A Note on a Multiplicity Result for the Mean Field Equation on Compact Surfaces

1 Introduction

In this paper, we consider the mean field equation

where Δ=Δg is the Laplace–Beltrami operator, ρ1,ρ2 are two nonnegative parameters, h:Σ→ℝ is a smooth positive function, and Σ is a compact orientable surface without boundary with Riemannian metric g and with total volume |Σ|. Throughout the paper, we assume that |Σ|=1 for the sake of simplicity.

Equation (1.1) arises in mathematical physics as the mean field equation of equilibrium turbulence with arbitrarily signed vortices and it was first introduced by Joyce and Montgomery [11] and by Pointin and Lundgren [17]. These vortices are composed of positive and negative intensities with the same value, where u and ρ1/ρ2 are associated with the stream function of the fluid and the ratio of the numbers of the signed vortices, respectively. The case ρ1=ρ2 has a close relationship with geometry and is related to the study of constant mean curvature surfaces, see [20].

Problem (1.1) is variational and its solutions correspond to critical points of the Euler–Lagrange functional Jρ:H1⁢(Σ)→ℝ, ρ=(ρ1,ρ2), given by

A fundamental tool in dealing with this kind of functionals is the Moser–Trudinger inequality (1.8) and its version for the two-parameter case (obtained in [16])

where u¯ denotes the average of u. It follows directly that the functional Jρ is bounded from below and is coercive whenever both ρ1 and ρ2 are less than 8⁢π. Therefore, one deduces the existence of a solution by a minimization technique. On the other hand, when one of the ρi exceeds the value 8⁢π, the functional is unbounded from below and the problem becomes more involved.

To describe the features of the problem and the general strategy to attack this kind of equations, it is first convenient to discuss its one-parameter counterpart, namely, the standard Liouville equation

Equation (1.4) concerns the problem of prescribing the Gauss curvature of a surface in conformal geometry. More precisely, letting g~=e2⁢v⁢g, the Laplace–Beltrami operator of the deformed metric is given by Δg~=e-2⁢v⁢Δg and the evolution of the Gauss curvature is given by the equation

- Δ g ⁢ v = K g ~ ⁢ e 2 ⁢ v - K g ,

where Kg and Kg~ are the Gauss curvatures of (Σ,g) and (Σ,g~), respectively.

An important feature of problem (1.4) is the lack of compactness, as its solutions might blow-up. In this case, a quantization phenomenon was proved in [13]. Indeed, a blow-up point x¯ for a sequence (un)n of solutions relatively to (ρn)n, i.e., there exists a sequence xn→x¯ such that un⁢(xn)→+∞ as n→+∞, satisfies

Somehow, each blow-up point has a quantized local mass. Furthermore, the limit profile of solutions is close to a bubble, namely, a function Uλ,x¯ defined as

U λ , x ¯ ⁢ ( y ) = log ⁡ ( 4 ⁢ λ ( 1 + λ ⁢ d ⁢ ( x ¯ , y ) 2 ) 2 ) ,

where y∈Σ, d⁢(x¯,y) stands for the geodesic distance, and λ is a large parameter. In other words, the limit function is the logarithm of the conformal factor of the stereographic projection from S2 onto ℝ2, composed with a dilation.

In the general case when ρ2≠0, namely, for problem (1.1), the refined blow-up analysis is not carried out in full depth. Still, one can show that equation (1.1) inherits some features from the Liouville case. In fact, in [10], the authors proved an analogous quantization property; for a blow-up point x¯ and a sequence (un)n of solutions relatively to (ρ1,n,ρ2,n), one gets

lim r → 0 ⁡ lim n → + ∞ ⁡ ρ 1 , n ⁢ ∫ B r ⁢ ( x ¯ ) h ⁢ e u n ⁢ 𝑑 V g ∫ Σ h ⁢ e u n ⁢ 𝑑 V g ∈ 8 ⁢ π ⁢ ℕ   and   lim r → 0 ⁡ lim n → + ∞ ⁡ ρ 2 , n ⁢ ∫ B r ⁢ ( x ¯ ) h ⁢ e - u n ⁢ 𝑑 V g ∫ Σ h ⁢ e - u n ⁢ 𝑑 V g ∈ 8 ⁢ π ⁢ ℕ .

