SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS MARCELLO LUCIA1 AND MARGHERITA NOLASCO2
Abstract. Motivated by the study of the asymptotic properties of “nontopological” condensates in the nonabelian Chern-Simons vortex theory (see [25]), we analyze the SU (3) Toda system:
P 8 > <−∆zj = λj R e P k z k z > :R z = 0 jM=e 1, 2; M j 2 −1 2 i=1
(P )λ1 ,λ2
2 i=1
where M = R2 /Z2 , K = (kij ) =
−1
2
!
ij i
−1
on M
ij i
is the SU (3) Cartan matrix and
λj are positive parameters. We study the variational problem associated to the system (P )λ1 ,λ2 in a range of parameters where the trivial solution is a strict local minimum and the corresponding Sobolev-type inequality fails to apply. In this situation, a lack of compactness may occur due to concentration phenomena. Nonetheless, we are able to establish the existence of a nontrivial solution for (P )λ1 ,λ2 which is not a minimizer.
1. Introduction In recent years the Chern-Simons vortex theory has been extensively studied, in view of its possible application to the physics of high critical temperature superconductivity (see e.g. [16] and [17]). In particular, the existence and multiplicity of different type of vortices (e.g. topological, non-topological, periodically constrained, etc.) have been established in the abelian theory (see [29], [28], [4], [33], [26], [35] and [5]. More recently, in [36], [25] and [34], some progress has been made in the study of vortices for the nonabelian self-dual Chern-Simons theory proposed by Dunne in [10], [12] (see also [19], [22] and [21]). For the SU (3) Chern-Simons self-dual theory, the existence of “non-topological” periodic vortices was established in [25]. There, it was proved that the asymptotic behavior of those vortices (as the Chern-Simons coupling constant goes to zero) can be described in terms of solutions of the following system of Liouville type in the two dimensional (flat) torus M = R2 /Z2 (see Proposition 5.5 in [25]):
(1.1)
−∆w1 =
8π 3 (2N1
h1 (x)ew1 4π h (x)ew2 4πN1 R2 + N2 ) R − (2N + N ) − 2 1 w w 1 2 3 |M | h e h e M 1 M 2
h2 (x)ew2 4π h1 (x)ew1 4πN2 8π R R (2N + N ) −∆w = − (2N + N ) − 2 1 1 2 2 3 w w 2 1 3 |M | h e h e M 2 M 1 R w = 0, i = 1, 2; M i
where N1 , N2 are the vortex numbers and h1 , h2 are given bounded nonnegative functions depending on the prescribed vortex points and on the geometry of M . 1 Supported by TMR network FMRX-CT98-0201. 2 Supported by M.U.R.S.T. 40 % Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”.
1
2
M. LUCIA AND M. NOLASCO
In this paper, we study (1.1) in the particular case h1 (x) = h2 (x) = 1. Namely, setting λi = 4π 3 (2Ni + Nj ) (i 6= j), we consider the system wi 1 −∆w = P2 k λ R e − on M j i=1 ij i |M | e wi (P )λ1 ,λ2 M R w = 0, j = 1, 2. M j 2 −1 where K = (kij ) = is the Cartan matrix of the SU (3) Lie group. −1 2 P2 In fact, letting wj = i=1 kij zi (j = 1, 2), we may reduce to analyze ! 2 kij zi 1 −∆z = λ R e i=1 on M − j j 2 |M | (1.2) e i=1 kij zi M R z = 0, j = 1, 2. M j
P P
We refer to system (P )λ1 ,λ2 (or (1.2)) as the two-dimensional SU (3)-Toda system with periodic boundary conditions. The system (1.2) is a particular case of a Liouville type system (see e.g. [8]). For results concerning Liouville-type systems we refer to [6] and [8], where the authors consider the corresponding Dirichlet problem and matrices K with nonnegative entries. The two dimensional Toda system has interest also in connection with some problems in differential geometry (see [15] and reference therein) and in statistical physics (see e.g. [14] and reference therein). In particular, for M = R2 , it provides an example of two-dimensional completely integrable system (see e.g. [20] and [24]). Recently, in [18], some analytic aspects of the system (1.2) have been analyzed in connection with a Moser-Trudinger type inequality. Indeed, in [18], the authors study the minimization problem for the associated action functional and prove the following result: Theorem 1 ([18]): Let M be a closed surface with |M | = 1, λi ≥ 0, i = 1, 2 and (kij ) the 2 × 2 Cartan matrix of the SU (3) Lie group. The functional Z 2 Z 2 2 X X 1 X Iλ1 ,λ2 (z1 , z2 ) = kij (∇zi ∇zj + 2λi zj ) − λi ln exp( kij zj ) 2 i,j=1 M M i=1 j=1 admits a lower bound in H 1 (M ) × H 1 (M ) if and only if (1.3)
max{λ1 , λ2 } ≤ 4π.
In particular, if max{λ1 , λ2 } < 4π, then the functional Iλ1 ,λ2 has a minimizer which satisfies (1.2). 3 Note that, for 2λ1 = λ and λ2 = 0, the result stated in Theorem 1 reduces to the well known Moser-Trudinger inequality which, for functions w ∈ H 1 (M ) with mean value zero, takes the following form: Z Z 1 w (1.4) e ≤ C(M ) exp[ |∇w|2 ] 16π M M with C(M ) a positive constant depending only on M (see e.g. [13]). The aim of this paper is to study the SU(3) Toda system (P )λ1 ,λ2 in the range of parameters where the associate action functional Iλ1 ,λ2 is unbounded from below (namely for max{λ1 , λ2 } > 4π). In this situation, we look for solutions of (P )λ1 ,λ2 as the critical points of the action functional Iλ1 ,λ2 which are not minimizers. In particular, we restrict our attention to the set of parameters (λ1 , λ2 ) for which the
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
3
trivial zero solution for (P )λ1 ,λ2 corresponds to a strict local minimum for Iλ1 ,λ2 . Namely, we require that the matrix 1 λ1 0 (1.5) Hλ1 λ2 = K −1 − 2 0 λ2 4π (where K is the SU (3) Cartan matrix) is positive definite (see lemma 2.1). For these values of the parameters the functional Iλ1 ,λ2 has the geometry of the Mountain Pass Theorem (see [1]). However, in such a situation, we do not have a global compactness result, and a lack of compactness may occur due to concentration phenomena. After a delicate blow-up analysis (see Sections 4 and 5), we obtain the following result: Theorem 2: Let M be the two dimensional (flat) torus with |M | = 1 and Hλ1 λ2 be given by (1.5). Then, for any (λ1 , λ2 ) ∈ R+ × R+ satisfying: (1.6)
(i) max{λ1 , λ2 } > 4π
and
min{λ1 , λ2 } = 6 4π,
(ii) Hλ1 λ2 is positive definite,
problem (P )λ1 ,λ2 admits a nontrivial solution. 3 Let us point out that it is still an open question to know whether or not Theorem 2 remains valid also in case min{λ1 , λ2 } = 4π. On the other hand, notice that max{λ1 , λ2 } = 4π corresponds to the critical case of the Moser-Trudinger inequality as stated in Theorem 1. Also in this situation, the existence of a minimizer remains as an open question. The result stated in Theorem 2 is largely inspired by a paper by Struwe and Tarantello [32], where they analyze the single Liouville equation on the two dimensional (flat) torus M = R2 /Z2 :
(1.7)
w −∆w = λ R e −1 ew M R w = 0. M
on M
They establish the existence of a nontrivial solution for (1.7) in the range of parameter λ ∈ (8π, 4π 2 ). The result given in Theorem 2 is in fact an extension, to a 2 × 2 Liouville-type system, of the result obtained in [32]. Indeed, for 2λ1 = λ and λ2 = 0, the system (P )λ1 ,λ2 reduces to the equation (1.7) and the conditions (i) and (ii) of Theorem 2 correspond to require λ1 ∈ (4π, 2π 2 ). Let us mention that also for the single equation (1.7) the problem of finding nontrivial solutions at λ = 8π (namely, the critical value for (1.4)) is still open. On the contrary, for M = R2 /(aZ × bZ), with a and b satisfying 4π 2 ab < 8π, the existence of a nontrivial solution for (1.7) at λ = 8π was obtained in [27] by a minimization argument. The proof of Theorem 2 is divided in two main steps. First, by the mountainpass geometry of the functional Iλ1 ,λ2 , we can use variational methods to define a minimax level which has a monotonicity property with respect to the parameters λi . Following an argument introduced by Struwe in [30] and [31], this monotonicity property allows us to obtain a priori bounds on the corresponding Palais-Smale sequences, and hence to get solutions for (P )λ1 ,λ2 , for almost all values of (λ1 , λ2 ). Then, by a compactness result, we are able to obtain the existence of a solution for (P )λ1 ,λ2 for all values of (λ1 , λ2 ) satisfying (i)-(ii), as stated in Theorem 2. This second step requires an appropriate version of a Brezis-Merle type result (see [3]) for the Liouville systems considered here (see Theorem 4.2), together with a delicate blow-up analysis (see section 5).
