MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS ANDREA MALCHIODI

To Paul with gratitutde, for so much inspiring mathematics Abstract. This paper surveys some recent results on Toda systems of Liouville equations. These systems model self-dual non abelian Chern-Simons vortices, and arise in the study of holomorphic curves. Suitable min-max schemes are employed, leading to existence of solutions in several situations. We will use in particular properties about concentration of exponential functions in order to describe low-energy levels of the Euler-Lagrange energy.

1. Introduction The Toda system is a coupled system of Liouville equations having the following expression N

X 1 − ∆ui (x) = aij euj (x) , 2 j=1

(1)

x ∈ Σ, i = 1, . . . , N.

Here ∆ = ∆g is Laplace-Beltrami operator and A = (aij )ij the Cartan matrix of the group SU (N + 1), namely   2 −1 0 . . . . . . 0  −1 2 −1 0 . . . 0     0 −1 2 −1 . . . 0  .  A=   ... ... ... ... ... ...   0 . . . . . . −1 2 −1  0 ... ... 0 −1 2 Such system appears in the study of self-dual non-abelian Chern-Simons models, see for example [22, 44, 45], but it also has interest in geometry, as it describes to the Frenet frame of holomorphic curves in CPn , see [9], [11], [16], [26]. We will consider here the following 2 × 2 version on a compact Riemannian surface (Σ, g) with no boundary.      u2  −∆u1 = 2ρ1 ´ h1 euu1 ´ h2 eu − 1 − ρ − 1 , 2 1 2 h e dV h e dV  Σ 1 u g    Σ 2 u g (2) 2 1  −∆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. A flat torus would describe for example a periodic physical pattern in the plane. The system (2) has variational structure, with Euler-Lagrange functional Jρ : H 1 (Σ) × H 1 (Σ) → R given by ˆ  ˆ ˆ 2 X ui (3) Jρ (u1 , u2 ) = Q(u1 , u2 ) dVg + ρi ui dVg − log hi e dVg ; ρ = (ρ1 , ρ2 ). Σ

i=1

Σ

Σ

Here Q(u1 , u2 ) is the positive-definite quadratic form  1 (4) Q(u1 , u2 ) = |∇u1 |2 + |∇u2 |2 + ∇u1 · ∇u2 . 3 One basic tool for analysing the above functional is a vectorial version of the Moser-Trudinger inequality, obtained in [26]. Theorem 1.1. ([26]) For ρ = (ρ1 , ρ2 ) the functional Jρ : H 1 (Σ) × H 1 (Σ) is bounded from below if and only if both ρ1 and ρ2 satisfy ρi ≤ 4π. 2000 Mathematics Subject Classification. 35B33, 35J35, 53A30, 53C21. Key words and phrases. Geometric PDEs, Variational Methods, Min-max Schemes. 1

2

ANDREA MALCHIODI

By the above theorem it turns out that when both ρ1 , ρ2 < 4π the functional Jρ is coercive, and solutions of (2) can be found by direct minimization (see also [28] for the case when maxi ρi = 4π, which is studied by approximation). It is indeed sufficient that one of the ρi ’s exceeds 4π to have that the energy becomes unbounded from below. In this case it is no longer possible to minimize the energy, but one might hope to still obtain solutions as saddle points, possibly using min-max methods. The aim of this paper is to describe some recent techniques for doing this, allowing sometimes the parameters ρi to be arbitrarily large. Min-max methods combine topological arguments and deformation lemmas, and a fundamental tool for applying them is compactness. To describe them for our case, it is convenient to first recall the scalar version of (2), namely   h(x)eu −1 on Σ. (5) −∆u = 2˜ ρ ´ h(x)eu dVg Σ Here ρ˜ ∈ R and h : Σ → R is smooth and positive. Equation (5) arises from the abelian version of (2) and, in geometry, from the problem of conformally prescribing the Gaussian curvature of a compact surface, see [1]. Also the latter equation is variational, with Euler-Lagrange functional functional Iρ˜ : H 1 (Σ) → R, ˆ  ˆ ˆ 1 (6) Iρ˜(u) = |∇g u|2 dVg + 2˜ ρ udVg − log h(x)eu dVg , u ∈ H 1 (Σ). 2 Σ Σ Σ The counterpart of Theorem 1.1 in the scalar case is the classical Moser-Trudinger inequality (u stands for the average of u on Σ) ˆ ˆ 1 (7) log e(u−u) dVg ≤ C + |∇u|2 dVg , 16π Σ Σ which gives coercivity of Iρ˜ for ρ˜ < 4π (see also [18], [40] for the borderline case ρ = 4π). Concerning compactness for (5), a useful result was proved in [10], [30], giving an alternative behavior on solutions. Define then the blow-up set S for a sequence of functions un as S = {x ∈ Ω : ∃xn → x such that un (xn ) − un → +∞} . Theorem 1.2. ([10], [30]) Suppose | log ρn | is uniformly bounded, and consider a sequence of solutions of (5). Then, up to a subsequence, one of the following two possibilities holds true (i) un − un is bounded in C 2,α (Σ); (ii) the blow-up set S of (un ) is finite, and moreover X h(x)eun ρn ´ * βi δxi u n h(x)e dVg Σ x ∈S i

with βi = 4ki π for every xi ∈ S and for some ki ∈ N. The finiteness of S was proved using potential theory jointly with Jensen’s inequality. One of the main extra ingredients relies on the blow-up analysis of solutions, rescaling the un ’s near their local maxima. Using the dilation invariance of the limit equation (8)

−∆u = 2ρ0 eu

in R2 ;

ρ0 = lim ρn , n

it is proved that, in the limit, a scaled un solves (8) and has a uniformly bounded global maximum at zero. Solutions of (8) with such property were classified in [14]: it was proved that, up to a dilation, the limit profile must have the form 1 + Cρ0 , (9) U0 (x) = 2 log 1 + |x|2 ´ for some explicit constant Cρ0 ∈ R. Such functions satisfy ρ0 R2 eU0 dx = 4π independently of ρ0 , and hence one finds multiples of 4π in the second alternative of Theorem 1.2. Some delicate work is then needed to show that no residual volume accumulates near the bubbling points. Using a moving plane technique in [29] it was also shown that ki ≡ 1 provided h is smooth enough. As a consequence of the previous result, one obtains compactness of solutions for almost all values of ρ. Corollary 1.3. Consider problem (5). Then for any compact set K of R \ 4πN there exists a positive constant CK such that all solutions of (5) are bounded in C 2,α (Σ) by CK whenever ρ ∈ K.

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

3

We next consider system (2): in this case there are different types of blow-ups, but it is still possible to obtain compactness results under rather neat assumptions on the ρi ’s. We first consider a sequence of solutions to a counterpart of (2), where the coefficients ρi ’s are allowed to vary, namely      u2,n 1,n  −∆u1,n = 2ρ1,n ´ h1 euu1,n ´ h2 eu − 1 − ρ − 1 , 2,n 2,n dV h e dV h e g g  Σ 1   Σ 2 u  (10) 2,n 1,n h e 1  −∆u2,n = 2ρ2,n ´ h2 euu2,n − 1 − ρ1,n ´ h1 eu1,n dVg − 1 . h2 e dVg Σ

Σ

Define the blow-up set S˜ = {x ∈ Σ : ∃xn → x such that ui,n (xn ) − ui,n → +∞ for some i = 1, 2} . For a point x ∈ S˜ we then define the local limit masses Ai (x) as ˆ hi eui,n Ai (x) = lim lim ρi,n ´ . r→0 n h eui,n dVg Br (x) Σ i Then one has the following result. Theorem 1.4. ([25]) Suppose x is a blow-up point for (10). Then only one among the following five possibilities may occur for (A1 (x), A2 (x)) (4π, 0);

(0, 4π);

(4π, 8π);

(8π, 4π);

(8π, 8π).

The first possibility (for the second one just exchange u1 and u2 ) occurs when the first component u1,n blows-up and the second does not. Then the situation is quite similar to the scalar case described before, with a blow-up profile given by (9). The third possibility (or similarly, the fourth one) occurs when the blow-up rate of the first component is faster than that of the second one. In this case, rescaling u1,n so that its maximal value becomes zero, the role of the second component u2,n will be irrelevant, so the limiting profile of u1,n will still be given by (9). On the other hand, looking atthe equation satisfied by u2,n and rescaling so that u2,n has maximal  height equal to zero, the term ρ1,n

´

Σ

h1 eu1,n h1 eu1,n dVg

− 1 in the right-hand side of the equation will resemble

a Dirac mass, with factor −4π. Therefore the profile of u2,n (after subtracting a suitable logarithmic function) will be given by the solution of the singular problem −∆u = 2ρ2 eu − 4πδ0

(11)

in R2 .

