Solutions for Toda systems on Riemann surfaces

Solutions for Toda systems on Riemann surfaces

JIAYULI ANDYUXIANG LI

Dedicated to Professor Ding Weiyue on the occasion of his 60th birthday

Abstract. In this paper we study the solutions of Toda systems on Riemann surface in the critical case, proving a sufficient condition for existence.

Mathematics Subject Classification (2000):35J60 (primary); 58G03, 35J45 (secondary).

1.

Introduction

Let (,g) be a compact Riemann surface with unit area 1. Ding-Jost-Li-Wang [8] studied the differential equation u = 8π −8πhe^u on(,g), the so-called Kazdan-Warner problem [17] related to the Abelian Chern-Simons model (see [3, 4, 6, 5, 7, 1, 2, 9, 10, 11, 23, 13, 14, 24, 21], etc). They pursued a variational approach to the problem, trying to minimize the functional

J(u)=1 2

|∇u|²d V_g+8π

ud V_g−8πlog

he^ud V_g≥C, inH^1,2() (1.1) for some constant C > 0. Because it is the critical case of the Moser-Trudinger inequality (1.1), the analysis is subtle.

LetK denote the Cartan matrix for SU(N +1),i.e.

K =(a_{i j})=









2 −1 0 . . . . 0

−1 2 −1 0 · 0 0 −1 2 −1. . . 0 . . . .

0 . . . . −1 2 −1 0 . . . . 0 −1 2







.

The research was supported by NSFC.

Pervenuto alla Redazione il 27 aprile 2005 e in forma definitiva il 2 novembre 2005.

In this paper we consider the Toda systems on(,g)which are related to the non- Abelian Chern-Simons model [22]:

−u_i = M_i

exp( ^N_j=1a_{i j}u_j)

exp( ^N_j=1a_{i j}u_j)−1

, for 1≤i ≤N.

If M_i <4πJost-Wang [16] proved the existence of solutions and in the case where is a torus, N = 2, max{M₁,M₂} > 4π and min{M₁,M₂} = 4π, Marcello- Margherita [20] proved the same result.

They studied the problem by considering foru₁, . . . ,u_N ∈H^1,2()the func- tional

₍M₁,... ,M_N)(u₁, . . . ,u_N) = 1 2

N i,j=1

a_{i j}(∇u_i∇u_j +2M_iu_j)d V_g

−^N

i=1

M_ilog

exp _N

j=1

a_{i j}u_j

d V_g. (1.2) Jost-Wang [16] proved that the functional has a lower bound if and only if

M_i ≤4π, fori =1,2, . . . ,N.

Marcello-Margherita [20] obtained a non-minimizing critical point of the functional motivated by an earlier paper of Struwe-Tarantello [23]. The idea was later also used by Djadli and Malchiodi [12] to study the existence of conformal metrics with constant Q-curvature. It is clear thatM_i =4π is the critical case of the functional.

Whether it admits minimizer is subtle. In this paper we study this problem. For simplicity, we consider only the case thatN =2, the general case need only more calculations. In our case the functional is

(u₁,u₂)= 1 2

2 i,j=1

a_{i j}(∇u_i∇u_j+8πu_j)d V_g

− 2 i=1

4πlog

exp ₂

j=1

a_{i j}u_j

d V_g,

and the Toda systems is

−u_i =4π



 exp ²_j=1a_{i j}u_j

exp ²_j=1a_{i j}u_j −1



, for 1≤i ≤2, (1.3)

wherea₁₁=a₂₂=2 anda₁₂=a₂₁= −1. The systems is equivalent to

−u_i =4π 2

j=1

a_{i j}

exp(u_j)

exp(u_j)−1

, for 1≤i ≤2, wherea₁₁=a₂₂=2 anda₁₂=a₂₁= −1.

Our main result is as follows:

Main Theorem. Let be a compact Riemann surface with area1. If the Gauss curvature K ofsatisfies that

maxp∈K(p) <2π, (1.4) then(u₁,u₂)has a minimizer.