4
M. LUCIA AND M. NOLASCO
Acknowledgements: The authors wish to thank G. Tarantello for useful discussion. M. Lucia is grateful to Prof. X. Cabr´e for fruitful discussions and his nice hospitality at the University Politecnica de Catalunya. 2. Variational setting and preliminary results Let Ω be the periodic cell domain [− 12 , 12 ] × [− 12 , 12 ] for the 2-dimensional flat torus M = R2 /Z2 , with |M | = 1. We denote by H1 (Ω) the space of doublyR periodic 1 functions v ∈ Hloc (R2 ) with periodic cell domain Ω. Let E = {v ∈ H1 (Ω); Ω v dx = R 1 0}, endowed with the standard norm kvk = ( Ω |∇v|2 dx) 2 . Then in E × E we P 1 consider the norm k(v1 , v2 )k = ( i=1,2 kvi k2 ) 2 . It can be easily checked that the (weak) solutions of (P )λ1 ,λ2 correspond to the critical points of the action functional Iλ1 ,λ2 : E × E → R given by (2.1)
Iλ1 ,λ2 (w1 , w2 ) =
1 2
Z
X
bij ∇wi ∇wj −
Ω i,j=1,2
X
Z λi ln
e wi ,
Ω
i=1,2
1 2 1 −1 . with (bij ) = K = 3 1 2 The functional Iλ1 ,λ2 is smooth. For wi , hi ∈ E (i = 1, 2), let us compute: Iλ0 1 ,λ2 (w1 , w2 )[h1 , h2 ] =
X
< Di Iλ1 ,λ2 (w1 , w2 ), hi >=
i=1,2
(2.2)
R w e i hi = bij ∇wi ∇hj − λi RΩ w ; e i Ω i,j=1,2 Ω i=1,2 Z
X
X
and Iλ001 ,λ2 (w1 , w2 )[(h1 , h2 ), (k1 , k2 )] =
X
2 < Dij Iλ1 ,λ2 (w1 , w2 ), (hi , kj ) >=
i,j=1,2
(2.3)
Z =
X
bij ∇hi ∇kj −
Ω i,j=1,2
X
R λi
e wi Ω
i=1,2
R Ω
R R ewi hi ki − Ω ewi hi Ω ewi ki R . ( Ω ewi )2
Note that I 0 (0, 0) = 0, hence the origin is a critical point for Iλ1 λ2 . Moreover, we have: Lemma 2.1. (w1 , w2 ) = (0, 0) is a strict local minimum for Iλ1 ,λ2 if and only if λ 0 1 Hλ1 λ2 = K −1 − 4π1 2 is positive definite. 0 λ2 R 2 Proof. Recalling that 4π is the first eigenvalue for −∆ on E, we have |∇h|2 ≥ Ω R 4π 2 Ω h2 for any h ∈ E, and hence for hi ∈ E (i = 1, 2), we have Z X X Z 00 Iλ1 ,λ2 (0, 0)[(h1 , h2 ), (h1 , h2 )] = bij ∇hi ∇hj − λi h2i ≥ Ω i,j=1,2
(2.4) ≥
2 X i,j=1
(bij −
i=1,2
Ω
λi δi,j )∇hi ∇hj . 4π 2
λi 1 2 1 where B = (bij ) = 3 = K −1 . Let define Hλ1 λ2 = (bij − 4π 2 δi,j ), then 1 2 00 Iλ1 ,λ2 (0, 0) is a positive definite operator whenever Hλ1 λ2 is a positive definite matrix.
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
let αi ∈ R and hi = αi ψ (i = 1, 2), with R Conversely, 2 |∇ψ| = 1 and −∆ψ = 4π 2 ψ, then Ω
(2.5)
Iλ001 ,λ2 (0, 0)[(h1 , h2 ), (h1 , h2 )] =
X i,j=1,2
(bij −
R Ω
5
ψ = 0, kψk2 =
λi δij )αi αj = 4π 2
= < (α1 , α2 ), Hλ1 λ2 (α1 , α2 ) > . Hence if Hλ1 λ2 is not positive definite we can find (α1 , α2 ) ∈ R2 for which Iλ001 ,λ2 (0, 0)[(h1 , h2 ), (h1 , h2 )] ≤ 0. Remark 2.2. Hλ1 λ2 is a positive definite matrix, if and only if ( Tr Hλ1 λ2 = 34 − 4π1 2 (λ1 + λ2 ) > 0 (2.6) +λ2 ) Det Hλ1 λ2 = (4π12 )2 λ1 λ2 − 23 (λ14π + 13 > 0. 2 The set of (λ1 , λ2 ) satisfying condition (2.6) is a branch of an hyperbolic curve, symmetric w.r.t. the two variables λi , which intersects the positive axis at 2π 2 (see fig. 1). As already shown in [18], if max{λ1 , λ2 } > 4π then the functional Iλ1 ,λ2 is unbounded below. Indeed, let δ ∈ (0, 21 ) and Bδ = {x ∈ Ω : |x| ≤ δ} ⊂ Ω, we define
(2.7)
2 + π|x|2 )2 u+ = 2 ln 2 ( + π|δ|2 )2 ln
(2
x ∈ Bδ x ∈ Ω \ Bδ
2 x ∈ Bδ 1 + + π|x|2 )2 (2.8) u− = − u = 2 2 x ∈ Ω \ Bδ . − 12 ln 2 ( + π|δ|2 )2 R ± R ± ± Then, we set wi, = u± i, − Ω ui, dx (i = 1, 2), so that Ω wi, = 0. By a direct computation, we get − 12 ln
(2.9)
(2
Iλ1 ,λ2 (w+ , w− ) = 2(λ1 − 4π)ln + O(1) Iλ1 ,λ2 (w− , w+ ) = 2(λ2 − 4π)ln + O(1)
with O(1) bounded as → 0. Hence, for max{λ1 , λ2 } > 4π, Iλ1 ,λ2 takes arbitrarily large negative values. 3. Mountain Pass solutions In this section we prove the existence of a mountain pass critical point for Iλ1 ,λ2 for almost all (λ1 , λ2 ) in a suitable region Λ where Iλ1 ,λ2 has a mountain pass geometry ([1]). In the following, let us fix a pair (λ1 , λ2 ), w.l.o.g. we can assume λ1 ≥ λ2 , so that we require λ1 > 4π. By (2.9) and noting that kw+ k → +∞, as → 0, there exists ¯ = (λ1 ) > 0 such that for (w ¯1 , w ¯2 ) = (w¯+ , w¯− ) we have kw ¯1 k2 + kw ¯2 k2 > 1 and (3.1)
Iλ1 ,λ2 (w ¯1 , w ¯2 ) < Iλ1 ,λ2 (0, 0) = 0.
Note that for any µ1 ≥ λ1 we have Iµ1 ,λ2 (w ¯1 , w ¯2 ) ≤ Iλ1 ,λ2 (w ¯1 , w ¯2 ) < 0.