Solutions of (11) were classified in [41], and have the expression (12)

 U (x) = −2 log 1 − 2|x|2 cos 2(θ − θ0 ) tanh ξ + |x|4 + Cζ,ρ2

where ξ > 0 and where θ0 ∈ [0, 2π). We notice that, compared to (9), the solution is not unique, and it is not radial. Indeed, if one considers the singular equation (11) but with a general weight −α in front of the Dirac mass, it turns out that solutions are always radial if α is not a positive multiple of 4π while, as already noticed in [13], there always exist non radial solutions in the complementary case. However, ´ as it happens for U0 in (9), it turns out that ρ2 R2 eU dx = 8π independently of ρ2 > 0. This fact yields the third alternative in Theorem 1.4. 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 solution of the entire system  −∆U1 = 2ρ1 eU1 − ρ2 eU2 ; (13) in R2 . −∆U2 = 2ρ2 eU2 − ρ1 eU1 Solutions of (13) were classified in [26]: their structure is quite involved, as they depend indeed on eight parameters, however one always has the quantization property ˆ ˆ ρ1 eU1 dx = 8π; ρ2 eU2 dx = 8π. R2

R2

This yields the fifth possibility for the mass accumulation values in Theorem 1.4. Using Green’s representation formulas, in [7] it was proved that in case of blow-up at least one component ui must accumulate at a finite number of points, and therefore the corresponding limiting parameter ρi must be quantized, according to Theorem 1.4. As a consequence one finds the following result.

4

ANDREA MALCHIODI

Theorem 1.5. ([25], [7]) Consider the problem (2). Then for any compact set K of R \ 4πN there exists a positive constant CK such that all solutions of (5) are bounded in C 2,α (Σ) by CK whenever ρi ∈ K. With these compactness results at hand, one is able to use variational schemes to find existence of solutions. These rely on analysing the exponentials of the two components of the system, and in particular their distribution over the surface Σ and their relative concentration scales. We will describe the following three results, that are consequences of this analysis. Theorem 1.6. ([36]) Suppose m is a positive integer, and let h1 , h2 : Σ → R be smooth positive functions. Then for ρ1 ∈ (4πm, 4π(m + 1)) and for ρ2 < 4π problem (2) is solvable. Theorem 1.7. ([8]) Suppose ρi 6∈ 4πN for both i = 1, 2 and that Σ has positive genus. Then system (2) is solvable. Theorem 1.8. ([37], [24]) Let h1 , h2 be two positive smooth functions and let Σ be any compact surface. Suppose that ρ1 ∈ (4kπ, 4(k + 1)π), k ∈ N and ρ2 ∈ (4π, 8π). Then problem (2) is solvable. Remark 1.9. (a) The case m = 1 in Theorem 1.6 was proved in [25] for surfaces with positive genus. (b) Another approach to attack the existence problem (for both (5) and (2)) relies on computing the Leray-Schauder degree of the equation, see [29], [15], [31]. A relation between this and our approach will be discussed in the last section. We discuss next the role of improved Moser-Trudinger inequalities for studying (5) when ρ˜ > 4π. Roughly, see [14], the improvement states that if the function eu spreads into two separate regions of Σ, then the constant in (7) can be nearly halved. In [19] equation (5) was studied for ρ˜ ∈ (4π, 8π), and the u improvement in [14] was used to show that if Iρ˜ is large negative, then the probability measure ´ eeu dVg Σ has 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 when Σ has 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 general case ρ˜ ∈ (4kπ, 4(k + 1)π). An extension of the argument in [14] and [19], u with a more involved topological construction, allowed to show that for low energy the measure ´ eeu dVg Σ concentrates near at most k points of the surface. This induces to consider the family Σk of probability measures on Σ (denoted by P(Σ)) Σk = {µ ∈ P(Σ) : card(supp(µ)) ≤ k} .

(14)

The above set of measures, which is naturally endowed with the weak topology of distributions, does not have a smooth structure for k ≥ 2 (while Σ1 is homeomorphic to Σ): it is indeed a stratified set, namely union of open manifolds of different dimensions (see [27] for further characterizations, especially for what concerns the Betti numbers). The basic property used in [21] and [20] is that Σk is non contractible, which allows to define proper min-max schemes to attack the existence problem. In [36] it was shown that this latter approach can be extended to study (2) in the situation described in Theorem 1.6. When ρ1 ∈ (4kmπ, 4(m + 1)π), k ∈ N, and ρ2 < 4π the Euler-Lagrange energy Jρ is virtually coercive in the second component u2 , and the set Σk again appears in the distributional description of the function eu1 when Jρ is low enough. Using the non-contractibility of Σk , one can then prove Theorem 1.6 using variational techniques. When both the ρi ’s exceed the threshold coercivity value 4π, still using improved inequalities in the spirit of [14], it is possible to show that if ρ1 < 4(k + 1)π, ρ2 < 4(l + 1)π, k, l ∈ N, and if Jρ (u1 , u2 ) is sufficiently low, then either eu1 is close to Σk or eu2 is close to Σl in the distributional sense. This (non-mutually exclusive) alternative can be expressed by means of the topological join of Σk and Σl . 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]) {(a, b, s) : a ∈ A, b ∈ B, s ∈ [0, 1]} , E where E is an equivalence relation given by A∗B =

(15)

E

(a1 , b, 1) ∼ (a2 , b, 1) ∀a1 , a2 ∈ A, b ∈ B

and

E

(a, b1 , 0) ∼ (a, b2 , 0) ∀a ∈ A, b1 , b2 ∈ B.

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

5

This construction allows to map low sublevels of Jρ into Σk ∗ Σl , with the join parameter s expressing whether distributionally eu1 is closer to Σk or whether eu2 is closer to Σl . However, while for (5) the description in terms of Σk is nearly optimal, for (2) other improved inequalities play a role. These latter ones have the feature of being scaling invariant, and resemble the quantization values given by Theorem 1.4, depending on different situations. analytically, these are due to the interaction term in the quadratic form Q, see (4), which penalizes simultaneous concentration at the same point. In Theorem 1.7 the strategy is to try to avoid these interactions exploiting the positivity of the genus. In Theorem 1.8 they are analysed in detail, but only for special values of ρ2 . Each of the following sections will be devoted to giving a sketch of the last three theorems. Notation Given points x, y ∈ Σ, d(x, y) will stand for the metric distance between x and y on Σ. Similarly, for any p ∈ Σ, Ω, Ω0 ⊆ Σ, we set: d(Ω, Ω0 ) = inf {d(x, y) : x ∈ Ω, y ∈ Ω0 } .

d(p, Ω) = inf {d(p, x) : x ∈ Ω} ,

The symbol Bs (p) stands for the open metric ball of radius s and centre p, and the complement of a set Ω in Σ will be denoted by Ωc . Given a function u ∈ L1 (Σ) and Ω ⊂ Σ, the average of u on Ω is denoted by the symbol ˆ 1 u dVg . u dVg = |Ω| Ω Ω We denote by u the average of u in Σ: we will assume without loss of generality that |Σ| = 1, so ˆ u= u dVg = u dVg . Σ

Σ

The sub-levels of the functional Jρ will be indicated as  Jρa := u = (u1 , u2 ) ∈ H 1 (Σ) × H 1 (Σ) : J(u1 , u2 ) ≤ a . Throughout the paper the letter C will stand for a large constants that is allowed to vary among different formulas or even within the same lines. We denote P(Σ) the set of probability measures on Σ, and introduce a distance using duality versus Lipschitz functions, that is, we set: ˆ ˆ f dµ1 − f dµ µ1 , µ2 ∈ P(Σ). (16) d(µ1 , µ2 ) = sup 2 ; kf kLip(Σ) ≤1

Σ

Σ

Acknowledgements The author is supported by the PRIN project Variational and perturbative aspects of nonlinear differential problems. The author is also a member of G.N.A.M.P.A., which is part of INdAM.