We consider the sequence of minimizersu = (u₁,u₂) of₍4π−,4π−) for small >0. Thenusatisfies a Toda type systems. Ifuconverges tou⁰ =(u⁰₁,u⁰₂) in H₂ := H^1,2()×H^1,2(), then it is clear that(u⁰)=inf_u∈H2(u),i.e.,u⁰ is a minimizer of. Ifu does not converge in H₂, in this case, we say thatu blows up. Then there are two cases happened according to Jost-Wang’s result. For each case, we derive a delicate lower bound ofwhich is one of the main points in this paper. We apply capacity to calculate the lower bound, so that we need not know details in the neck. Such a trick has been used by the second author of this paper in [18], [19] to prove the existence of extremal functions for the clas- sical Moser-Trudinger inequality on a compact manifold. Another main point of this paper is the delicate constructions of blowing up sequencesφ in both cases, so that (φ) are strictly less than the lower bound derived before, and conse- quently we get a contradiction to the assumption thatublows up, which proves our main theorem.

After submitting the present article we were informed of the existence of the paper [15], where a result similar to our one is established. As a matter of fact, Jost, Lin and Wang consider the more general equation

−u_i =4π²

j=1

a_{i j}

h_jexp(u_j)

h_jexp(u_j)−1

for 1≤i ≤2,

where a₁₁ = a₂₂ = 2 and a₁₂ = a₂₁ = −1, and h₁, h₂ are positive smooth functions on. Our equation corresponds to the case whereh₁ ≡ h₂is constant.

Our approach to the proof is completely different from theirs, and it appears that the more general case considered by them could also be treated with our method. In addition, there are hopes that our techniques could be used to face other non-linear existence problems.

ACKNOWLEDGEMENTS. We thank the referee for his many helpful comments.

2.

Review of known results

For anyu=(u₁,u₂)∈ H^1,2()×H^1,2(), we set (u)= 1

3

|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂+3(4π−)u₁+3(4π−)u₂

d V_g

−(4π −)log

e^u1d V_g−(4π−)log

e^u2d V_g. It is not difficult to check that

₍4π−,4π−)(v)=(u), if we setv1= ^2u1₃^+u2 andv2= ^u1+₃^2u2.

By Jost-Wang’s result ([16] Corollary 4.6), one sees that has a minimizer u of the functional(u),i.e.we can findu ∈ H^1,2()×H^1,2()such that

(u)=inf(u).

Without loss of generality, we may assume that

e^u1d V_g =

e^u2d V_g =1. Then we have the following equations:





−u₁=(8π−2)e^u1−(4π −)e^u2−(4π−)

−u₂=(8π−2)e^u2−(4π −)e^u1−(4π−) Fori =1,2, let

S_i = {x ∈: there is a sequencey →xs.t.u_i(y)→ +∞}.

Jost-Wang [16] (section 5) proved that there will be two possibilities:

Case 1. S₁= {p₁}, andS₂= {p₂}, where p₁ p₂are two different points in. In this case, we set, fori =1,2,

m_i =u_i(x_i)=maxu_i, (r_i)²=e^−mi, u¯_i =

u_id V_g.

Let(i,x = (x¹,x²))be an isothermal coordinate system around p_i (i = 1,2), and we assume the metric to be

g|_i =e^ϕi((d x¹)²+(d x²)²) withϕi(0)=0,i =1,2.

We set, fori = 1,2,_i = {x ∈

R

R

−0(u₁(x₁+r₁x)−m₁) = e^{−ϕ1(x1+r1x)}

(8π −2)e^{u1(x1+r1x)−m1}

−(r₁)²(4π−)e^u2(x1+r1x)−(r₁)²(4π−) , where−0= _∂^∂2²x¹+_∂^∂2²x². Sinceu₂are bounded from above in₁, it follows from the Harnack inequality and the elliptic estimates thatu₁converges inC_loc^k (

R





−w=8πe^w, ∀x ∈

R

R²e^wd x ≤1. Hence, by the result in [7], we know that

w= −2 log(1+π|x|²).

In the same way,u₂(x₂+r₂x)−m₂converges tow. Setting u¯_i =

u_id V_g, we have the following proposition (see Lemma 5.6, and the proof of Theorem 3.1 in [16]).

Proposition 2.1. We haveu¯_j → −∞for j = 1,2. Furthermore, for any q ∈ (1,2), we have

u_j − ¯u_j converges to G_j in H^1,q(), where G₁and G₂satisfy







−G₁ = 8πδp₁−4πδp₂−4π,

−G₂ = 8πδp₂−4πδp₁−4π,

G_jd V_g = 0, for j =1,2 whereδyis the Dirac distribution. Moreover,

u_j − ¯u_j converges to G_j in C_loc² (\{p₁,p₂}).