6
M. LUCIA AND M. NOLASCO
Now, we define the set (3.2)
Γ = {(γ1 , γ2 ) ∈ C([0, 1], E × E) : γi (0) = 0, γi (1) = w ¯i (i = 1, 2)}
and the set Λ = {(µ1 , µ2 ) ∈ (4π, +∞) × [0, +∞) : such that (i) µ1 > λ1
(3.3)
(ii) Hµ1 µ2 is positive definite}, where Hµ1 µ2 is defined in lemma 2.1. It is easy to check that Λ is not empty. Then, for any (µ1 , µ2 ) ∈ Λ, we define (3.4)
c(µ1 , µ2 ) =
inf
max Iµ1 ,µ2 (γ1 (t), γ2 (t)).
(γ1 ,γ2 )∈Γ t∈[0,1]
By lemma 2.1, we have c(µ1 , µ2 ) ≥ m(Hµ1 µ2 ), where we denote by m(Hµ1 µ2 ) the smallest eigenvalue of the matrix Hµ1 µ2 . Moreover, we have Lemma 3.1. Let A ⊆ Λ be such that for (µ1 , µ2 ) ∈ A there exists finite 1 lim (c(µ1 + , µ2 + ) − c(µ1 , µ2 )) ≡ α(µ1 , µ2 ) ∈ R. →0
(3.5)
Then, m(A) = m(Λ) (where m( · ) is the Lebesgue measure). Proof. It is convenient to introduce the following change of variables: z1 = 12 (µ1 + µ2 ) and z2 = 12 (µ1 − µ2 ). Then, the map T : Λ → T (Λ) = {(z1 , z2 ) ∈ R2 : (µ1 , µ2 ) ∈ Λ} defined by T (µ1 , µ2 ) = (z1 , z2 ) is a bijection. It is easy to check that for any given (w1 , w2 ) and z2 ∈ R, the map z1 → Iz1 ,z2 (w1 , w2 ) is monotone decreasing. Hence, in particular, given z2 ∈ R, the function c˜(z1 , z2 ) = c(T −1 (z1 , z2 )) is monotone non-increasing w.r.t. the variable z1 . Therefore, given z2 ∈ R, for a.e. z1 , with (z1 , z2 ) ∈ T (Λ) there exists finite (3.6)
1 1 c(z1 + , z2 ) − c˜(z1 , z2 )) = lim (c(µ1 + , µ2 + ) − c(µ1 , µ2 )). lim (˜ →0
→0
Moreover, the set A˜ = {(z1 , z2 ) ∈ T (Λ) : (3.6) holds} is measurable and, by ˜ = m(T (Λ)). Therefore, setting A = T −1 (A), ˜ we have Fubini’s theorem, m(A) m(A) = m(Λ). Remark 3.2. Let us remark that the monotonicity property of the function c(µ1 , µ2 ) w.r.t. each variables implies that c(µ1 , µ2 ) is in BVloc (Λ). Hence, the lemma 3.1 can be also obtained as a consequence of the general properties of BVloc functions. Now, let (µ1 , µ2 ) ∈ A (as defined in lemma 3.1). We consider a sequence n ↓ 0, such that, setting µi,n = µi + n (i = 1, 2), (µ1,n , µ2,n ) ∈ Λ. Moreover, let γn (t) = (γ1,n (t), γ2,n (t)) ∈ Γ be a sequence of paths such that (3.7)
max Iµ1 ,µ2 ((γn (t)) ≤ c(µ1 , µ2 ) + n .
t∈[0,1]
Then, as a consequence of lemma 3.1, we get the following bound: Lemma 3.3. There exists N1 ∈ N and a constant C1 > 0 such that for γn satisfying (3.7) and for any t ∈ [0, 1] satisfying (3.8)
Iµ1,n ,µ2,n (γn (t)) ≥ c(µ1,n , µ2,n ) − 2n
we have kγ1,n (t)k + kγ2,n (t)k ≤ C1 , for any n ≥ N1 ,
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
7
Proof. By lemma 3.1, for (µ1 , µ2 ) ∈ A and for any δ > 0 there exists N1 ∈ N such that ∀n ≥ N1 , c(µ1,n , µ2,n ) − c(µ1 , µ2 ) ≥ α(µ1 , µ2 ) − δ, n where α(µ1 , µ2 ) < 0 is defined in lemma 3.1. Hence, setting β = 2+δ−α(µ1 , µ2 ) > 2 and Jn = {t ∈ [0, 1] : (3.8) holds}, for any t ∈ Jn , we get (3.9)
Iµ1,n ,µ2,n (γn (t)) ≥ c(µ1,n , µ2,n ) − 2n ≥
(3.10)
≥ c(µ1 , µ2 ) − βn .
Then, by (3.7) and (3.10), for any t ∈ Jn , we get 0≤ (3.11)
X Z Iµ1 ,µ2 (γn (t)) − Iµ1,n ,µ2,n (γn (t)) = ln eγi,n (t) ≤ n Ω i=1,2
c(µ1 , µ2 ) + n − (c(µ1 , µ2 ) − βn ) = β + 1. n By (3.11), for any n ≥ N1 and t ∈ Jn , we have Z X X kγi,n (t)k2 ≤6Iµ1 ,µ2 (γn (t)) + 6µi ln eγi,n (t) ≤ Ω (3.12) i=1,2 i=1,2 ≤
≤ 6(c(µ1 , µ2 ) + n + (µ1 + µ2 )(β + 1)) ≤ C1 , for some constant C1 > 0 (independent on n ∈ N). Then, we have Lemma 3.4. For any (µ1 , µ2 ) ∈ A there exists a bounded Palais P Smale sequence for Iµ1 ,µ2 . Namely, there exists a sequence (w1,n , w2,n ), with i=1,2 kwi,n k2 ≤ C1 (where C1 > 0 is the constant given in lemma 3.3) such that (3.13)
Iµ1 ,µ2 (w1,n , w2,n ) → c(µ1 , µ2 ) Iµ0 1 ,µ2 (w1,n , w2,n )
→0
as n → +∞
as n → +∞.
Proof. On the contrary, let us suppose that there exists δ > 0 such that for any P w = (w1 , w2 ) ∈ E × E satisfying i=1,2 kwi k2 ≤ C1 and |Iµ1 ,µ2 (w) − c(µ1 , µ2 )| < δ, we have kIµ0 1 µ2 (w)k > δ. To simplify the notation, we set Iµ1,n ,µ2,n ( · ) = Iµn ( · ) and Iµ ( · ) = Iµ1 ,µ2 ( · ). Let γn ∈ Γ satisfying (3.7) and define (3.14)
γ˜i,n (t) = γi,n (t) −
√
n φn (γn (t))
Di Iµ (γn (t)) kIµ0 (γn (t))k
i = 1, 2,
where n = µ1,n − µ1 = µ2,n − µ2 and Iµn (u) − c(µ1,n , µ2,n ) + 2), n with φ ∈ C ∞ (R, [0, 1]), φ(s) = 0 if s ≤ 0 , φ(s) = 1 if s ≥ 1. Now, we consider the path (3.15)
(3.16)
φn (u) = φ(
γ˜n (t) = (˜ γ1,n (t), γ˜2,n (t)).
It is easy to check that γ˜n ∈ Γ. Note that, setting Jn = {t ∈ [0, 1] : (3.8) holds}, we have (3.17)
max Iµn (˜ γn (t)) = max Iµn (˜ γn (t)),
t∈[0,1]
t∈Jn
8
M. LUCIA AND M. NOLASCO
and, by lemma 3.3, kγn (t)k ≤ C1 , for any t ∈ Jn . √ D Iµ (γn (t)) For any t ∈ Jn , we set vi = n φn (γn (t)) kIi0 (γ , i = 1, 2. Then, letting n (t))k µ 2 1 (bij ) = 13 = K −1 , we have 1 2 X Iµn (˜ γn (t)) = Iµn (γn (t)) − < Di Iµn (γn (t)), vi > + i=1,2
(3.18)
1 + 2
Z bij ∇vi ∇vj − Ω
X
R µi,n (ln(
i=1,2
By Jensen’s inequality (with measure dµ =
Ree
Ω R
(3.19)
R γ (t) eγi,n (t)+vi e i,n vi ) − RΩ γ (t) ). γ (t) i,n e e i,n Ω Ω
γi,n (t)
Ω
R
Ω R
γi,n (t)
) we have
R γ (t) e i,n vi eγi,n (t)+vi RΩ ≥ exp( ). eγi,n (t) eγi,n (t) Ω Ω
Hence, we get X
Iµn (˜ γn (t)) ≤Iµn (γn (t)) −
< Di Iµn (γn (t)), vi > +
i=1,2
(3.20) +
X
kvi k2 .
i=1,2
Now, let us compute < Dk Iµn (γn (t)), vk >, for k = 1, 2: < Dk Iµn (γn (t)), Dk Iµ (γn (t)) >≥ kDk Iµ (γn (t))k2 − − | < Dk Iµn (γn (t)) − Dk Iµ (γn (t)), Dk Iµ (γn (t)) > | ≥ 1 1 ≥ kDk Iµ (γn (t))k2 − kDk Iµn (γn (t)) − Dk Iµ (γn (t))k2 . 2 2 We claim that for t ∈ Jn there exists a positive constant L such that
(3.21)
(3.22)
kDk Iµn (γn (t)) − Dk Iµ (γn (t))k ≤ L|µk,n − µk |
(k = 1, 2).