2. Proof of Theorem 1.6 Theorem 1.6 has a counterpart for (5): in [20] (see also [21]) it was shown that the latter equation is always solvable provided ρ˜ is not an integer multiple of 4π. Apart from the compactness that is deduced from Theorem 1.2, a main ingredient is an improved version of the Moser-Trudinger inequality. This occurs when the exponential of u is sufficiently spread over Σ, see [14]. We present next the counterpart for system (2). 2.1. Improved Moser-Trudinger inequalities. We present now a first inequality that improves Theorem 1.1 under suitable assumptions on u1 and u2 . Basically, if the mass of both h1 eu1 and h2 eu2 is 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 Section 4 in [8] for details) that are useful for localizing the Moser-Trudinger inequality in Theorem 1.1.

6

ANDREA MALCHIODI

  e ⊂ Σ be such that d Ω, ∂ Ω e ≥ δ. Then, for any ε > 0 there exists Lemma 2.1. Let δ > 0 and Ω b Ω C = C(ε, δ) such that for any u = (u1 , u2 ) ∈ H 1 (Σ) × H 1 (Σ) ˆ ˆ ˆ ˆ ffl ffl 1 log eu2 − Ωe u2 dVg dVg ≤ Q(u1 , u2 )dVg + ε eu1 − Ωe u1 dVg dVg + log Q(u1 , u2 )dVg + C. 4π Ω e Ω Ω Σ Lemma 2.2. Let δ > 0, θ > 0, k, l ∈ N with k ≥ l, fi ∈ L1 (Σ) be non-negative functions with kfi kL1 (Σ) = 1 for i = 1, 2 and {Ω1,i , Ω2,j }i∈{0,...,k},j∈{0,...,l} ⊂ Σ such that d(Ω1,i , Ω1,i0 ) ≥ δ

∀ i, i0 ∈ {0, . . . , k} with i 6= i0 ;

d(Ω2,j , Ω2,j 0 ) ≥ δ

∀ j, j 0 ∈ {0, . . . , l} with j 6= j 0 ,

ˆ

and

f1 dVg ≥ θ

∀ i ∈ {0, . . . , k};

f2 dVg ≥ θ

∀ j ∈ {0, . . . , l}.

Ω1,i

ˆ

Ω2,j

Then, there exist δ > 0, θ > 0, independent of fi , and {Ωn }kn=1 ⊂ Σ such that ∀ n, n0 ∈ {0, . . . , k} with n 6= n0

d(Ωn , Ωn0 ) ≥ δ and

∀ n ∈ {0, . . . , k};

|Ωn | ≥ θ

ˆ

f1 dVg ≥ θ

∀ n ∈ {0, . . . , k};

f2 dVg ≥ θ

∀ n ∈ {0, . . . , l}.

ˆΩn Ωn

We then have the following result: it says qualitatively that the more the components (u1 , u2 ) of the system are spread over Σ, the more effectively Q(u1 , u2 ) controls the exponential integrals. Proposition 2.3. ([8]) Let δ > 0, θ > 0, k, l ∈ N and {Ω1,i , Ω2,j }i∈{0,...,k},j∈{0,...,l} ⊂ Σ be such that ∀ i, i0 ∈ {0, . . . , k} with i 6= i0 ;

d(Ω1,i , Ω1,i0 ) ≥ δ

d(Ω2,j , Ω2,j 0 ) ≥ δ ∀ j, j 0 ∈ {0, . . . , l} with j 6= j 0 . Then, for any ε > 0 there exists C = C (ε, δ, θ, k, l, Σ) such that any u = (u1 , u2 ) ∈ H 1 (Σ) × H 1 (Σ) satisfying ˆ ˆ h1 eu1 dVg ≥ θ Ω1,i

h1 eu1 dVg

∀ i ∈ {0, . . . , k};

h2 eu2 dVg

∀ j ∈ {0, . . . , l}

Σ

ˆ

ˆ h2 eu2 dVg ≥ θ

Ω2,j

verifies

Σ

ˆ

ˆ h1 eu1 −u1 dVg + (l + 1) log

(k + 1) log Σ

h2 eu2 −u2 dVg ≤ Σ

1+ε 4π

ˆ Q(u1 , u2 )dVg + C. Σ

Proof. In the proof we assume that u1 = u2 = 0. After relabelling the indexes, we can suppose k ≥ l ui and apply Lemma 2.2 with fi = ´ hhiieeui dVg to get {Ωj }kj=0 ⊂ Σ with Σ

d(Ωi , Ωj ) ≥ δ

∀ i, j ∈ {0, . . . , k} with i 6= j ˆ

ˆ

and

h1 eu1 dVg ≥ θ ˆ

h1 eu1 dVg

∀ i ∈ {0, . . . , k};

h2 eu2 dVg

∀ j ∈ {0, . . . , l}.

ˆ

Ωi

Σ

h2 eu2 dVg ≥ θ Ωj

Σ

ˆ

Notice that:

ˆ ui

log

hi e dVg = Σ

ui dVg + log ej Ω

hi e Σ

ui −

ffl e Ω j

ui dVg

dVg , i = 1, 2.

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

7

e j can be estimated by Poincar´e’s inequality: The average on Ω

(17)

1 ui dVg ≤ ej ej Ω Ω

ˆ

ˆ |ui |dVg ≤ C

Σ

2

ˆ

1/2

|∇ui | dVg

|∇ui |2 dVg , i = 1, 2.

≤C +ε

Σ

Σ

n o e =Ω e j := x ∈ Σ : d(x, Ωj ) < δ : We now apply, for any j ∈ {0, . . . , k} Lemma 2.1 with , Ω = Ωj and Ω 2 for j ∈ {0, . . . , l} we get ˆ log

(18)

h1 e

ffl u1 − Ω e u1 dVg j

ˆ dVg + log

h2 e

u2 −

ffl e Ω j

u2 dVg

dVg

ˆ ˆ ffl ffl 1 u1 − Ω u2 − Ω e u1 dVg e u2 dVg j j + log h1 e dVg + log h2 e dVg θ Ωj Ωj ˆ ˆ ffl ffl u1 − Ω u2 − Ω e u1 dVg e u2 dVg j j C + log e dVg + log e dVg Σ



Σ

2 log



Ωj

1 C+ 4π



ˆ

ˆ Q(u1 , u2 )dVg + ε

ej Ω

Ωj

Q(u1 , u2 )dVg , j = 1, . . . l. Σ

For j ∈ {l + 1, . . . , k} we have ˆ ˆ ffl ffl 1 u1 − Ω u1 − Ω e u1 dVg e u1 dVg j j (19) log h1 e dVg ≤ log + kh1 kL∞ (Σ) + log e dVg θ Σ Ωj ˆ ˆ ˆ ffl 1 u2 − Ω e u2 dVg j dVg + e ≤ C − log Q(u1 , u2 )dVg + ε Q(u1 , u2 )dVg . 4π ej Ωj Ω Σ The exponential term on the second component can be estimated by using Jensen’s inequality: ˆ ffl ffl u2 − Ω u2 − Ω e u2 dVg e u2 dVg j j (20) dVg = log |Ωj | + log e dVg log e Ωj

Ωj



log |Ωj | ≥ −C.

Putting together (20) and (19), we have: ˆ ˆ ˆ ffl 1 u1 − Ω e u1 dVg j (21) log h1 e dVg ≤ Q(u1 , u2 )dVg + ε Q(u1 , u2 )dVg + C, j = l + 1 . . . k. 4π Ω ej Σ Σ Summing over all j ∈ {0, . . . , k} and taking into account (18), (21), we obtain the result, renaming ε appropriately. The above result can be applied to analyze the behaviour of functions whose Euler-Lagrange energy is low enough, see Corollary 2.6. To deduce it, we recall the following technical result from [21] Lemma 2.3. Lemma 2.4. Let f ∈ L1 (Σ) be a non-negative function with kf kL1 (Σ) = 1. We also fix an integer ` and suppose that the following property holds true. There exist ε > 0 and r > 0 such that ˆ f dVg ≤ 1 − ε for all the `-tuples p1 , . . . , p` ∈ Σ. ∪`i=1 Br (pi )

Then there exist ε > 0 and r > 0, depending only on ε, r, ` and Σ (and not on f ), and ` + 1 points p1 , . . . , p`+1 ∈ Σ (which depend on f ) satisfying ˆ ˆ f dVg > ε, . . . , f dVg > ε; B2r (pi ) ∩ B2r (pj ) = ∅ for i 6= j. Br (p1 )

Br (p`+1 )

Next we characterize the functions in H 1 (Σ)×H 1 (Σ) for which the value of Jρ is large negative, assuming ρ2 < 4π.

