this were not true,we would havefor someε>0.Also,By(R0σ(y,f))cannot coincide with Σ,sostands for the open annulus centered at y with radii r1,r2)is non-empty and open.This implies that

By interchanging x and y,we also obtain the opposite inequality,which proves(3.7).

Next,setting

we make the following claim.

Claim 3.1If x0∈ Σsatisfies,then σ(x0,f)<3σ(x,f)for any other

To see this,fix x ∈ Σ and ε>0.First,reasoning as above we find thatand similarly that Bx(R0σ(x,f)+ε)cannot becontained inFrom the triangular inequality,one has

so by the arbitrariness ofεwe get thatThe claim follows from the fact that R0>3.

Using a covering argument,one also has that there exists aτ>0(independent of f)such that

Let us now fix x0∈Σsuch thatBy the above claim,for any x ∈one has

Taking a finite covering of the form

(where k can be chosen depending only onΣand R),we find

Considering the continuous function

and givenτas in(3.8),define

Claim 3.1 and(3.8)imply that if x0∈Σmaximizes T(x,f),then x0∈S(f).Hence for any f∈A S(f)is non-empty and open.Moreover,(3.7)implies

EmbeddingΣin R3and identifying it with its image,we define the center of mass

Forδ>0 small,let P be an orthogonal projection from aδ-neighbourhood ofΣ onto the surface,and define

To conclude the proof,we check that ψ(f)=(β(f),σ(f))satisfies the desired condition.If σ(f) ≤ δ,then d(β(f),S(f))<(R0+1)σ(f).Taking p ∈ S(f),recalling that R0=3R and thatσ(f)≤ 3σ(x,f)<3σ(f)for any x ∈ S(f),we then deduce both(a)and(b).

The next result provides a lower bound on the functional in terms of the functionψ.Its proof relies on using Kelvin inversions,which preserve the integral of the quadratic form Q.

Proposition 3.3(see[37])Given any ε>0,there exist R=R(ε)>1 and ψ as in Proposition 3.2 for which,if

then there exists C=C(ε)such that

The previous result states roughly that if the two components have the same scale of concentration and near the same point,then the Moser-Trudinger constant improves.The next proposition applies instead to the case in which one component(u1)is much more concentrated than the other.

and

then

The requirement in(3.12)means qualitatively that eu2is well distributed around the concentration point(with smaller scale)of eu1.The result is inspired from a similar situation regarding the singular Liouville problem,whose relation to(1.2)was discussed in Section 1.In this case,one has the following improved inequality.

Proposition 3.5(see[4])Let p∈Σand r>0,τ0>0.Then,for anyε>0,there exists C=C(ε,r)such that

for every functionsuch that

In[4],the assumption in the last proposition was also expressed in terms of the angular moments of the function d(x,p)2evaround the singular point p.

4 Weighted Barycentric Sets and Topological Join

In this section,we characterize the low-energy levels of Jρin supercritical regimes.We show first that the improved inequality in Proposition 3.1 leads naturally to consider barycentric set ofΣand their topological join.Then,thescaling-invariant improved inequalitiesin Propositions 3.3–3.4 are used to provide further properties of low-energy functions.

4.1 General description of low energy levels

We now state a technical result that gives sufficient conditions to apply Proposition 3.1.Its proof relies on a covering argument.

Lemma 4.1(see[36,20])Let f ∈ L1(Σ)be a non-negative function withand let m∈N be such that there existε>0,s>0 with

Then there existnot depending on f,andsatisfying

Applying this result to both h1(x)eu1and h2(x)eu2(once normalized in L1),jointly with Proposition 3.1,we obtain the following concentration alternative for the exponential functions.

Lemma 4.2Suppose ρ1∈ (4kπ,4(k+1)π)and ρ2∈ (4lπ,4(l+1)π).Then,for any ε>0,s>0,there exists L=L(ε,s)>0 such that for anythere are either somesuch that

or somesuch that

An immediate consequence of the previous lemma is that at least one of the two’s(once normalized in L1)has to be distributionally close respectively to the sets of k-barycenters or l-barycenters ofΣ.

Corollary 4.1Suppose ρ1∈ (4kπ,4(k+1)π)and ρ2∈ (4lπ,4(l+1)π).Then,for any ε>0,there exists L>0 such that anyverifies either

We can now see the role of the topological join of the barycentric sets ofΣ.

Proposition 4.1Supposeand letΦλbe as in(5.2).Then for L sufficiently large,there exists a natural continuous map

from low-energy levels of Jρinto the topological join ofΣkandΣl.