It suffices to prove (3.22) for k = 1. For h1 ∈ E, let us compute R γ (t) e 1,n h1 R (3.23) < D1 Iµn (γn (t)) − D1 Iµ (γn (t)), h1 >= (µ1 − µ1,n ) Ω , eγ1,n (t) Ω by H¨ older’s inequality, we get R γ (t) R 1 R 1 e 1,n h1 ( Ω e2γ1,n (t) ) 2 ( Ω h21 ) 2 Ω R (3.24) |(µ1 − µ1,n ) R γ (t) | ≤ |µ1 − µ1,n | . e 1,n eγ1,n (t) Ω Ω R R Recalling that Ω γ1,n (t) = 0 = Ω h1 and kγ1,n (t)k ≤ C1 , by Moser-Trudinger inequality (1.4) and Poincar´e inequality, we conclude (3.25)
C1
| < D1 Iµ (γn (t)) − D1 Iµn (γn (t)), h1 > | ≤ C(Ω)e 8π |µ1 − µ1,n |kh1 k,
with C(Ω) a positive constant (depending only on Ω). Then, by (3.22), for any t ∈ Jn and k = 1, 2 we get 1 1 kDk Iµ (γn (t))k2 − L2 |µk − µk,n |2 . 2 2 Therefore, by (3.26), for t ∈ Jn and k = 1, 2 we obtain
(3.26)
< Dk Iµn (γn (t)), Dk Iµ (γn (t)) >≥
(3.27) < Dk Iµn (γn (t)), vk > ≥
√
n φn (γn (t))
(kDk Iµ (γn (t))k2 − L2 |µk − µk,n |2 ) . 2kIµ0 (γn (t))k
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
9
Inserting (3.27) into (3.20) we get Iµn (˜ γn (t)) ≤Iµn (γn (t))− −
(3.28)
√
2 2 k=1,2 (kDk Iµ (γn (t))k − L |µk 2kIµ0 (γn (t))k
P n φn (γn (t))
− µk,n |2 )
+
+ n φn (γn (t))2 . Hence, recalling that for t ∈ Jn , we have kIµ0 (γn (t))k ≥ δ, then, for n sufficiently large such that L|µk − µk,n | < 2δ , we get √
δ n φn (γn (t)) + 4 + n φn (γn (t))2 ≤ (3.29) √ δ ≤ Iµn (γn (t)) − n φn (γn (t)) . 8 Hence, by (3.17), (3.29), (3.7), (3.10) and recalling that Iµn (w1 , w2 ) ≤ Iµ (w1 , w2 ), we obtain a contradiction as follows: (3.30) δ√ c(µ1,n , µ2,n ) ≤ max Iµ1,n ,µ2,n (˜ γn (t)) ≤ max Iµ1 ,µ2 (γn (t)) − n ≤ 8 t∈[0,1] t∈[0,1] δ√ δ √ ≤c(µ1 , µ2 ) + n − n ≤ c(µ1,n , µ2,n ) + (β − 1)n − n < 8 16
0 (where β > 2 is defined in for n ∈ N sufficiently large to have 16 the proof of lemma 3.3). Iµn (˜ γn (t)) ≤Iµn (γn (t)) −
Proposition 3.5. Let (µ1 , µ2 ) ∈ A (see lemma 3.1). Then c(µ1 , µ2 ) defines a critical value for Iµ1 ,µ2 . Namely, there exists a solution of (P )µ1 ,µ2 for a.e. values (µ1 , µ2 ) ∈ Λ. Proof. By lemma 3.4 there exists a bounded PS sequence (w1,n , w2,n ) for Iµ1 ,µ2 , for any (µ1 , µ2 ) ∈ A. Hence, wi,n → wi weakly in E and strongly in Lp (Ω), for p ≥ 2 (up to subsequences). Moreover, by Moser-Trudinger inequality (see (1.4)), we have that ewi,n is a bounded sequence in L2 (Ω). Thus, setting wn = (w1,n , w2,n ) and w = (w1 , w2 ), by H¨ older inequality we get X X kDi Iµ1 ,µ2 (wn )kkwi,n − wi k ≥ < Di Iµ1 ,µ2 (wn ), wi,n − wi > ≥ i=1,2
(3.31)
i=1,2
Z X 1 1 X ≥ kwi,n − wi k2 + o(1) − µi ( e2wi,n ) 2 kwi,n − wi kL2 (Ω) . 3 i=1,2 Ω i=1,2
Namely, (3.32)
o(1)
X
kwi,n − wi k ≥
i=1,2
1 X kwi,n − wi k2 + o(1). 3 i=1,2
So that, we may conclude (3.33)
kwi,n − wi k → 0,
as
n → +∞,
(i = 1, 2)
and (3.34)
Iµ0 1 ,µ2 (wn ) → Iµ0 1 ,µ2 (w) = 0
as
n → +∞;
that is, w = (w1 , w2 ) is a critical point at the critical level c(µ1 , µ2 ).
10
M. LUCIA AND M. NOLASCO
4. Uniform estimates and blow-up behavior for solutions of Liouville-type systems In Proposition 3.5 we have established the existence of a solution (w1 , w2 ) for (P )λ1 ,λ2 for almost every (λ1 , λ2 ) satisfying the conditions (i) and (ii) of Theorem 2. In particular, for any given (µ1 , µ2 ) ∈ Λ there exist sequences µi,n → µi , (µ1,n , µ2,n ) ∈ A, for which (P )µ1,n ,µ2,n has a solution (w1,n , w2,n ). The aim of this and the next section is to study the compactness property of the sequence (w1,n , w2,n ). In particular, in this section we analyze the asymptotic behavior of a sequence of solutions for a Liouville-type system of the following form: (4.1)
−∆ui,n =
X
kij Vn(j) (x)euj,n
on D
(i = 1, 2)
j=1,2
2
on a bounded domain D ⊂ R , with K = (kij ) =
2 −1 , and −1 2
0 ≤ Vn(i) (x) ≤ b (i = 1, 2), Z eui,n ≤ C (i = 1, 2),
(4.2)
D
for some positive constants b and C. We follow essentially the main ideas contained in [3] (see also [23]), where the authors analyze the behavior of a sequence of solutions for the single Liouville-type equation on a bounded domain D ⊂ R2 , namely: −∆vn = Vn (x)evn
(4.3)
on D
with 0 ≤ Vn (x) ≤ b Z evn ≤ C,
(4.4)
D
for b and C positive constants. In [3], among other results, the authors proved the following concentrationcompactness theorem. Theorem 4.1. ([3]-Theorem 3, with p = +∞) Let vn be a sequence of solutions for (4.3) and satisfying (4.4). Then vn admits a subsequence vnk satisfying one of the following alternative: (i) vnk is a bounded sequence on L∞ loc (D); (ii) for any compact set K ⊂ D, supK vnk → −∞, as k → +∞; (iii) there exists a finite set S = {a1 , . . . , as } ⊂ D (blow up set) and sequences {xink } ⊂ D such that, xink → ai , vnk (xink ) → +∞ (i = 1, . . . , s), and supK vnk → −∞, k → +∞, for every compact set K ⊂ D \ S. In addition, Vnk evnk →
s X
αi δai ,
as k → +∞,
in the sense of measure,
i=1
where δai is the Dirac measure concentrated at the point ai and αi ≥ 4π (i = 1, . . . , s).