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ANDREA MALCHIODI

Lemma 2.5. Suppose ρ1 ∈ (4πm, 4π(m + 1)) and that ρ2 < 4π. Then for any ε > 0 and any r > 0 there exists´ a large positive L = L(ε, r) such that for every (u1 , u2 ) ∈ H 1 (Σ) × H 1 (Σ) with Jρ (u) ≤ −L and with Σ eui dVg = 1, i = 1, 2, there exists m points p1,u1 , . . . , pm,u1 ∈ Σ such that ˆ eu1 dVg < ε. (22) Σ\∪m i=1 Br (pi,u1 )

Proof. Suppose by contradiction that the statement is not true. Then we can apply Lemma 2.4 with ` = m + 1 and f = eu1 to obtain δˆ0 , γˆ0 and sets Sˆ1 , . . . Sˆm+1 such that d(Sˆi , Sˆj ) ≥ δˆ0 , ˆ

ˆ u1

ˆi S

i 6= j;

eu1 dVg ,

e dVg > γˆ0

i = 1, . . . , m + 1.

Σ

Now, one can use Proposition 2.3 and ρ1 < 4π(m + 1), ρ2 < 4π to get Jρ (u1 , u2 ) ≥ −C, for some C depending on m, δˆ0 , γˆ0 . This would be a contradiction if L in the statement is chosen large enough. This concludes the proof. As a consequence of Lemma 2.5 we have the following result, regarding the distance of the functions eu1 (suitably normalized) from Σm , see (16). Corollary 2.6. Let ε be a (small) arbitrary positive number,´ and let ρ1 ∈ (4πm, 4π(m + 1)), ρ2 < 4π. Then there exists L > 0 such that, if Jρ (u1 , u2 ) ≤ −L and if Σ eu1 dVg = 1, we have d(eu1 , Σm ) < ε. Proof. We consider ε and r small and positive (to be fixed later), and we let L be the corresponding constant given by Lemma 2.5. We let p1 , . . . , pm denote the corresponding points. Now we define σ ∈ Σm by ˆ m X σ= tj δpj ; where tj = eu1 dVg , Ar,j := Br (pj ) \ ∪j−1 k=1 Br (pk ). Ar,j

j=1

Notice that all the sets Ar,j ’s are disjoint by construction. Now, given ϕ with kϕkLip(Σ) ≤ 1, (using also m (22)) we have that ∪m j=1 Br (pj ) = ∪j=1 Ar,j and that ˆ ˆ eu1 ϕdVg < ε; ϕeu1 dVg − tj ϕ(pj ) ≤ CΣ rkϕkC 1 (Σ) ≤ CΣ r. Σ\∪m Ar,j j=1 Br (pj ) By (16) then it follows that  ˆ  d(eu1 , Σm ) ≤ sup eu1 ϕdVg − hσ, ϕi | kϕkC 1 (Σ) = 1 ≤ ε + mCΣ r. Σ

Now it is sufficient to choose ε and r such that ε + mCΣ r < ε. This concludes the proof. The closeness of eu1 to Σm allows to construct a continuous natual map from low sublevels into Σm , as specified by the following proposition,proved in [20], [21]. Proposition 2.7. Suppose m is a positive integer, and suppose that ρ1 ∈ (4πm, 4π(m + 1)), and that ρ2 < 4π. Then there exists a large L > 0 and a continuous map Ψm from {Jρ ≤ −L} into Σm such that u1,n if ´ h1he1 eu1,n * σ for some σ ∈ Σm , then d(Ψm (un ), σ) → 0. Σ

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

9

2.2. Min-max scheme. The next step consists in mapping Σm into arbitrarilyP negative sublevels Pm of Jρ . m In order to do this, we need some preliminary notation. Given σ ∈ Σm , σ = i=1 ti δxi ( i=1 ti = 1) and λ > 0, we define the function ϕλ,σ : Σ → R by 2  m X λ , (23) ϕλ,σ (y) = log ti 1 + λ2 d2i (y) i=1 where we have set di (y) = d(y, xi ), xi , y ∈ Σ. We point out that, since the distance from a fixed point of Σ is a Lipschitz function, ϕλ,σ (y) is also Lipschitz in y, and hence it belongs to H 1 (Σ). Proposition 2.8. Suppose m is a positive integer, that ρ1 ∈ (4πm, 4π(m + 1)), and that ρ2 < 4π. For λ > 0 and for σ ∈ Σm , we define Φ : Σm → H 1 (Σ) × H 1 (Σ) as   1 (24) (Φ(σ))(·) = (Φ(σ)1 (·), Φ(σ)2 (·)) := ϕλ,σ (·), − ϕλ,σ (·) , 2 where ϕλ,σ is given in (23). Then for L sufficiently large there exists λ > 0 such that (i): Jρ (Φ( σ)) ≤ −L uniformly in σ ∈ Σm ; (ii): Ψm ◦ Φ is homotopic to the identity on Σm , where Ψm is given by Proposition 2.7, and where we assume L to be so large that Ψm is well defined on {Jρ ≤ −L}. We will next prove Theorem 1.6 employing a minimax scheme based on the set Σm , see Lemma 2.9. We then define a modified functional Jtρ1 ,tρ2 for which we can prove existence of solutions in a dense set of the values of t. Following an idea of Struwe, this is done proving the a.e. differentiability of the map t 7→ αtρ , where αtρ is the minimax value for the functional Jtρ1 ,tρ2 given by the scheme. Let Km be the topological cone over Σm , i.e. KΣm = (Σm × [0, 1]) /∼ ,

(25)

where the equivalence relation means that the set Σm × {1} is collapsed to a single point. First, let L be so large that Proposition 2.7 applies with L4 , and choose then Φ such that Proposition 2.8 applies for L. Fixing L and Φ, we define the class of maps  (26) ΠΦ = π : Km → H∗1 (Σ) × H∗1 (Σ) : π is continuous and π|Σm (=∂Km ) = Φ , where H∗1 =



ˆ

 eu dVg = 1 .

u ∈ H 1 (Σ) : Σ

Then we have the following properties. Lemma 2.9. The set ΠΦ is non-empty and moreover, letting αρ = inf

sup Jρ1 ,ρ2 (π(m)),

π∈ΠΦ m∈Km

there holds

L αρ > − . 2

Proof. To prove that ΠΦ 6= ∅, we just notice that the following map ˆ  (27) π(σ, t) = tΦ(σ) − log etΦ(σ) dVg ; σ ∈ Σm , t ∈ [0, 1]

((σ, t) ∈ Km )

Σ

belongs to ΠΦ . Assuming by contradiction that αρ ≤ − L2 , there would exist a map π ∈ ΠΦ with supσ˜ ∈Km II(π(˜ σ )) ≤ − 38 L. Then, since Proposition 2.7 applies with L4 , writing σ ˜ = (σ, t), with σ ∈ Σm , the map t 7→ Ψm ◦ π(·, t) would be an homotopy in Σm between Ψm ◦ Φ and a constant map. But this is impossible since Σm is non-contractible (see [27]) and since Ψm ◦ Φ is homotopic to the identity on Σm , by Proposition 2.8. Therefore we deduce αρ > − L2 .

10

ANDREA MALCHIODI

Proof of Theorem 1.6 concluded. We introduce a variant of the above minimax scheme, following [42] and [19] (see also [34]). For t close to 1, we consider the functional ˆ ˆ ˆ 1X ij Jtρ1 ,tρ2 (u) = a ∇ui · ∇uj dVg + tρ1 u1 dVg + tρ2 u2 dVg 2 i,j Σ Σ Σ ˆ ˆ u1 − tρ1 log h1 e dVg − tρ2 log h2 eu2 dVg . Σ

Σ

Repeating the previous estimates, one easily checks that the above min-max scheme applies uniformly for t ∈ [1 − t0 , 1 + t0 ] with t0 sufficiently small. α Therefore it follows easily that also the function t 7→ ttρ is non-increasing, and hence is almost everywhere differentiable. Using Struwe’s monotonicity argument, see for example [19], one can see that α at the points where ttρ is differentiable Jtρ1 ,tρ2 admits a bounded Palais-Smale sequence at level αtρ , which converges to a critical point of Jtρ1 ,tρ2 . Therefore, since the points with differentiability fill densely the interval [1 − t0 , 1 + t0 ], there exists tk → 1 such that the following system has a solution (u1,k , u2,k )   N X hj euj,k (28) −∆ui,k = −1 , i = 1, 2. tk ρj aij ´ h euj,k dVg Σ j j=1 Now it is sufficient to apply Theorem 1.2 to obtain a limit (u1 , u2 ) which is a solution of (2). This concludes the proof.