Now, we let

Combining the local volume quantization with some further analysis, we get that the set of solutions is compact for ρi bounded away from multiples of 8⁢π. This is the main reason why one has to restrict oneself to parameters (ρ1,ρ2)∉Λ. In fact, in order to utilize variational methods, some compactness property is required, usually the Palais–Smale condition. Unfortunately, it is not known whether the latter holds or not for this equation. However, there is a way around this by using the monotonicity argument from [18] along with the compactness result.

We briefly illustrate here the role played by the study of sublevels of the energy functional in the existence issue. Let us first consider the Liouville case (1.4) with associated functional

From the standard Moser–Trudinger inequality

we get boundedness and coercivity provided ρ<8⁢π. For larger values of the parameter, a general existence result was obtained in [5], where an improved Moser–Trudinger inequality is presented. Roughly speaking, the more the function eu is spread over the surface, the better the constant in the inequality and, as a consequence, one gets new lower bounds on the functional (1.7). Basically, if ρ<8⁢(k+1)⁢π, k∈ℕ, and if Iρ⁢(u) is large negative, i.e., lower bounds fail, then eu has to be concentrated around at most k points of Σ. To represent this scenario, it is then natural to consider the family of unit measures Σk which are supported in at most k points of Σ, known as formal barycenters of Σ of order k, i.e.,

The authors indeed proved a homotopy equivalence between the latter set and the low sublevels of Iρ. The existence of solutions follows then from the noncontractibility of Σk and from suitable min-max schemes.

Concerning the general case (1.1), the semicoercive case ρ1∈(8⁢k⁢π,8⁢(k+1)⁢π), k∈ℕ, and ρ2<8⁢π was considered in [21]. The author exploited the condition ρ2<8⁢π to characterize the low sublevels of Jρ by means of the component eu only, which has the same concentration behavior that occurs in the one-parameter case (1.4).

For parameters above the threshold value (8⁢π,8⁢π), the existence problem gets more involved and still has to be examined in depth due to the nontrivial interaction of the two components eu and e-u. It turns out that there is some analogy between this problem and the Toda system of Liouville equations arising from Chern–Simons theory.

The first step was done in [9], where the author derived an existence result for the first nontrivial interval, i.e., (ρ1,ρ2)∈(8⁢π,16⁢π)2. The proof relies on an improved Moser–Trudinger inequality. One can indeed show that when both eu and e-u concentrate around the same point and with the same rate, the constant in the left-hand side of (1.3) can be basically doubled.

The general case with (ρ1,ρ2)∉Λ was then considered in [2] under the assumption that the surface Σ is not homeomorphic to S2. The strategy goes as follows. Exploiting improved Moser–Trudinger inequalities it is possible to show that if ρ1<8⁢(k+1)⁢π and ρ2<8⁢(l+1)⁢π, k,l∈ℕ, then either eu is close to Σk or e-u is close to Σl in the distributional sense. This alternative can be expressed by means of the topological join of Σk and Σl. We recall that, given two sets A and B, the join A*B is defined as the family of elements of the form

where R is an equivalence relation such that

Roughly speaking, the join parameter s expresses which of the above alternatives is more likely to be fulfilled.

To minimize the interaction of the two components eu and e-u, the assumption on Σ is needed. One can indeed construct two disjoint simple noncontractible curves γ1,γ2 such that Σ retracts on each of them through continuous maps Π1,Π2. Taking into account the retractions Πi, starting from Σk*Σl one can restrict himself to targets in (γ1)k*(γ2)l only. The final step is then to gain some nontrivial topological information of the low sublevels of Jρ in terms of (γ1)k*(γ2)l.