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
11
3 In the following theorem we generalize the result contained in [3] to a sequence of solutions for (4.1), satisfying (4.2). Results in this direction have been obtained also in [18]. Theorem 4.2. Let (u1,n , u2,n ) be a sequence of solutions for (4.1) and satisfying (4.2). Define (4.5) Si = {p ∈ D : ∃ a sequence xn → p such that ui,n (xn ) → +∞}
i = 1, 2.
Then (u1,n , u2,n ) admits a subsequence, that we still denote by (u1,n , u2,n ), satisfying one of the following alternative: • (i) ui,n is a bounded sequence in L∞ loc (D), for i = 1, 2; • (ii) either, for some i ∈ {1, 2}, ui,n is a bounded sequence in L∞ loc (D) and, for j 6= i, uj,n → −∞ or, for every i ∈ {1, 2}, ui,n → −∞, uniformly on any compact subset of D; • (iii) the blow up set S1 ∪S2 is finite, not empty and for i ∈ {1, 2} either ui,n is bounded in L∞ loc (D \ S1 ∪ S2 ) or ui,n → −∞ uniformly on any compact subset of D \ (S1 ∪ S2 ). If in addition, for some i = 1, 2, we have Si \ (S1 ∩ S2 ) 6= ∅ , then ui,n → −∞ uniformly on any compact subset of D \ (S1 ∪ S2 ). Moreover, setting Si = {pi1 , . . . , pis }, then Vn(i) eui,n →
s X
αji δpij ,
as n → +∞,
in the sense of measure,
j=1
with αji ≥ 2π, j = 1, . . . , s. To prove Theorem 4.2 we need some preliminary results. First, let us recall the following estimate proved in [3]-Theorem 1. Lemma 4.3. Let D ⊂ R2 be a bounded domain, f ∈ L1 (D) and let u be a solution of ( −∆u = f on D (4.6) u = 0 on ∂D. Then, for any δ ∈ (0, 4π) we have Z 4π 2 (4π − δ)|u| ≤ (diam D)2 . (4.7) exp kf kL1 δ D 3 We have Lemma 4.4. Let (u1,n , u2,n ) be a sequence of solutions of (4.1), satisfying (4.2). Assume that for some i ∈ {1, 2} Z (4.8)
Vn(i) (x)eui ,n ≤ 0 < 2π,
D
then
ku+ i,n kL∞ loc (D)
is uniformly bounded.
Proof. W.l.o.g. we consider only the case i = 1. We set (4.9)
fn := 2Vn(1) (x)eu1,n ,
12
M. LUCIA AND M. NOLASCO
and we fix x0 ∈ D and a ball Br (x0 ) ⊂ D, for some r > 0. Let zn be the solution for ( −∆zn = fn on Br (x0 ) (4.10) zn = 0 on ∂Br (x0 ). Since fn > 0, by the maximum principle, we have zn ≥ 0 in Br (x0 ). By lemma 4.3, for any δ ∈ (0, 4π) we have Z (4π − δ)zn r2 ≤C (4.11) exp kfn kL1 δ Br (x0 ) for some constant C > 0 (independent on x0 ). Then, since 0 < 2π, we can choose δ > 0 to be such that 4π − δ > 20 (1 + δ) and we get, for any n ∈ N, Z (4.12) e(1+δ)zn ≤ C, Br (x0 )
for some constant C (independent of x0 ). Hence ezn is bounded in L1+δ (Br (x0 )). Now, we split u1,n as follows u1,n = zn + (u1,n − zn )
(4.13)
for x ∈ Br (x0 ).
Then (u1,n − zn ) is a subharmonic function on Br (x0 ) and, recalling that zn ≥ 0, we have Z Z Z (4.14) (u1,n − zn )+ ≤ u+ ≤ eu1,n ≤ C, 1,n Br (x0 )
Br (x0 )
D
for some positive constant C (independent of x0 ). Then, by the mean value theorem for subharmonic function, for any x ∈ B r2 (x0 ) we have Z Z (u1,n − zn )(x) ≤ C (u1,n − zn ) ≤ C (u1,n − zn )+ B r (x)
(4.15)
B r (x)
2
2
Z ≤C
+
(u1,n − zn ) Br (x0 )
where C is a positive constant depending only on r. Thus, from (4.13), (4.15) and (4.14), we deduce that eu1,n
is bounded in L1+δ (B r2 (x0 )),
and we may conclude that fn , as defined in (4.9), is bounded in L1+δ (B r2 (x0 )). Inserting this information in the Dirichlet problem (4.10) (considered now in B r2 (x0 )), by the mean value theorem (4.15) and standard elliptic estimates we derive (4.16)
+ ku+ 1,n kL∞ (B r (x0 ) ) ≤ Ck(u1,n − zn ) kL1 (B r (x0 ) ) + Ckfn kL1+δ (B r (x0 )) . 4
2
2
By (4.14) (with B r2 (x0 ) instead of Br (x0 )), and the fact that all the constants C ∞ are independent of x0 , we may conclude that u+ 1,n is a bounded sequence in Lloc (D). (i)
Now, since (Vn (x)eui,n ), for i = 1, 2, are bounded sequences in L1 (D), there exist two non-negative bounded measures η1 , η2 such that for any ψ ∈ Cc (D) Z Z Vi,n eui,n ψ → ψdηi (i = 1, 2). D
D
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
13
In view of lemma 4.4, it is convenient to introduce the following definition (see also [3]): Definition 4.5. We say that a point x0 ∈ D is a regular point with respect to ηi if there exists a function ψ ∈ Cc (D), 0 ≤ ψ ≤ 1, with ψ = 1 in some neighborhood of x0 , such that Z (4.17) ψ dηi < 2π. D
Then, we define the set Σi := {x ∈ D : x is not a regular point w.r.t. ηi },
i = 1, 2.
Now, let Si be the blow up set for ui,n , as defined in (4.5), (i = 1, 2). We have (see also [3]): Lemma 4.6. (i) Si = Σi , for i = 1, 2. (ii) card (Si ) is finite, for i = 1, 2. Proof. (i) Clearly Si ⊂ Σi , (i = 1, 2). Indeed, for any x 6∈ Σi , there exists r0 > 0 such that Z (4.18) dηi < 2π. Br0 (x)
Hence, by lemma 4.4 (with D = Br0 (x)) we get that u+ i,n is a bounded sequence in ∞ L (Br0 (x)), so that x 6∈ Si . On the other hand, suppose x0 6∈ Si , then there exists r0 > 0, a constant C > 0 and a subsequence ui,nk such that max ui,nk ≤ C,
Br0 (x0 )
∀k ∈ N.
Therefore, there exists a constant C such that Z Vn(i) (x)eui,nk ≤ Cr2 k Br (x0 )
for all r ≤ r0 , hence x0 6∈ Σi , namely Σi ⊂ Si . (ii) By definition x0 ∈ Σi if and only if ηi ({x0 }) ≥ 2π. Hence, since ηi are non-negative bounded measures it follows that Σi is a set of finite cardinality for i = 1, 2. Then, the conclusion follows from (i). Now, in order to obtain the alternatives stated in Theorem 4.2 we prove the following Harnack-type lemma essentially contained in [3]. Lemma 4.7. Let D ⊂ R2 be a bounded domain. Let (fn ) be a bounded sequence 1 in L∞ loc (D) ∩ L (D) and (un ) be such that −∆un = fn (u+ n)
in D.
L∞ loc (D),
If is bounded in then there exists a subsequence unk satisfying the following alternative: either (i) unk is bounded in L∞ loc (D), or (ii) unk → −∞ uniformly on any compact subset of D. Proof. We write un = zn + (un − zn ), where zn is the solution of ( −∆zn = fn on D (4.19) zn = 0 on ∂D.