3. Proof of Theorem 1.7 3.1. Use of the topological join. First, still as a consequence of Proposition 2.3, we have a counterpart of Lemma 2.5, which gives an alternative for the concentration properties of the functions hi eui . Lemma 3.1. Suppose 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 any u ∈ Jρ−L there are either some {xi }ki=1 ⊂ Σ such that ´ Sk h1 eu1 dVg i=1 Bs (xi ) ´ ≥1−ε h eu1 dVg Σ 1 or some {yj }lj=1 ⊂ Σ such that

´ Sl

´

j=1

Bs (yj )

Σ

h2 eu2 dVg

h2 eu2 dVg

≥ 1 − ε.

An immediate consequence of the previous Lemma is that at least one of the two hi eui ’s (once normalized in L1 ) has to be distributionally close respectively to the sets Σk and Σl . Corollary 3.2. Suppose ρ1 ∈ (4kπ, 4(k + 1)π) and ρ2 ∈ (4lπ, 4(l + 1)π). Then, for any ε > 0, there exists L > 0 such that any u ∈ Jρ−L verifies either     h1 eu1 h2 eu2 ´ d ´ , Σ < ε or d , Σ < ε. k l h eu1 dVg h eu2 dVg Σ 1 Σ 2 We can now see the role of the topological join, see (15). Proposition 3.3. Suppose ρ1 ∈ (4kπ, 4(k + 1)π), ρ2 ∈ (4lπ, 4(l + 1)π) and let Φλ be as in (33). Then for L sufficiently large there exists a natural continuous map Ψ : Jρ−L → Σk ∗ Σl from low-energy levels of Jρ into the topological join of Σk and Σl . By natural we mean that we will be able to construct a sort of right-inverse of this map, see Proposition 3.7. Proof. As for Proposition 2.7, if m ∈ N there exists a retraction ψm from a small neighbourhood of Σm (with respect to the distance d defined after (??)) onto Σm .

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

11

    h2 eu2 h1 eu1 ´ or ψ is well defined (or By Corollary 3.2 we know that either ψk ´ l h eu1 dVg h eu2 dVg Σ 1 Σ 2    h1 eu1 h2 eu2 ´ both), since either d ´ , Σ < ε or d , Σl < ε (or both). k h eu1 dVg h eu2 dVg Σ 1 Σ 2 We then set     h1 eu1 h2 eu2 ´ ´ d1 = d , Σk ; d2 = d , Σl , h eu1 dVg h eu2 dVg Σ 1 Σ 2 and consider the function se = se(d1 , d2 )   d1 (29) se(d1 , d2 ) = f , d1 + d2 where f satisfies   0 2z − f(z) =  1

(30)

1 2

if z ∈ [0, 1/4], if z ∈ (1/4, 3/4), if z ∈ [3/4, 1].

The desired map is then defined by  (31)

Ψ(u1 , u2 ) =

 ψk

h1 eu1 h eu1 dVg Σ 1

´



 , ψl

´

h2 eu2 h eu2 dVg Σ 2



 , se ,

where we are using the notation in (15). We next specialize to the case of surfaces with positive genus, in particular regarding Proposition 3.2. We begin with an easy topological result, whose proof is evident from the picture below. Lemma 3.4. Let Σ be a compact surface with positive genus. Then there exist two simple closed curves γ1 , γ2 ⊆ Σ such that (1) γ1 , γ2 do not intersect each other; (2) there exist global retractions Πi : Σ → γi , i = 1, 2.

Σ γ1

γ2

Figure 1 Consider the global retractions Π1 : Σ → γ1 and Π2 : Σ → γ2 given in Lemma 3.4: acting by push-forward, any probability measure on Σ is sent by (Π1 )∗ (respectively, by (Π2 )∗ ) into a probability measure on γ1 (respectively, on γ2 ). In this case Proposition 3.3 has the following variant. Proposition 3.5. Suppose ρ1 ∈ (4kπ, 4(k + 1)π), ρ2 ∈ (4lπ, 4(l + 1)π). Then for L sufficiently large there exists a natural continuous map ˜ : Jρ−L → (γ1 )k ∗ (γ2 )l Ψ from low-energy levels of Jρ into the topological join of (γ1 )k and (γ2 )l .

Remark 3.6. Since each γi is topologically a circle, it follows from Proposition 3.2 in [10] that (γ1 )k ' (S 1 )k is homeomorphic to S 2k−1 and (γ2 )l to S 2l−1 (in [27] it was proved previously a homotopy equivalence). As it is well-known, the topological join S m ∗ S n is homeomorphic to S m+n+1 (see for example [23]), and therefore (γ1 )k ∗ (γ2 )l is homeomorphic to the sphere S 2k+2l−1 .

12

ANDREA MALCHIODI

3.2. Min-max argument. For ρ1 ∈ (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 join (γ1 )k ∗ (γ2 )l , see Lemma 3.4. Let ζ = (σ1 , σ2 , r) ∈ (γ1 )k ∗ (γ2 )l , where: σ1 :=

k X

ti δxi ∈ (γ1 )k

and

σ2 :=

i=1

l X

sj δyj ∈ (γ2 )l .

j=1

Our goal is to define a test function modelled on any ζ ∈ (γ1 )k ∗ (γ2 )l , depending on a positive parameter λ and belonging to low sub-levels of Jρ for large λ, that is to find a map Φλ : (γ1 )k ∗ (γ2 )l → Jρ−L ;

L  0.

For λ > 0 large and r ∈ [0, 1], we define the parameters λ1,r = (1 − r)λ;

(32)

λ2,r = rλ.

We introduce next Φλ (ζ) = ϕλ,ζ whose components are defined by   2  2  Pk Pl   1 1 1 t − s log log 2 2 i j 2 2 i=1 j=1 ϕ1 (x) 2 1+λ1,r d(x,xi ) 1+λ d(x,yj )   2 2  .   2,r (33) = Pk Pl ϕ2 (x) 1 1 1 − 2 log i=1 ti 1+λ2 d(x,xi )2 + log j=1 sj 1+λ2 d(x,yj )2 1,r

2,r

Pl

Notice that when r = 0 one has also λ2,r = 0, and therefore, as j=1 sj = 1, the second terms in both rows are constant, independent of σ2 ; a similar consideration holds when r = 1. These arguments imply that the function Φλ is indeed well defined on (γ1 )k ∗ (γ2 )l (where equivalence relations are used). We have then the following result. Proposition 3.7. Suppose ρ1 ∈ (4kπ, 4(k + 1)π) and ρ2 ∈ (4lπ, 4(l + 1)π) and that Σ has positive genus. Then one has Jρ (ϕλ,ζ ) → −∞

as λ → +∞

uniformly in ζ ∈ (γ1 )k ∗ (γ2 )l .

˜ is as in Proposition 3.5, the composition ζ 7→ ϕλ,ζ 7→ Ψ(ϕ ˜ λ,ζ ) is homotopic to the identity Moreover, if Ψ on (γ1 )k ∗ (γ2 )l for λ large. We will not prove this result, referring to [8] for details, but we will only sketch some aspects of the construction and of the estimates. If σ1 is as above, it turns out that h1 (x)eϕ1 ´ * σ1 as λ1,r → +∞, h (x)eϕ1 dVg Σ 1 ˜ the latter fact allows and similarly for ϕ2 , replacing h1 by h2 and λ1,r by λ2,r . By the construction of Ψ, to deduce the second statement in Proposition 3.7. Concerning the estimate of Jρ (ϕλ,ζ ), the most delicate term to understand in (3) is the quadratic one in the gradient. Using direct algebraic inequalities it is possible to prove that     2 4 near γ1 ; |∇ϕ1 |(x) . min λ1,r , near γ2 , |∇ϕ1 |(x) . min λ1,r , d1,min (x) d2,min (x) and that, vice versa  |∇ϕ2 |(x) . min λ2,r ,

4 d2,min (x)



 near γ2 ;

|∇ϕ2 |(x) . min λ2,r ,

2 d1,min (x)

 near γ1 .