Remark 2.2. It is easy to see that, in1,

G₁= −4 logr+A₁(p₁)+ f₁, and G₂=2 logr+ A₂(p₁)+g₁ (2.1) where r² = x₁² + x₂², A_i(p₁) (i = 1,2) are constants, and f₁, g₁ are smooth functions which are zero at 0. Similarly, in2, we can write

G₁=2 logr +A₁(p₂)+ f₂, and G₂= −4 logr+A₂(p₂)+g₂ (2.2) where A_i(p₁)(i = 1,2) are constants, and f₂,g₂are smooth functions which are zero at 0.

Case 2. S₁= {p}, andS₂= ∅.

In this case,u₂are bounded from above. Let(;x)be an isothermal coordi- nate system around p, similar to the Case 1, we have

u₁(x₁+r₁x)−m₁→ −2 log(1+π|x|²).

We also have the following proposition (c.f.[16]):

Proposition 2.3. Letu¯₁be the average of u₁. We haveu¯₁ → −∞. Furthermore, for any q∈(1,2), we have

u₁− ¯u₁converges to G₁in H^1,q(), and

u₂converges to G₂in H^1,q(), where G₁and G₂satisfy









−G₁ = 8πδp−4πe^G2−4π,

−G₂ = 8πe^G2−4πδp−4π,

G₁d V_g = 0,

e^G2d V_g=1, sup

x∈G₂<+∞ (2.3) whereδyis the Dirac distribution. Moreover,

u₁− ¯u₁converges to G₁, and u₂converges to G₂in C_loc² (\{p}).

SinceG₂is bounded from above, we can deduce from the equation (2.3) thatG₂= 2 logr+hin, whereh ∈ H_loc^2,q()for anyq >0. Thene^G2 =r²e^h ∈C_loc¹ (), and then0h∈C_loc¹ (). Therefore, by the standard elliptic estimates,G₂−2 logr is smooth in. So, we can write

G₁= −4 logr+A₁(p)+ f, and G₂=2 logr+A₂(p)+g (2.4) wherer²= x₁²+x₂²,A_i(p)(i =1,2) are constants and f,gare smooth functions which are zero at 0.

3.

The lower bound for Case 1

We assume that 1∩2 = ∅, and B_r(p₁) ⊂ 1. We setv₂ = ¹₃(2u₂+u₁)−

1

3(2u¯₂+ ¯u₁). Then, in B_r(p₁), we have







−v₂ =(4π −)e^u2−(4π−)∈L^∞(B_r(p₁)) v₂|_∂B_r(p₁) → 1

3(2G₂+G₁) .

Sov₂C¹ ≤C, whereC is a constant depending only onr.

By a direct calculation one gets 1

3

B_δ(x₁)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g = 1 4

B_δ(xk)(|∇u₁|²+3|∇v₂|²)d V_g

= 1 4

B_δ(x₁)|∇u₁|²d V_g+O(δ²).

Recall thatu₁(x₁+r₁x)−m₁→winC^k(B_L(0)), for anyk, we have 1

4

B_δ(x₁)|∇u₁|²d V_g = 1 4

B_L

|∇w|²d x

+1 4

B_δ(x₁)\B_Lr

1(x₁)|∇u₁|²d V_g+o(1)+O(δ²).

Let

a₁ = inf

∂B_Lr

1(x₁)u₁, b₁ = sup

∂B_δ(x₁)

u₁.