14
M. LUCIA AND M. NOLASCO
1 ∞ Since (fn ) is bounded in L∞ loc (D) ∩ L (D), (zn ) is also bounded in Lloc (D). More+ over, note that un − zn is harmonic on D. Since (un ) is bounded in L∞ loc (D), then for any compact set K ⊂ D there exists a constant C such that un − zn ≤ C. Hence, on each compact subset K ⊂ D we can apply Harnack’s inequality to the positive harmonic function C − (un − zn ) and we may conclude that: either a subsequence
(unk − znk ) is bounded in L∞ loc (D),
(4.20) or (4.21)
(un − zn ) → −∞ uniformly on any compact subsets of D.
Recalling that (zn ) is a bounded sequence in L∞ loc (D), (4.20) and (4.21) in fact correspond, respectively, to the alternatives (i) and (ii) stated in the lemma. Proof of Theorem 4.2: If S1 ∪ S2 = ∅, we set fi,n :=
X
kij Vn(j) (x)euj,n ,
i = 1, 2.
j=1,2
Thus, we have −∆ui,n = fi,n , for i = 1, 2. Since u+ i,n (i = 1, 2) are bounded sequences in L∞ (D), we have that also f (i = 1, 2) are bounded sequences in i,n loc L∞ (D). Therefore, by lemma 4.7, we deduce that for each i ∈ {1, 2} there exists a loc subsequence, that we still denote by ui,n , satisfying the following alternative: either ui,n is bounded in L∞ loc (D), or ui,n → −∞ uniformly on any compact subset of D. Since we can always take a common subsequence for both i ∈ {1, 2}, we obtain the alternatives (i) and (ii) of the theorem. If S1 ∪ S2 6= ∅, by lemma 4.6, we know that Si is a finite set (i = 1, 2). Moreover, ∞ u+ i,n (i = 1, 2) are bounded sequences in Lloc (D \ (S1 ∪ S2 )). Hence, as above, by lemma 4.7 applied to the set D \ (S1 ∪ S2 ), we immediately conclude that for i = 1, 2 either a subsequence ui,nk is bounded in L∞ loc (D \ (S1 ∪ S2 )) or ui,n → −∞ uniformly on any compact subset of D \ (S1 ∪ S2 ). Now, let assume that for some i = 1, 2 we have Si \ (S1 ∩ S2 ) 6= ∅. W.l.o.g. we can suppose for instance p ∈ S1 \ S2 . Let r > 0 be such that Br (p) ⊂ D and Br (p) \ {p} ∩ (S1 ∪ S2 ) = ∅ and let zn be the solution for ( (2) −∆zn = Vn (x)eu2,n on Br (p) (4.22) zn = 0 on ∂Br (p). (2)
Since p 6∈ S2 , then Vn (x)eu2,n is a bounded sequence in L∞ (Br (p)). Hence, by standard elliptic estimates, we get that also zn is a bounded sequence in L∞ (Br (p)). Setting wn = u1,n + zn we have −∆wn = Un (x)ewn
(4.23) with 0 ≤ Un (x) =
(1) 2Vn (x)e−zn
(4.24)
on Br (p),
≤ C, and Z ewn ≤ C. Br (p)
Noting that p is a blow up point for wn , we can apply Brezis-Merle’s result, as stated in Theorem 4.1 (with D = Br (p)), to conclude that (4.25)
sup wn → −∞ K
for any compact subset K ⊂ Br (p) \ {p}.
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
15
Since zn is bounded, we get in particular that supK u1,n → −∞ for any compact subset K ⊂ Br (p) \ {p}. Consequently, by the alternative proved above, supK u1,n → −∞ for any compact subset K ⊂ D \(S1 ∪S2 ). Moreover, since in this (1) case, Vn (x)eu1,n → 0 in Lploc (D \ (S1 ∪ S2 )), for any p ≥ 1, and η1 (S2 \ S1 ) = 0, the measure η1 is supported on S1 . Hence, setting S1 = {p11 , . . . , p1s }, we have P η1 = j=1,...,s αj1 δp1j , and, by lemma 4.4, αj1 ≥ 2π. 2 5. Compactness Let (w1,n , w2,n ) be a solution for (P )µ1,n ,µ2,n , with (µ1,n , µ2,n ) ∈ A, µi,n → µi , as n → +∞, and (µ1 , µ2 ) ∈ Λ. In order to apply the general results obtained in Section 4 to the sequence (w1,n , w2,n ), let make the following remark. Let Si the blow up set of wi,n on M = R2 /Z2 (i = 1, 2). Since Si are finite set, we may consider the periodic cell domain Ω in such a way that (in this chart) wi,n admits no blow up points on the boundary ∂Ω and w.l.o.g. we can always choose Ω = (− 12 , 12 ) × (− 12 , 12 ). Then, in Ω, let define Z (5.1)
ui,n (x) = wi,n (x) − ln
ewi,n + ln µi,n + zi,n (x),
Ω
where zi,n ∈ C 2 (Ω) is given by (5.2)
zi,n =
1 X kij µj,n |x|2 . 4 j=1,2
Namely, zi,n is a solution for (5.3)
−∆zi,n =
X
kij µj,n
on Ω
(i = 1, 2).
j=1,2
Moreover, set (5.4)
Vn(i) (x) = exp{−zi,n (x)}
and rewrite the system (P )µ1 ,µ2 as follows ( (i) −∆ui,n = Vn (x)eui,n R (5.5) (i) V (x)eui,n = µi,n . Ω n
(i = 1, 2),
on Ω (i = 1, 2)
Namely, (u1,n , u2,n ) satisfies a system of the type (4.1) together with (4.2). Note that from (5.1) we have maxΩ ui,n ≥ C (i = 1, 2), for some constant C, hence, applying Theorem 4.2 to the sequence (u1,n , u2,n ) we have either (i) holds, namely ui,n is bounded in L∞ (Ω) for i = 1, 2, or S1 ∪ S2 6= ∅ and (iii) holds. Note that we take into account the additional information that (S1 ∪S2 )∩∂Ω = ∅. In the following we want to characterize the values of (µ1 , µ2 ) ∈ Λ for which S1 ∪ S2 = ∅. So, let us suppose that S1 ∪ S2 6= ∅. For p ∈ Ω, let us define (up to subsequence) Z σi (p) = lim lim µi,n Vn(i) (x)eui,n = r→0 n→+∞ B (p) Z r (5.6) wi,n = lim lim µi,n ewi,n −ln Ω e (i = 1, 2).
R
r→0 n→+∞
Br (p)
Note that 0 ≤ σi (p) ≤ µi (i = 1, 2), and we have Lemma 5.1. σi (p) 6= 0 if and only if p ∈ Si (for i = 1, 2).
16
M. LUCIA AND M. NOLASCO
Proof. If p 6∈ Si then there exists r > 0 such that ui,n is uniformly bounded in R (i) Br (p) (up to subsequence). Hence, in particular, Br (p) Vn (x)eui,n ≤ Cr2 , that implies σi (p) = 0. Now, let us suppose σi (p) = 0. Recalling the definition of a regular point (see 4.5), it is easy to see that σi (p) = 0 implies that p ∈ Ω is a regular point, and hence p 6∈ Σi . Since, by lemma 4.6, Si = Σi , we may conclude that p 6∈ Si . Moreover, we have R P Lemma 5.2. µi = p∈Si σi (p) if and only if wi,n − ln Ω ewi,n → −∞, uniformly on any compact subsets of Ω \ (S1 ∪ S2 ). Proof. W.l.o.g. we prove the lemma for i = 1. Let r > 0 be such that Br (p) \ {p} ∩ (S1 ∪ S2 ) = ∅, for any p ∈ S1 ∪ S2 , and Br (p) ⊂ Ω, then let us compute Z w1,n µ1 = lim µ1,n = lim µ1,n ew1,n −ln Ω e = n→+∞ n→+∞ Ω Z X X Z w1,n w1,n −ln Ω ew1,n = lim µ1,n ( e + ew1,n −ln Ω e +
R
R
n→+∞
p∈S1
Br (p)
Z
ew1,n
+
R −ln
Ω
R
p∈S2 \S1 ew1,n
Br (p)
).