In these formulas we defined d1,min (x) = min d(x, xi ) and d2,min (x) = min d(x, yj ). Moreover it i=1,...,k

j=1,...,l

turns out that

1 1 ∇ϕ2 ' − ∇ϕ1 near γ1 ; ∇ϕ1 ' − ∇ϕ2 near γ2 . 2 2 ´ These two ingredients allow to estimate the desired quantity Σ Q(ϕ1 , ϕ2 )dVg . We also notice that, near the peak points xi and yj , by the last formula the gradients of ϕ1 , ϕ2 point in opposite directions, and their proportionality turns out to be optimal for keeping Q as small as possible. We are now in position to introduce the variational scheme used to prove existence.

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

13

Proof of Theorem 1.7. By Proposition 3.7, given any L > 0, there exists λ so large that Jρ (ϕλ,ζ ) < −L for any ζ ∈ (γ1 )k ∗(γ2 )l . We choose L so large that Proposition 3.5 applies: we then have that the following composition (γ1 )k ∗ (γ2 )l

Φ

λ −→

˜ Ψ

Jρ−L −→

(γ1 )k ∗ (γ2 )l

is homotopic to the identity map. Since (γ1 )k ∗ (γ2 )l is not contractible, this implies that Φλ ((γ1 )k ∗ (γ2 )l ) is not contractible in Jρ−L . Moreover, we can take λ larger so that Φλ ((γ1 )k ∗ (γ2 )l ) ⊂ Jρ−2L . Define the topological cone with base (γ1 )k ∗ (γ2 )l via the equivalence relation C=

(γ1 )k ∗ (γ2 )l × [0, 1] : (γ1 )k ∗ (γ2 )l × {0}

notice that, since (γ1 )k ∗ (γ2 )l ' S 2k+2l−1 , then C is homeomorphic to a Euclidean ball of dimension 2k + 2l. We now define the min-max value: m = inf max J(ξ(u)), ξ∈Γ u∈C

where (34)

Γ = {ξ : C → H 1 (Σ) × H 1 (Σ) : ξ(ζ) = ϕλ,ζ ∀ ζ ∈ ∂C}.

Observe that tΦλ : C → H 1 (Σ) × H 1 (Σ) belongs to Γ, so this is a non-empty set. Moreover, sup Jρ (ξ(ζ)) =

Jρ (ϕλ,ζ ) ≤ −2L.

sup ζ∈(γ1 )k ∗(γ2 )l

ζ∈∂C

We now show that m ≥ −L. Indeed, ∂C is contractible in C, and hence in ξ(C) for any ξ ∈ Γ. Since ∂C is not contractible in Jρ−L , we conclude that ξ(C) is not contained in Jρ−L . Being this valid for any arbitrary ξ ∈ Γ, we conclude that m ≥ −L. Then one can argue as in the final steps of the proof of Theorem 1.6.

4. Proof of Theorem 1.8 4.1. Scaling-invariant improved inequalities. We next introduce two new improved inequalities that have the feature of being scaling-invariant. As we already remarked, this means that can be applied to functions which might be indefinitely concentrated near a single point, differently from Section 2. We begin by considering the family of normalized functions   ˆ A = f ∈ L1 (Σ) : f > 0 a. e. and f dVg = 1 , Σ

We have the following result about functions that are sufficiently concentrated near a single point. Proposition 4.1. Fix R > 1: then there exists δ = δ(R) > 0 and a continuous map ψ : A ∩ {d(·, Σ1 ) < δ(R)} → Σ × R+ ,

ψ(f ) = (β, σ),

satisfying the following property. Given f ∈ A there exists p ∈ Σ such that a) d(p, β) ≤ C 0 σ for C 0 = max{3R + 1, δ −1 diam(Σ)}; b) there exists τ > 0 depending only on R and Σ such that ˆ ˆ f dVg > τ, f dVg > τ. Bp (σ)

Bp (Rσ)c

Proof. We will only sketch the main arguments, referring to [38] for full details. Take R0 = 3R, and let σ : Σ × A → (0, +∞) (well defined and continuous) be such that ˆ ˆ (35) f dVg = f dVg . Bx (σ(x,f ))

Bx (R0 σ(x,f ))c

14

ANDREA MALCHIODI

We notice that σ satisfies d(x, y) ≤ R0 max{σ(x, f ), σ(y, f )} + min{σ(x, f ), σ(y, f )}.

(36)

In fact, if this were not true we would have Bx (R0 σ(x, f )) ∩ By (σ(y, f ) + ε) = ∅ for some ε > 0. Also, By (R0 σ(y, f )) cannot coincide with Σ, so Ay (σ(y, f ), σ(y, f ) + ε) (Ay (r1 , r2 ) stands for the open annulus centred at y with radii r1 , r2 ) is non-empty and open. This implies that ˆ ˆ ˆ ˆ f dVg = f dVg ≥ f dVg > f dVg . Bx (R0 σ(x,f ))c

Bx (σ(x,f ))

By (σ(y,f )+ε)

By (σ(y,f ))

By interchanging x and y, we also obtain the opposite inequality, which proves (36). Next, setting ˆ T : Σ × A → R, T (x, f ) = f dVg , Bx (σ(x,f ))

we make the following Claim. If x0 ∈ Σ satisfies T (x0 , f ) = maxy∈Σ T (y, f ), then σ(x0 , f ) < 3σ(x, f ) for any other x 6= x0 . To see this, fix x ∈ Σ and ε > 0. First, reasoning as above we find that Bx (R0 σ(x, f ) + ε) ∩ Bx0 (σ(x0 , f )) 6= ∅, and similarly that Bx (R0 σ(x, f ) + ε) cannot be contained in Bx0 (R0 σ(x0 , f )). From the triangular inequality one has 2(R0 σ(x, f ) + ε) > (R0 − 1)σ(x0 , f ), so by the arbitrariness of ε we get that σ(x, f ) ≥ R0 > 3.

R0 −1 2R0 σ(x0 , f ).

The claim follows from the fact that

Using a covering argument, one also has that there exists a τ > 0 (independent of f ) such that (37)

max T (x, f ) > τ > 0 x∈Σ

for all f ∈ A.

Let us now fix x0 ∈ Σ such that T (x0 , f ) = maxx∈Σ T (x, f ). By the above claim, for any x ∈ Ax0 (σ(x0 , f ), Rσ(x0 , f )), one has ˆ ˆ f dVg ≤ f dVg ≤ T (x0 , f ). Bx (σ(x0 ,f )/3)

Bx (σ(x,f ))

Taking a finite covering of the form Ax0 (σ(x0 , f ), Rσ(x0 , f )) ⊂ ∪ki=1 Bxi (σ(x0 , f )/3) (where k can be chosen depending only on Σ and R) we find ˆ ˆ ˆ k ˆ X 1= f dVg ≤ f dVg + f dVg + Σ

Bx0 (Rσ(x0 ,f ))c

Bx0 (σ(x0 ,f ))

i=1

f dVg ≤ (k + 2)T (x0 , f ).

Bxi (σ(x0 ,f )/3)

Considering the continuous function σ : A → R,

σ(f ) = 3 min{σ(x, f ) : x ∈ Σ},

and given τ as in (37), define (38)

S(f ) = {x ∈ Σ : T (x, f ) > τ, σ(x, f ) < σ(f )} .

The claim and (37) imply that if x0 ∈ Σ maximizes T (x, f ), then x0 ∈ S(f ). Hence for any f ∈ A S(f ) is non-empty and open. Moreover, (36) implies diam(S(f )) ≤ (R0 + 1)σ(f ).