By natural,we mean that we are able to construct a sort of right-inverse of this map(see the next section for details).

Proof of Proposition 4.1It was proved in[20–21]that if m ∈ N,then there exists a retraction ψmfrom a small neighbourhood ofΣm(with respect to the distance d defined after(1.10))ontoΣm.

We then set

and consider a functionwith the expression

where f is such that

The desired map is then defined by

where we are using the notation in(1.9).

4.2 Surfaces with positive genus

In this section,we consider the case of positive genus,where the map from Proposition 4.1 will be specialized.We begin with an easy topological result,whose proof is evident from the picture below.

Lemma 4.3LetΣbe a compact surface with positive genus.Then,there exist two simple closed curvessatisfying

(1) γ1,γ2do not intersect each other,

(2)there exist global retractions

Figure 1 A compact surface with positive genusΣ.

Consider theglobal retractionsandgiven in Lemma 4.3.Acting by push-forward,any probability measure on Σ is sent by(resp.,byinto a probability measure on γ1(resp.,on γ2).In this case,Proposition 4.1 has the following variant,that is quite useful for our purposes.

Proposition 4.2SupposeThen for L sufficiently large,there exists a natural continuous map

from low-energy levels of Jρinto the topological join of(γ1)kand(γ2)l.

Remark 4.1Since eachγiis homeomorphic to S1,it follows from[9,Proposition 3.2]that(γ1)kishomeomorphic to S2k?1and(γ2)lto S2l?1(in[27],it wasproved previously a homotopy equivalence).As it is well-known,the topological join Sm?Snis homeomorphic to Sm+n+1(see,for example,[23]),and thereforeis homeomorphic to the sphere S2k+2l?1.

4.3 Constraints from scaling-invariant inequalities

In this subsection,we make use of the scaling-invariant improved inequalities from the previous section in order to find some constraints on the maps from low-energy levels into the topological join of the barycentric sets.

We first consider the case(ρ1,ρ2) ∈ (4π,8π).We perform a construction similar to(4.3),but taking the scales of concentration of the ui’s(as defined in Proposition 3.2)into account.Notice that the scaleσ is only defined in aδ(R)(the choice of R will be made before Proposition 4.3)neighbourhood(with respect to thedistance d).To extend thisscaleto arbitrary functions,we set

and then

with the convention of choosingwheneveris not well defined.

If f is as in(4.1),we define a modified mapas

By meansof Proposition 3.3,wethen deducethefollowing result,which imposessomeconstraint on the map into the topological join(the numberεin Proposition 3.3 will be taken sufficiently small,and R in Proposition 3.2 taken as R(ε)).

Proposition 4.3Forletbe as in(4.4).Then for L sufficiently largesendswhere

When k>1 in Theorem 1.4,forδ>0 small,we define the set

The counterpart of the above proposition becomes the following one.

Proposition 4.4Letρ1,ρ2be as in Theorem 1.4,with k≥2.Letbe as in(4.5)and letThen,for L>0 large there exists a continuous mapfrom the low sublevelsinto the set Y.

We limit ourselves to give just few ideas beyond this result,referring to[24,Section 3]for the details.Under the assumptions of Theorem 1.4,when k≥1,we have that for low values of Jρ(u)either eu1is concentrated near at most k points ofΣ or eu2is concentrated near a single point.From the construction in the proof of Proposition 4.1,the join parameter is chosen depending on the d-distances of the exponentials from Σkand

In a situation when both components are concentrated,we would also like to take into account the relative scales of the two components,as it was done in(4.4).For u2,which is concentrated near a single point,a natural scale to use is the functionβfrom Proposition 3.2.For u1,which might be concentrated near multiple points(recall that now k≥2),there is a way to localize this quantity near each peak,and to choose the one for the peak closest to that of u2.The latter definition might sound ambiguous because of possible multiple choices,but there is a rigorous way to define a scale of u1near the peak of u2by an averaging process.The choice of the join parameter should then take also into account the ratios of the two scales(absolute for u2and local near the peak of u2for u1).

Now,two competing effects might take place in determining the join parameter.On the one hand,a small local scale of u1relative to that of u2would tend the join parameter to approach 0.On the other hand,having d(eu1,Σk)not that small would make the join parameter approach 1.This is precisely the situation in the assumption of Proposition 3.4.u1has a peak sharper than that of u2(and near the same point),but at the same time starts to split(at a macroscopic level)into k+1 regions(see Lemma 4.1).One can then combine(a localized version of)Proposition 3.4 and Proposition 3.1 to get a lower bound on the energy.