Ω\∪p∈S1 ∪S2 Br (p)
Then, passing to the limit as r → 0, the conclusion immediately follows. In particular, in view of Theorem 4.2, we have: Corollary 5.3. If for some i = 1, 2 we have Si \ (S1 ∩ S2 ) 6= ∅, then µi = P p∈Si σi (p). Moreover, X wi,n µi,n ewi,n −ln Ω e → σi (p)δp , as n → +∞,
R
p∈Si
in the sense of measure on Ω. We want to characterize the possible values of σi (p) which give rise to the concentration phenomena. To begin with we prove the following convergence result. Lemma 5.4. Let (w1,n , w2,n ) be a solution for (P )µ1,n ,µ2,n , and µi,n → µi . Then, R 2 (Ω \ (S1 ∪ S2 )), 1 < q < 2, Ω Gi = 0, , i = 1, 2, there exists Gi ∈ W 1,q (Ω) ∩ Cloc doubly periodic on ∂Ω such that (5.7)
wi,n → Gi ,
as n → +∞,
(i = 1, 2)
2 in Cloc (Ω \ (S1 ∪ S2 )) and weakly in W 1,q (Ω) for 1 < q < 2. Moreover, if S1 ∪ S2 = {p1 , p2 , . . . pN }, then for r > 0 such that (Br (pk ) \ {pk }) ∩ (S1 ∪ S2 ) = ∅ (k = 1, . . . , N ), we have 1 X 1 (5.8) Gi (x) = kij σj (pk )ln + gi (x) for x ∈ Br (pk ) \ {pk } 2π j=1,2 |x − pk |
with gi ∈ C 2 (Br (pk )) (for i = 1, 2, and k = 1, . . . N ). Proof. First of all we prove that wi,n (i = 1, 2) is a bounded sequence in W 1,q (Ω), with 1 < q < 2. Let q 0 > 2 be such that 1q + q10 = 1. By definition, we have that Z Z 0 (5.9) k∇wi,n kLq ≤ sup{| ∇wi,n ∇φ| : φ ∈ W 1,q (Ω), φ = 0, kφkW 1,q0 = 1}. Ω
Ω
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
17
Since by the Sobolev embedding theorem we have kφkL∞ ≤ ckφkW 1,q0 , for some constant c > 0, we have Z (5.10) |
Z ∇wi,n ∇φ| = |
Ω
Z ∆wi,n φ| ≤ kφk
L∞
Ω
|
X
Ω j=1,2
kij µj,n ( R
ewj,n − 1)| ≤ C ewj,n Ω
with C some positive constant. Therefore, we may conclude that wi,n → Gi weakly R in W 1,q (Ω), and Ω Gi = 0. Let ηi (i = 1, 2) be the non-negative bounded measures such that (5.11)
µi,n ewi,n R → ηi ewi,n Ω
in the sense of measure, as n → +∞ (i = 1, 2)
R and Ω ηi = µi (i = 1, 2). Recalling Theorem 4.2 we have that either R (i) wi,n − ln Ω ewi,n is bounded in L∞ loc (Ω \ (S1 ∪ S2 )) or R (ii) wi,n − ln Ω ewi,n → −∞, uniformly on any compact subsets of Ω \ (S1 ∪ S2 ). wi,n In both cases ((i) and (ii)), ewi,n −ln Ω e (i = 1, 2) are bounded sequences p respectively in L (Ω \ S ), for any p ≥ 1, hence we claim that there exist functions i loc R ui such that Ω eui = 1 (i = 1, 2) and, for i = 1, 2, (5.12) X wi,n µi,n ewi,n −ln Ω e → (µi − σi (p))eui , as n → +∞, in Lploc (Ω \ Si ).
R
R
p∈Si
Indeed, if (i) holds, then (5.12) follows by using system (P )λ1 ,λ2 and Sobolev embedding theorem. −ln Ω ewi,n If, otherwise, (ii) holds, then µi,n ewi,nP → 0 and (5.12) becomes trivial in view of lemma 5.2, which gives µi = p∈Si σi (p). In view of (5.12), we may conclude that (in the sense of distributions) X X (5.13) ηi = (µi − σi (p))eui + σi (p)δp (i = 1, 2).
R
p∈Si
p∈Si
Now, we claim that Gi satisfies (in the sense of distributions) the following equation
(5.14)
( P −∆Gi = j=1,2 kij (ηj − µj ) i = 1, 2 R G = 0, doubly periodic on ∂Ω. Ω i
Indeed, for any φ ∈ C ∞ (Ω) (doubly periodic in ∂Ω) , we have (5.15) Z Z | ∇(wi,n −Gi )∇φ| = | (wi,n − Gi )∆φ| ≤ Ω Ω Z X wj,n ≤ |kij |(| (µj,n ewj,n −ln Ω e − ηj )φ| + |µj,n − µj |) → 0
R
j=1,2
Ω
as n → +∞ (i = 1, 2). Moreover, in view of (5.12), by standard elliptic estimates and Sobolev embed2 ding theorem, we may conclude that wi,n → Gi in Cloc (Ω \ (S1 ∪ S2 )) (i = 1, 2). 1,q ∞ Now, let Gp ∈ W (Ω) ∩ C (Ω \ {p}), (1 < q < 2) be the solution of ( −∆Gp = δp − 1 on Ω R (5.16) G = 0, Gp doubly periodic on ∂Ω. Ω p
18
M. LUCIA AND M. NOLASCO
P P We define γi (x) = Gi (x) − ( j=1,2 kij p∈Sj σj (p))Gp , i = 1, 2. It is easy to check that γi satisfies an equation of the type −∆γi = fi with X X σj (p))(euj − 1) ∈ Lp (Ω), ∀p > 2, (i = 1, 2). (5.17) fi = kij (µj − j=1,2
p∈Sj
Therefore, by elliptic estimates, we can infer that in fact γi ∈ C 2 (Ω). Then, (5.8) follows recalling that the solution Gp of (5.16) can be written as 1 1 Gp = 2π ln |x−p| + γp (x) with γp ∈ C ∞ (Ω) (see [2]). Remark 5.5. Let ui,n be given by (5.1). Then, since zi,n is convergent in C 2 (Ω), by lemma 5.4, Z ˜ i (x) (5.18) ui,n − ui,n → G i = 1, 2 Ω 2 in Cloc (Ω \ (S1 ∪ S2 )) and weakly in W 1,q (Ω) for 1 < q < 2. Moreover, if S1 ∪ S2 = {p1 , p2 , . . . pN }, then for any r > 0 such that (Br (pk ) \ {pk }) ∩ (S1 ∪ S2 ) = ∅ (k = 1, . . . , N ), we have that X 1 ˜ i (x) = 1 kij σj (pk )ln + g˜i (x), for x ∈ Br (pk ) \ {pk }, (5.19) G 2π j=1,2 |x − pk |
with g˜i ∈ C 2 (Br (pk )), for k = 1, . . . N and i = 1, 2. As a consequence of lemma 5.4 and, in particular, of remark 5.5, we get the following relation between the blow-up masses σi (p): Lemma 5.6. If p ∈ S1 ∪ S2 , then we have σ1 (p)2 + σ2 (p)2 − σ1 (p)σ2 (p) = 4π(σ1 (p) + σ2 (p)).
(5.20) (see fig.2)
Proof. W.l.o.g. we can assume p = 0 and we set σi (0) = σi . Then, let r > 0 be such that (Br (0) \ {0}) ∩ (S1 ∪ S2 ) = ∅ and Br (0) ⊂ Ω. From system (5.5) we have the following Pohozaev-type identities: Z
Z 1 2 − r(|∂r u1,n | − |∇u1,n | ) = 2 rVn(1) (x)eu1,n − 2 ∂Br (0) ∂Br (0) Z Z (5.21) (1) u1,n −4 Vn (x)e − Vn(2) (x)eu2,n x · ∇u1,n ; 2
Br (0)
− (5.22)
Br (0)
Z
Z
1 r(|∂r u2,n |2 − |∇u2,n |2 ) = 2 rVn(2) (x)eu2,n − 2 ∂Br (0) ∂Br (0) Z Z (2) u2,n −4 Vn (x)e − Vn(1) (x)eu1,n x · ∇u2,n . Br (0)
Now, from (5.5) we obtain the following system: ( (1) −∆(2u1,n + u2,n ) = 3Vn (x)eu1,n (5.23) (2) −∆(2u2,n + u1,n ) = 3Vn (x)eu2,n
Br (0)
on Ω on Ω.