(39)

Embedding Σ in R3 and identifying it with its image we define the center of mass ˆ + (T (x, f ) − τ )+ (σ(f ) − σ(x, f )) x dVg Σ ∈ RN . η(f ) = ˆ + + (T (x, f ) − τ ) (σ(f ) − σ(x, f )) dVg Σ

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

15

For δ > 0 small, let P be a orthogonal projection from a δ-neighbourhood of Σ onto the surface, and define β : {f ∈ A : σ(f ) ≤ δ} → Σ, β(f ) = P ◦ η(f ). 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 that σ(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 function ψ in Proposition 4.1: its proof relies on using Kelvin inversions, which preserve the integral of the quadratic form Q, as well as suitable harmonic replacements. Proposition 4.2. ([37]) Given any ε > 0, there exist R = R(ε) > 1 and ψ as in Proposition 4.1 for which, if     eu2 eu1 =ψ ´ u , ψ ´ u e 1 dVg e 2 dVg Σ Σ then there exists C = C(ε) such that  ˆ  ˆ ˆ (1 + ε) Q(u1 , u2 ) dVg ≥ 8π log eu1 −u1 dVg + log eu2 −u2 dVg + C. Σ

Σ

Σ

The previous result states roughly that if the two components have the same scale of concentration near the same point, then the Moser-Trudinger constant from Theorem 1.1 improves by nearly doubling. The next proposition applies instead to the case in which one component (u1 ) is much more concentrated than the other. Proposition 4.3. ([24]) Let  r > 0, γ0 > 0 and τ0 > 0. For any ε > 0 there exists C = C(ε, r, τ0 , γ0 ) such that, if for some σ ∈ 0, Cr2 and z ∈ Σ one has ˆ ˆ eu1 dVg eu2 dVg r Bσ/2 (z) Az (Cσ, C ) ˆ ˆ (40) > γ0 , > γ0 u1 e dVg eu2 dVg Σ

Σ

and

ˆ eu2 dVg

(41)

B

ˆ τ0 d(y,z)

sup

(y)

r y∈Az (Cσ, C )

< 1 − τ0 ,

eu2 dVg r Az (Cσ, C )

then

ˆ

ˆ eu1 −u1 dVg + 8π log

4π log Σ

ˆ eu2 −u2 dVg ≤

Σ

ˆ Q(u1 , u2 ) dVg + ε

Br (z)

Q(u1 , u2 ) dVg + C. Σ

Condition (41) qualitatively means that eu2 is well distributed around the concentration point (with smaller scale) of eu1 . The result is inspired from a similar situation regarding the singular Liouville equation, whose relation to (2) has been discussed in the Introduction: in this case one has the following improved inequality. Proposition 4.4. ([4]) Let p ∈ Σ and let r > 0, τ0 > 0. Then, for any ε > 0 there exists C = C(ε, r) such that ˆ ˆ 1+ε log d(x, p)2 ev dVg ≤ |∇v|2 dVg + C, 32π Br (p) Br (p) for every function v ∈ H01 (Br (p)) such that ˆ d(x, p)2 ev dVg sup

Bτ0 d(y,p) (y)

ˆ

d(x, p)2 ev dVg

y∈Br (p); y6=p Br (p)

< 1 − τ0 .

16

ANDREA MALCHIODI

In [4], the assumption of the last proposition was also expressed in terms of the angular moments of the function d(x, p)2 ev around the singular point p. Remark 4.5. Propositions 4.2 and 4.3 reflect the situation described by Theorem 1.4 about blow-up of solutions. The main point here is that the latter propositions apply to arbitrary functions, that are not satisfying any equation. Anyway, the same constants as in the quantization of volume appear here. We will next make use of the above scaling-invariant improved inequalities in order to find some extra constraints on the maps from low-energy levels of Jρ into the join Σk ∗ Σl , see Proposition 3.5. We first consider the case (ρ1 , ρ2 ) ∈ (4π, 8π): we construct a map similar to (31), but taking the scales of concentration of the ui ’s (as defined in Proposition 4.1) into account. Notice that the scale σ is only defined in a δ(R) (the choice of R will be made before Proposition 4.6) neighbourhood of Σ1 (with respect to the distance d): to extend this scale to arbitrary functions we set   1 σ ˆ1 = inf σ(f ) : d(f, Σ1 ) ≤ δR , 2 and then

 hi eui , σ(ui ) = min σ ˆ1 , σ h eui dVg Σ i   ui with the convention of choosing σ ˆ1 whenever σ ´ hhiieeui dVg is not well defined. Σ ˆ as If s˜ is as in (29), we define a modified map Ψ 

(42)



´

ˆ 1 , u2 ) = (β(u1 ), β(u2 ), s˜(σ(u1 ), σ(u2 ))) . Ψ(u

By means of Proposition 4.2 we then deduce the following result, which imposes some constraint on the map into the topological join Σ1 ∗ Σ1 ' Σ ∗ Σ (the number ε in Proposition 4.2 will be taken sufficiently small, and R in Proposition 4.1 taken as R(ε)). ˆ be as in (42). Then for L sufficiently large Ψ ˆ sends Jρ−L Proposition 4.6. For (ρ1 , ρ2 ) ∈ (4π, 8π), let Ψ ˆ where into (Σ ∗ Σ) \ S,  Sˆ := (x, x, 12 ) : x ∈ Σ ⊆ Σ ∗ Σ. When k > 1 in Theorem 1.8, for δ > 0 small we define the set (43) ( )   k X 1 Sˇ = ν, δz , ∈ Σk ∗ Σ 1 : ν = ti δxi ; d(xi , xj ) ≥ δ ∀i 6= j, δ ≤ ti ≤ 1 − δ ∀i ; z ∈ supp (ν) . 2 i=1 The counterpart of the above proposition becomes the following one Proposition 4.7. Let ρ1 , ρ2 be as in Theorem 1.8, with k ≥ 2. Let Sˇ be as in (43) and let Y = ˇ Then, for L > 0 large there exists a continuous map Ψ ˇ from the low sublevels Jρ−L into the (Σk ∗ Σ1 ) \ S. set Y . We will just sketch the main ideas needed to prove this result, referring to Section 3 in [24] for details. Under the assumptions of Theorem 1.8, when k ≥ 2 we have that for low values of Jρ (u) either eu1 is concentrated near at most k points of Σ or eu2 is concentrated near a single point. From the construction in the proof of Proposition 3.3, the join parameter is chosen depending on the d-distances of the exponentials from Σk and Σ1 ' Σ. 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 (42). For u2 , which is concentrated near a single point, a natural scale to use is the function β from Proposition 4.1. 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 u1 near the peak of u2 by

MIN-MAX SCHEMES FOR SU(3) TODA SYSTEMS

17

an averaging process. The choice of the join parameter should then take also into account the ratios of these two scales (absolute for u2 and local for u1 ). In doing this, two competing effects might take place in determining the join parameter. On one hand, a small local scale of u1 relative to that of u2 would 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 4.3: u1 has 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 2.4). One can then combine (a localized version of) Proposition 4.3 and Proposition 2.3 to get a lower bound ˆ maps a couple (u1 , u2 ) into S, ˇ see (43). on the energy when Ψ 4.2. Min-max argument. We consider now the assumptions of Theorem 1.8, and for simplicity here we limit ourselves to the case k = 1, referring to [24] for when k ≥ 2. ˆ see Proposition 4.6. When k = 1 we wish to parametrize the test functions by the set (Σ ∗ Σ) \ S, Indeed, as the latter set is not compact, it is convenient to consider, for ν > 0 small, a deformation retract of (Σ ∗ Σ) \ Sˆ onto the compact set Xν , corresponding to (Σ ∗ Σ) with a ν-neighbourhood of Sˆ removed. For ζ = (δx1 , δx2 , r) ∈ Xν , and λ > 0 define λ1,r , λ2,r as in (32), and then the test functions ϕˆλ,ζ := (ϕˆ1 , ϕˆ2 ) whose expression is (44)

ϕˆ1 (y) = log

2 1 + λ−2 2,r d(x2 , y)

(1 + λ1,r d(x1

2, , y)2 )

ϕˆ2 (y) = log

1 + λ1,r d(x1 , y)2 (1 + λ2,r d(x2 , y)2 )

2.

ˆ λ := (ϕˆ1 , ϕˆ2 ) is well defined on Xν , and moreover, when one of the parameters By construction, this map Φ ti is greater than δ these test functions resemble the ones constructed in the previous section. We have next the counterpart of Proposition 3.7. Proposition 4.8. Suppose ρ1 , ρ2 ∈ (4π, 8π). Then one has Jρ (ϕλ,ζ ) → −∞

as λ → +∞

uniformly in ζ ∈ Xν .