The goal of this paper is to present the first multiplicity result for this class of equations.

Here, by generic choice of (g,h) we mean that it can be taken in an open dense subset of ℳ2×C 2⁢(Σ)+, where ℳ2 stands for the space of Riemannian metrics on Σ equipped with the C 2 norm, see Proposition 2.5.

The proof is carried out by means of Morse theory in the spirit of [1], where the problem of prescribing conformal metrics on surfaces with conical singularities and the Toda system are considered, respectively. The argument is based on the analysis developed in [2]. In particular, we will exploit the topological descriptions of the low sublevels of Jρ to get a lower bound on the number of solutions to (1.1). Indeed, it will turn out that the high sublevels of Jρ are contractible, while the low sublevels carry some nontrivial topology. We will finally apply the weak Morse inequalities to deduce the estimate on the number of solutions by means of the latter change of topology. Somehow, one expects that the more involved the topology of the surface Σ, the higher the number of solutions. In fact, we will exploit the genus of Σ to describe the topology of the low sublevels of Jρ by means of some bouquet of circles, see Lemma 3.1 and Proposition 3.4. In this way, we will capture the topological information of Σ and we will provide a better bound on the number of solutions to (1.1).

The paper is organized as follows. In Section 2, we introduce some notation and we collect some known results that we will use in the following. In particular, we first focus on a compactness result of equation (1.1) and we introduce a deformation lemma for the functional Jρ in order to use Morse arguments. The second part is concerned with a classical result in Morse theory, the Morse inequalities. In Section 3, we finally prove our main result, Theorem 1.1.

2 Preliminaries

Here, we give some notation and some known results which we will use throughout the paper.

2.2 A Compactness Result and a Deformation Lemma

We state now a compactness result of the set of solutions of equation (1.1). Recall the definition of the set Λ given in (1.6). As mentioned in Section 1, the blow-up phenomenon yields a quantization property of the local volume and with some standard analysis, see [2], we deduce the following theorem.

Theorem 2.1

Theorem 2.1 ([2, 10])

For (ρ1,ρ2) in a fixed compact set of ℝ2∖Λ, the family of solutions to (1.1) is uniformly bounded in C2,α for some α>0.

We will need the latter compactness property to bypass the Palais–Smale condition since it is not known whether it holds or not for this class of equations. More precisely, one can adapt the strategy in [14], where a deformation lemma for the Liouville equation (1.4) was presented; for our framework, see also [15]. One has the alternative that either there exists a critical point of the functional Jρ inside some interval or there is a deformation retract between the relative sublevels. Recall the notation for the sublevels Jρa given in Section 2.1.

Lemma 2.2

If ρ=(ρ1,ρ2)∉Λ and if a<b∈ℝ are such that Jρ has no critical levels inside the interval [a,b], then Jρa is a deformation retract of Jρb.

Here, by deformation retract of a space X onto some subspace A⊆X we mean a continuous map R:[0,1]×X→X such that R⁢(t,a)=a for all (t,a)∈[0,1]×A and such that the final target of R is contained in A, i.e., R⁢(1,⋅)∈A.

Notice now that by the compactness result of Theorem 2.1 it follows that Jρ has no critical points above some high level b≫0. Therefore, one can obtain a deformation retract of the whole Hilbert space H1⁢(Σ) onto the sublevel Jρb by following a suitable gradient flow; see, for example, [15, Corollary 2.8] (with minor adaptations). Somehow, the absence of critical points of Jρ above the level b prevents us from having obstructions while following the flow, see Figure 1.

Figure 1

The deformation retract onto the sublevel Jρb.

Proposition 2.3

Suppose that ρ=(ρ1,ρ2)∉Λ. Then, there exists b>0 sufficiently large such that the sublevel Jρb is a deformation retract of H1⁢(Σ). In particular, it is contractible.