We seta₁−b₁ =m₁− ¯u₁+d₁. It is clear that, for fixedLandδ, d₁ →w(L)− sup

∂B_δ(p1)G₁ as →0. Let f₁ =max{min{u₁,a₁},b₁}. We get

B_δ(x₁)\B_Lr

1(x₁)|∇u₁|²d V_g ≥

B_δ(x₁)\B_Lr

1(x₁)|∇f₁|²d V_g

=

B_δ(x₁)\B_Lr

1(x₁)|∇0f₁|²d x

≥ inf

|∂BLr

1(0)=a₁,|∂Bδ(0)=b₁

B_δ(0)\B_Lr

1(0)|∇0|²d x. Here,|∇0g|²= |_∂^∂_x^g₁|²+ |_∂^∂_x^g₂|². It is well-known that

|∂BLr inf

1=a₁,|∂Bδ=b₁

B_δ\B_Lr 1

|∇0|²d x

is uniquely attained by the functionφwhich satisfies the equation −0φ=0

φ|_∂B_Lr

1 =a₁, φ|_∂B_δ =b₁. Hence,

φ= a₁−b₁

−logLr₁+logδlogr+a₁logδ−b₁logLr_k

−logLr₁+logδ , and then

B_δ(0)\B_Lr

1(0)|∇0φ|²d x = 4π(a₁−b₁)²

−log(Lr₁)²+logδ². Therefore, we have

B_δ(x₁)\B_Lr

1(x₁)|∇u₁|²d V_g≥ 4π(m₁− ¯u₁+d₁)²

−logL²−log(r₁)²+logδ². Recalling that−log(r₁)²=m₁, we get

B_δ(x₁)\B_Lr

1(x₁)|∇u₁|²d V_g ≥4π(m₁− ¯u₁+d₁)² m₁

1−logL²−logδ² m₁

₋₁

≥4π(m₁− ¯u₁+d₁)² m₁

1+logL²−logδ²

m₁ + A

(m₁)²

≥4π(m₁− ¯u₁)²

m₁ +8πd₁

1− u¯₁ m₁

+4π

1− u¯₁ m₁

2

(logL²−logδ²)+ Au¯₁ (m₁)², where AandAare constants which depend only onδandL.

Then we have 1

3

B_δ(x₁)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g ≥ 1 4

B_L

|∇w|²d x

+π(m₁− ¯u₁)²

m₁ +2πd₁

1− u¯₁ m₁

+π

1− u¯₁

m₁ 2

(logL²−logδ²) +A(δ,L)u¯₁

(m₁)² +o(1)+O(δ²).

Similarly, we have 1

3

B_δ(x₂)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g

≥ 1 4

B_L

|∇w|²d x +π(m₂− ¯u₂)²

m₂ +2πd₁

1− u¯₂ m₂

+π

1− u¯₂ m₁

2

(logL²−logδ²)+ Au¯₂

(m₂)² +o(1)+O(δ²).

It follows that 1

3

B_δ(x₁)∪B_δ(x₂)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g+(4π−)u¯₁+(4π−)u¯₂

≥ 1 3

B_δ(x₁)∪B_δ(x₂)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g+4πu¯₁+4πu¯₂

≥ 1 2

B_L|∇w|²d x +

i=1,2

π(m_i + ¯u_i)²

m_i +2πd_i

1− u¯_i m_i

+π

1− u¯₂

m₁ 2

(logL²−logδ²)

+

i=1,2

Au¯_i

(m_i)² +o(1)+O(δ²)

≥ 1 2

B_L

|∇w|²d x

+

i=1,2

πm_i

1+ u¯_i

m_{i 2}

+2πd_i

1− u¯_i m_i

+π

1− u¯₂

m_{1 2}

(logL²−logδ²)

+

i=1,2

Au¯_i

(m_i)² +o(1)+O(δ²).

We now sets_i =1+_m^u¯i

i. Then, for fixedL,δ, we have 1

3

(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g+(4π−)u¯₁+(4π−)u¯₂

≥

i

m_i

s_i+O 1

m_{i 2}

+C. Since(u)≤C, we see that

|s_i| = O 1

m_i

.

Hence for bothi =1,2,s_i→0 as→0. So, 1

3

B_δ(x₁)∪B_δ(x₂)(|∇u₁|²+ |∇u₂|²+∇u₁∇u₂)d V_g+(4π−)u¯₁+(4π−)u¯₂

≥ 1 2

B_L|∇w|²d x+4πd₁+4πd₂+8π(logL²−logδ²)+o(1)+O(δ²)

= 1 2

B_L

|∇w|²d x+8πw(L)+8π(logL²−logδ²)

−4π sup

∂B_δ(p₁)G₁−4π sup

∂B_δ(p₂)G₂+o(1)+O(δ²). (3.1) By a direct calculation, we obtain

B_L|∇w|²d x =16πlog(1+πL²)− 16π²L²

1+πL². (3.2) Moreover, by (2.1) and (2.2), we have

1 3

B_δ^c(x₁)∩B_δ^c(x₂)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g

= 1 3

B_δ^c(x₁)∩B_δ^c(x₂)(|∇G₁|²+ |∇G₂|²+ ∇G₁∇G₂)d V_g+o(1)

= −1 3

i=1,2

∂B_δ(pi)