ˆ is as in Proposition 4.6, the composition ζ 7→ ϕλ,ζ 7→ Ψ(ϕ ˆ λ,ζ ) is homotopic to the identity Moreover, if Ψ on Xν for λ large. Proof of Theorem 1.8 continued. We proceed next similarly to the previous case, restricting ourselves to considering k = 1. Let X ν denote the topological cone over Xν , namely Xν × [0, 1] Xν = . Xν × {1} We choose L > 0 so large that Proposition 4.6 applies and then λ so large that, by Proposition 4.8, the ˆ λ (see the notation after (44)) is less than −2L. supremum of Jρ on the image of Φ Consider then the class of maps  (45) Γ = η : X ν → H 1 (Σ) × H 1 (Σ) : η is continuous and η(· × {0}) = ϕ(ϑ1 ,ϑ2 ) on Xν . Similarly to the previous case we have that the set Γ is non-empty and moreover, letting α = inf

η∈Γ

sup Jρ (η(m)),

one has

α > − 32 L.

m∈X ν

Indeed, assuming by contradiction that α ≤ − 32 L, there would be η ∈ Γ such that supm∈X ν Jρ (η(m)) ≤ − 56 L. Then, letting Rν denote a retraction of (Σ ∗ Σ) \ Sˆ onto Xν , writing m = (ϑ, s) (ϑ ∈ Xν ) the map ˆ ◦ η(·, s) s 7→ Rν ◦ Ψ ˆ ◦ ϕ(ϑ ,ϑ ) and a constant map. would be a homotopy in Xν between Rν ◦ Ψ 1 2 This fact is indeed impossible since Xν is non-contractible: the proof of this fact is given in the appendix of [38], while here we limit ourselves to describe the case when Σ is a sphere. Indeed in this situation the set Σ ∗ Σ ' S 2 ∗ S 2 is homeomorphic to S 5 , see Remark 3.6, while Xν is homeomorphic to S 5 , with a two-dimensional sphere removed. This latter set has a non-vanishing second homology group. Finally, we find that α > − 23 L, which is the desired conclusion. As before, the reasoning applies when slightly varying ρ, so we can then apply the compactness result from Theorem 1.4.

18

ANDREA MALCHIODI

Remark 4.9. In [17] it was proved that there exist blowing-up solutions to (2) for which only one component concentrates near finitely-many points, while the other does not. Therefore, compactness holds true provided the couple (ρ1 , ρ2 ) stays bounded away from the grid Λ := {(ρ1 , ρ2 ) : ρi ∈ 4πN for some i = 1, 2} , and not only from the squared lattice of points {(ρ1 , ρ2 ) : ρi ∈ 4πN for both i = 1, 2} . When (ρ1 , ρ2 ) 6∈ Λ, the Leray-Schauder degree of (2) is well defined for (ρ1 , ρ2 ) 6∈ Λ. Some degreecomputations can be found in [39], [31] (and in [33] for other Liouville systems). We speculate that there might be a relation to these degree formulas with the set Y in Proposition 4.7, as it was proved for equation (5) in [35] concerning the sets Σk and the formulas derived in [15]. 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 (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], [6], [8] (and [3], [4], [12], [37] in the scalar case) for some particular situations. References [1] T. Aubin, Some Nonlinear Problems in Differential Geometry, Springer, (1998). [2] A. Bahri, and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294. [3] D. Bartolucci, F. De Marchis and A. Malchiodi, Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not. (2011), Vol. 2011(24), 5625-5643. [4] D. Bartolucci and A. Malchiodi, An improved geometric inequality via vanishing moments, with applications to singular Liouville equations, Comm. Math. Phys., 322 (2013), no. 2, 415-452. [5] L.Battaglia, Existence and multiplicity result for the singular Toda system, J. Math. Anal. Appl. 424 (2015), no. 1, 49-85. [6] L. Battaglia and A. Malchiodi, Existence and non-existence results for the SU(3) singular Toda system on compact surfaces. J. Funct. Anal. 270 (2016), no. 10, 3750-3807. [7] L.Battaglia and G.Mancini, A note on compactness properties of the singular Toda system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 3, 299307. [8] L. Battaglia, A. Jevnikar, A. Malchiodi, D. Ruiz, A general existence result for the Toda System on compact surfaces, Adv. in Math. 285 (2015), 937979. [9] J. Bolton and L. M. Woodward, Some geometrical aspects of the 2-dimensional Toda equations, In Geometry, topology and physics (Campinas, 1996), 69-81. de Gruyter, Berlin, 1997. [10] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of −∆u = V (x)eu in two dimensions Commun. Partial Differ. Equations 16-8/9 (1991), 1223-1253. [11] E. Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58, 1-23, 1953. [12] A. Carlotto and A. Malchiodi, Weighted barycentric sets and singular Liouville equations on compact surfaces. J. Funct. Anal. 262 (2012), no. 2, 409-450. [13] S. Chanillo and M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys. 160 (1994), 217-238. [14] W. X. Chen and C. Li, Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal. 1-4, (1991), 359-372. [15] C.C Chen and C.S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math. 56-12 (2003), 1667-1727. [16] S.S. Chern and J. G. Wolfson, Harmonic maps of the two-sphere into a complex Grassmann manifold. II, Ann. of Math. (2) 125, no. 2, 301-335, 1987. [17] T. DAprile, A. Pistoia and D. Ruiz, Asymmetric blow-up for the SU (3) Toda System, preprint. [18] W. Ding, J. Jost, J. Li and G. Wang, The differential equation ∆u = 8π − 8πheu on a compact Riemann surface, Asian J. Math. 1 (1997), 230-248. [19] W. Ding, J. Jost, J. Li and G. Wang, Existence results for mean field equations, Ann. Inst. Henri Poincar´ e, Anal. Non Lin` eaire 16-5 (1999), 653-666. [20] Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genus, Comm. Contemp. Math. 10 (2008), no. 2, 205-220. [21] Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Annals of Math., 168 (2008), no. 3, 813-858. [22] G. Dunne, Self-dual Chern-Simons Theories, Lecture Notes in Physics, vol. 36, Berlin: Springer-Verlag, 1995. [23] A. Hatcher, Algebraic Topology, Cambridge University Press 2002. [24] A.Jevnikar, S.Kallel and A.Malchiodi, A topological join construction and the Toda system on compact surfaces of arbitrary genus, Anal. PDEs, 8 (2015), no. 8, 1963-2027.

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[25] J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system II. Bubbling behavior and existence of solutions, Comm. Pure Appl. Math. 59 (2006), 526-558. [26] J. Jost and G. Wang, Analytic aspects of the Toda system I. A Moser-Trudinger inequality, Comm. Pure Appl. Math. 54 (2001), 1289-1319. [27] S. Kallel and R. Karoui, Symmetric joins and weighted barycenters, Advanced Nonlinear Studies 11, 117-143, 2011. [28] J. Li and Y. Li, Solutions for Toda systems on Riemann surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 4, 703-728. [29] Y.Y. Li, Harnack type inequality: The method of moving planes, Commun. Math. Phys. 200-2, (1999), 421-444. [30] Y.Y. Li and I. Shafrir, Blow-up analysis for solutions of −∆u = V eu in dimension two, Indiana Univ. Math. J. 43-4, 1255-1270 (1994). [31] C.S. Lin, J. Wei and W. Yang, Degree counting and shadow system for SU (3) Toda system: one bubbling, preprint, 2014, arXiv http://arxiv.org/abs/1408.5802. [32] C.S. Lin, J. Wei and C. Zhao, Sharp estimates for fully bubbling solutions of a SU(3) Toda system. Geom. Funct. Anal. 22 (2012), no. 6, 1591-1635. [33] C.S. Lin and L. Zhang, A Topological Degree Counting for Some Liouville Systems of Mean Field Type, Comm. Pure Appl. Math. 64 (2011), 556-590. [34] M. Lucia, A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal. 30 (2007), no. 1, 113-138. [35] A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Diff. Eq., 13 (2008), 1109-1129. [36] A. Malchiodi and C. B. Ndiaye, Some existence results for the Toda system on closed surfaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007), no. 4, 391-412. [37] A. Malchiodi and D. Ruiz, New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, Geom. Funct. Anal. 21 (2011), no. 5, 1196-1217. [38] A. Malchiodi and D. Ruiz, A variational Analysis of the Toda System on Compact Surfaces, Comm. Pure Appl. Math., 66 (2013), no. 3, 332-371. [39] A. Malchiodi and D. Ruiz, On the Leray-Schauder degree of the Toda system on compact surfaces, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2985-2990. [40] M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Ration. Mech. Anal., 145 (1998), 161195. [41] J. Prajapat and G. Tarantello, On a class of elliptic problems in R2 : Symmetry and Uniqueness results, Proc. Roy. Soc. Edinburgh 131A (2001), 967-985. [42] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985) 558-581. [43] M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8)-1, (1998), 109-121. [44] G. Tarantello, Self-Dual Gauge Field Vortices: An Analytical Approach, PNLDE 72, Birkh¨ auser Boston, Inc., Boston, MA, 2007. [45] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag, 2001. Scuola Normale Superiore, Piazza dei Cavalieri 7, 50126 Pisa (Italy) E-mail address: [email protected]

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