Our aim will be then to show how rich the topological structure of the very low sublevels of Jρ is and to apply the Morse inequalities of Theorem 2.4 to deduce Theorem 1.1.

3 Proof of the Main Result

We have now all the tools in order to prove Theorem 1.1. Since the high sublevels of Jρ are contractible, see Proposition 2.3, the goal will be to describe the topology of the low sublevels.

This will be done by means of a bouquet of circles and its homology will then give a bound on the number of solutions to (1.1) by Theorem 2.4.

We recall that a bouquet ℬN of N circles (see Figure 2) is defined as ℬN=⋃i=1N𝒮i, where 𝒮i is homeomorphic to S1 and 𝒮i∩𝒮j={c}, where c is called the center of the bouquet. The first simple result we need is the following, see the proof of [1, Proposition 3.1].

Figure 2

The bouquet of g⁢(Σ) circles.

Figure 3

The curves γ1 and γ2.

We will now exploit the analysis developed in [2] to describe the topology of the low sublevels of the functional Jρ. As mentioned in Section 1, by means of improved Moser–Trudinger inequalities one can deduce that if ρ1<8⁢(k+1)⁢π and ρ2<8⁢(l+1)⁢π, then either eu is close to Σk or e-u is close to Σl. This alternative is then expressed using the notion of the topological join of Σk and Σl, see (1.10). Finally, applying the retractions Π1,Π2 introduced in the above lemma, low energy sublevels may be described in terms of (γ1)k*(γ2)l only.

In fact, one can project the low sublevels of Jρ onto the latter set, see the proof of [2, Proposition 4.7] and [2, Section 6]. For ρ1∈(8⁢k⁢π,8⁢(k+1)⁢π), ρ2∈(8⁢l⁢π,8⁢(l+1)⁢π), and for L sufficiently large, there exists a continuous map

Ψ : J ρ - L → ( γ 1 ) k * ( γ 2 ) l .

On the other hand, it is possible to do the converse, mapping (γ1)k*(γ2)l into the low sublevels using suitable test functions, see [2, Proposition 6.3], i.e.,

Φ : ( γ 1 ) k * ( γ 2 ) l → J ρ - L .

The above maps are somehow natural in the description of the low sublevels as we have the following important result, see [2, Proposition 4.7 and Section 6].

By the latter homotopy equivalence we directly deduce that the homology groups of (γ1)k*(γ2)l are mapped injectively into the homology groups of Jρ-L through the map induced by Φ.

As a consequence of the above result, we obtain a bound on the number of solutions to (1.1) by Theorem 2.4. One just has to observe that by Proposition 2.3, taking L≥b, the sublevel JρL is contractible and, therefore, by the long exact sequence of the relative homology it follows that

H q + 1 ⁢ ( J ρ L , J ρ - L ) ≅ H ~ q ⁢ ( J ρ - L ) , q ≥ 0 ,

and

H 0 ⁢ ( J ρ L , J ρ - L ) = 0 ,

where H~q⁢(X) of a topological set X is defined in Section 2.1. Recalling the definition of βq⁢(a,b) introduced in (2.1), (2.2) and the notation of β~q given in Section 2.1, the next result holds true by the above discussion and by taking a=-L in Theorem 2.4.

The next step is then to compute the homology groups of the topological join (γ1)k*(γ2)l. We recall that the two curves γ1 and γ2 were chosen such that they are homeomorphic to respectively two disjoint bouquets, see Lemma 3.1. The homology group of the barycenters over this object was computed in [1, Proposition 3.2].

Finally, it is well known that the homology groups of the topological join of two sets A and B are expressed in terms of the sum of the homology groups of each set, see [8].

We are now able to deduce the main Theorem 1.1. The proof will follow by applying the weak Morse inequality stated in Theorem 2.4 jointly with Proposition 3.4 and Propositions 3.5, 3.6. More precisely, we get

where the last three inequalities follow from Theorem 2.4, Proposition 3.4, and Propositions 3.5, 3.6, respectively, and the proof is concluded.