G₁∂G₁

∂n +G₂∂G₂

∂n + G₁^∂_∂^G_n² +G₂^∂_∂^G_n¹ 2

d S_g+o(1)

+

B_δ(p₁)+B_δ(p₂)2π(G₁+G₂)d V_g+o(1)

= −1 3

i=1,2

_2π

0

G₁∂G₁

∂r +G₂∂G₂

∂r + G₁^∂_∂^G_r² +G₂^∂_∂^G_r¹ 2

r dθ|r=δ+o(1)

+

B_δ(p₁)+B_δ(p₂)2π(G₁+G₂)d V_g+o(1)

= −16πlogδ−2πA₁(p₁)−2πA₂(p₂)+o(1)+O(δlogδ). (3.3) In the end, (3.1), (3.2) and (3.3) imply that

inf0(u)≥ −8πlogπ −8π −2π(A₁(p₁)+A₂(p₂)).

4.

Lower bound for Case 2

In this case we setv₂ = ¹₃(2u₂+u₁)−¹₃(2u¯₂+ ¯u₁). Then we have







−v₂=(4π−)e^u2−(4π −)

v₂ =0

By the standard elliptic estimates,||v2||C¹(M) <C.

Similarly to Case 1, we have 1

3

B_δ(x₁)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g= 1 4

B_δ(x₁)|∇u₁|²d V_g+O(δ²), and

1 4

B_δ(x₁)|∇u₁|²d V_g+(4π −)u¯₁ ≥ 1 4

B_L

|∇w|²d x

+

π(m₁+ ¯u₁)²

m₁ +2πd₁

1− u¯₁ m₁

+π

1− u¯₁

m_{1 2}

(logL²−logδ²)

+ Au¯₁

(m₁)² +o(1)+O(δ²).

By an argument similar to that used in Case 1, we can show that _m^u¯1

1 → −1, hence 1

3

B_δ(x₁)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g+(4π−)u¯₁

≥ 1 4

B_L |∇w|²d x +4πw(L) +4π(logL²−logδ²)−4π sup

∂B_δ(p₁)G₁+o(1)+O(δ²).

Set

G₁= −4 logr+A₁(p)+o(x), G₂=2 logr+A₂(p)+o(x).

Applying (2.4), we get 1

3

B_δ^c(x₁)(|∇u₁|²+ |∇u₂|²+ ∇u₁∇u₂)d V_g

= 1 3

B_δ^c(x₁)

|∇G₁|²+ |∇G₂|²+∇G₁∇G₂+ ∇G₂∇G₁ 2

d V_g+o(1)

=

∂B_δ(p)

G₁∂G₁

∂n +G₂∂G₂

∂n +G₁^∂_∂^G_n² +G₂^∂_∂^G_n¹ 2

+2π

B_δ(p)(G₁+G₂)d V_g−2π

G₂d V_g+o(1)+O(δlogδ)

= −8πlogδ−2πA₁(p₁)−2π

G₂d V_g+o(1)+O(δlogδ).

In the end, we obtain

inf0(u)≥ −4πlogπ−2πA₁(p)+2π

G₂d V_g.

5.

Test functions for Case 1

In this section we will construct a functionφ = (φ1, φ2)∈ H^1,2(M)× H^1,2(M), such that

0(φ) <−8πlogπ−8π−2π(A₂(p₁)+A₁(p₂)), whenever (1.4) holds. So, under assumption (1.4), Case 1 will not occur.

Let(i;(x,y))be an isothermal coordinate system around p_i (i =1,2). We set

r(x,y)=

x²+y², and B_δ = {(x,y):x²+y²< δ²}.

We assume that near p_i (i =1,2), for eachk =1,2,

G_k = a_k(p_i)logr+ A_k(p_i)+λk(p_i)x+µk(p_i)y

+αk(p_i)x²+βk(p_i)y²+γk(p_i)x y+h(x,y)+O(r⁴).

We havea₁(p₁)=a₂(p₂)= −4, anda₁(p₂)=a₂(p₁)=2. Moreover, we assume that

g|_i =e^ϕi(d x²+d y²), and

ϕi =b₁(p_i)x+b₂(p_i)y+c₁(p_i)x²+c₂(p_i)y²+c₁₂(p_i)x y+O(r³).

It is well known that

K(p_i)= −(c₁(p_i)+c₂(p_i)),

|∇u|²d V_g= |∇u|²d xd y,