5 Proofs of the Theorems

In this section,we sketch the proofs of Theorems 1.3–1.4.We first construct suitable test functions modelled on(proper subsets of the)the topological joins,and then introduce variational min-max schemes in order to find solutions as saddle points of the Euler-Lagrange energy.

5.1 Test functions

We first consider the case of positive genus.Forρ1∈ (4kπ,4(k+1)π)and ρ2∈ (4lπ,4(l+1)π),we wish to build a family of test functions modelled on the topological jointhe barycentric sets of the curves γ1,γ2(see Lemma 4.3).

Letwhere

Our goal is to define a test function modelled on any ζ∈ (γ1)k?(γ2)l,depending on a positive parameterλ and belonging to low sub-levels of Jρfor largeλ,that is,to find a map

Forλ>0 large and r∈[0,1],we define the parameters

We introduce next Φλ(ζ)= φλ,ζwhose components are defined by

Proposition 5.1Suppose thatand ρ2∈ (4lπ,4(l+1)π)and thatΣ has positive genus.Then one has

Moreover,ifis as in Proposition 4.2,the compositionis homotopic to the identity onfor λ large.

We will not prove this result(referring to[7]for details),but we will limit ourselves to discuss some aspects of the construction and of the estimates.Ifσ1is as above,it turns out that

and similarly forφ2,replacing h1by h2andλ1,rbyλ2,r.By the way eΨis constructed,the latter fact allows to deduce the second statement in Proposition 5.1.

Concerning the estimate ofthe most delicate term to understand in(1.3)is the quadratic one in the gradient.Using direct algebraic inequalities,it is possible to prove thatand that,vice versa

We consider now the assumptions of Theorem 1.4,and for simplicity here we limit ourselves to the case k=1,referring to[24]when k≥2.

When k=1,wewish to parametrizethe test functionson the set(see Proposition 4.3).Indeed,as the latter set is not compact,it is convenient to consider,forν>0 small,a deformation retract ofonto the compact set Xν,corresponding to(Σ?Σ)with a ν-neighbourhood ofremoved.

Forand λ >0 define λ1,r,λ2,ras in(5.1),and then the test functionswhose expression is

By construction,this mapis well defined on Xν,and moreover,when one of the parameters tiis greater thanδthese resemble the previous ones.

We have next the counterpart of Proposition 5.1.

Proposition 5.2Suppose ρ1,ρ2∈ (4π,8π).Then one has

Moreover,ifis as in Proposition 4.3,the compositionis homotopic to the identity on Xνforλlarge.

5.2 The min-max argument

We are now in position to introduce the variational scheme used to prove existence.

Proof of Theorem 1.3By Proposition 5.1,given any L>0,there existsλso large thatfor anyWe choose L so large that Proposition 4.2 applies.We then have that the following composition:

is homotopic to the identity map.In this situation,it is said that the setdominatesSinceis not contractible,this implies that

Moreover,we can takeλlarger so that

Define the topological cone with basisvia the equivalence relation

Notice that,sinceis homeomorphic to a Euclidean ball of dimension 2k+2l.

We now define the min-max value

where

Observe thatbelongs to Γ,so this is a non-empty set.Moreover,

We now show that m≥ ?L.Indeed,?C is contractible in C,and hence inξ(C)for anyξ∈ Γ.Since?C is not contractible inwe conclude thatξ(C)is not contained inBeing this valid for any arbitraryξ∈ Γ,we conclude that m ≥ ?L.The above argument applies when slightly varyingρ,so we can then apply Proposition 2.1.

Proof of Theorem 1.4Weproceed next similarly to thepreviouscase,restricting ourselves to considering k=1.Letdenote the topological cone over Xν,namely

We choose L>0 so large that Proposition 4.3 applies and thenλso large that,by Proposition 5.2,the supremum of Jρon the image of(see the notation after(5.3))is less than?2L.

Consider then the class of maps

Similarly to the previous case,we have that the setΓis non-empty and moreover,letting

one has

Indeed,assuming by contradiction thatthere would beη∈Γsuch thatThen,letting Rνdenote a retraction ofonto Xν,writingthe map

would be a homotopy in Xνbetweenand a constant map.

This fact is indeed impossible since Xνis non-contractible.The proof of this fact is given in the appendix of[38],while here we limit ourselves to describe the case whenΣis a sphere.Indeed in thissituation thesetishomeomorphic to S5,while Xνishomeomorphic to the product S5with a two-dimensional sphere removed.This latter set has a non-vanishing second homology group.Finally,we find thatwhich is the desired conclusion.

As before,the reasoning applies when slightly varyingρ,so we can then apply Proposition 2.1.

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