From (5.23) we get the following Pohozaev-type identities: (5.24) Z −
Z 1 r(|∂r (2u1,n +u2,n )|2 − |∇(2u1,n + u2,n )|2 ) = 6 rVn(1) (x)eu1,n − 2 ∂Br (0) ∂Br (0) Z Z (1) u1,n − 12 Vn (x)e +3 Vn(1) (x)eu1,n x · ∇u2,n ; Br (0)
Br (0)
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
19
(5.25) Z −
Z 1 r(|∂r (2u2,n +u1,n )|2 − |∇(2u2,n + u1,n )|2 ) = 6 rVn(2) (x)eu2,n − 2 ∂Br (0) ∂Br (0) Z Z − 12 Vn(2) (x)eu2,n + 3 Vn(2) (x)eu2,n x · ∇u1,n . Br (0)
Br (0)
By (5.21)-(5.22) and (5.24)-(5.25) we finally get the following identity : (5.26) X Z i=1,2
Z 1 1 2 r(|∂r ui,n | − |∇ui,n | ) + r(∂r u1,n ∂r u2,n − ∇u1,n ∇u2,n ) = 2 2 ∂Br (0) ∂Br (0) Z Z X X 3 =− r Vn(i) (x)eui,n + 3 Vn(i) (x)eui,n . 2 ∂Br (0) i=1,2 Br (0) i=1,2 2
By lemma 5.4 (and remark 5.5), passing to the limit as n → +∞ in (5.26), we get Z Z X 1 2 (5.27) α r(|∂r G0 (r)| = 3 lim Vn(i) (x)eui,n + o(r) n→+∞ B (0) 2 ∂Br (0) r i=1,2 P P P where α = i=1,2 ( j=1,2 kij σj )2 + Πi=1,2 ( j=1,2 kij σj ) and G0 is the solution of (5.16) for p = 0. It is easy to compute Z (5.28) r(|∂r G0 (r)|2 = 2π. ∂Br (0)
Then, taking the limit as r → 0 in (5.27), we finally get (5.20). From lemma 5.1 and lemma 5.6, it is immediate to conclude the following result: Corollary 5.7. If for some i = 1, 2 there exists p ∈ Si \ (S1 ∩ S2 ), then σi (p) = 4π. Now, we prove that in the range of parameters Λ we have in fact S1 ∩ S2 = ∅. Indeed, we have Lemma 5.8. If p ∈ S1 ∩ S2 then max{σ1 (p), σ2 (p)} ≥ 8π. Proof. Let p ∈ S1 ∩ S2 . W.l.o.g. we can assume p = 0. Hence, there exist two sequences xi,n → 0 (i = 1, 2) as n → +∞, such that ui,n (xi,n ) → +∞ (i = 1, 2) as n → +∞. Let us define 1 i = 1, 2. (5.29) i,n = exp[− ui,n (xi,n )] 2 W.l.o.g. we can assume (up to subsequences) 1,n ≤ 2,n and x1,n = 0. Let r > 0 be such that Br (0) \ {0} ∩ (S1 ∪ S2 ) = ∅. For x ∈ Bn = {x ∈ R2 : 1,n x ∈ Br (0)}, we define u ˜1,n (x) = u1,n (1,n x) − u1,n (0) (5.30) u ˜2,n (x) = u2,n (1,n x) − u1,n (0). We get
(5.31)
P (j) −∆˜ ui,n = j=1,2 kij Vn (1,n x)eu˜j,n on Bn R (i) u ˜i,n V ( x)e ≤ µ i = 1, 2 1,n i,n Bn n u ˜ (x) ≤ u ˜ (0) = 0 ∀x ∈ Bn 1,n 1,n u ˜2,n (x) ≤ u2,n (x2,n ) − u1,n (0) ≤ 0 ∀x ∈ Bn .
Then, according to Theorem 4.2, we have the following alternative: either 2 1) u ˜i,n bounded in L∞ loc (R ) for i = 1, 2;
20
M. LUCIA AND M. NOLASCO
or 2 2) u ˜1,n bounded in L∞ ˜2,n → −∞, as n → +∞, uniformly on any loc (R ) and u 2 compact set of R . So, let us analyze the two cases separately: 2 Case 1): In this case u ˜i,n → u ¯i (up to subsequence), as n → +∞, in L∞ loc (R ) (i) (i = 1, 2). Moreover, since Vn (i,n x) → 1, as n → +∞, uniformly on compact sets of R2 , we may conclude that (¯ u1 , u ¯2 ) satisfies P ui = j=1,2 kij eu¯j on R2 −∆¯ (5.32) max{¯ u1 , u ¯2 } = 0 R u ¯i e ≤ σi (0) i = 1, 2. R2
We claim that Z (5.33)
eu¯i > 4π
(i = 1, 2).
R2
To prove the claim we follow essentially an argument used in [7]-(Lemma 1.2). Since max{¯ u1 , u ¯2 } < +∞, we can define (5.34)
wi (x) =
1 2π
Z
(ln |x − y| − ln (|y| + 1))eu¯i
(i = 1, 2).
R2
Then, it is easy to check that (5.35)
wi (x) 1 → ln |x| 2π
Z
eu¯i ,
as |x| → +∞,
uniformly
(i = 1, 2)
R2
P Moreover, ∆wi = eu¯i on R2 . Therefore, if we define vi (x) = u ¯i (x)+ j=1,2 kij wj , we have ∆vi = 0, for i = 1, 2. By (5.35) and recalling that max{¯ u1 , u ¯2 } < +∞, we get that (5.36)
vi (x) ≤ C1 + C2 ln |x|
i = 1, 2
for |x| sufficiently large, with C1 , C2 positive constants. Therefore, by Liouville’s Theorem on harmonic functions, vi has to be constant (for i = 1, 2), and hence we get Z X u ¯i (x) 1 →− kij eu¯j as |x| → +∞, uniformly (i = 1, 2). (5.37) ln |x| 2π R2 j=1,2 R R P Then, by (5.37), recalling that R2 eu¯i < +∞ we can infer R2 j=1,2 kij eu¯j > 4π for i = 1, 2, and we get (5.33). Hence, in particular, we obtain σi (0) > 4π for both i = 1, 2. Now, since σi (0) satisfy (5.20) (see fig. 2), we get in fact max{σ1 (0), σ2 (0)} ≥ 8π. 2 Case 2): If otherwise, u ˜1,n bounded in L∞ ˜2,n → −∞, as n → +∞, loc (R ) and u 2 uniformly on any compact set of R . We get that u ˜1,n → ξ (up to subsequence), as 2 n → +∞, in L∞ loc (R ), where ξ satisfies ( −∆ξ = 2eξ on R2 R (5.38) eξ ≤ σ1 (0) R2
In this case,R by a result of Chen-Li (see [7]) we know that a solution of (5.38) has to satisfy R2 eξ = 4π, hence we get σ1 (0) ≥ 4π. If σ1 (0) = 4π and hence, by lemma 5.6, σ2 (0) = 0, we conclude, by lemma 5.1, that 0 6∈ S2 , a contradiction. If otherwise, σ1 (0) > 4π, we define β = σ1 (0) − 4π > 0. Let us suppose β < 4π, as otherwise we already would have σ1 (0) > 8π .
SU(3) CHERN-SIMONS VORTEX THEORY AND TODA SYSTEMS
Note that (up to subsequences) we have Z (5.39) β = lim lim lim r→0 R→+∞ n→+∞
21
eu1,n .
Br (0)\BR1,n (0)
Let us fix sequences rk → 0 and Rk → +∞, as k → +∞, and for any k ∈ N let nk ∈ N be such that rk > 1,nk Rk . We denote Dk = Brk (0) \ BRk 1,nk (0). Then, (1)
let yk large
(1)
∈ Dk be such that u1,nk (yk ) = maxDk u1,nk . We have, for k sufficiently
(5.40)
(1)
eu1,nk (yk
)
≥
1 πrk2
Z
eu1,nk ≥
Dk
(1) (1) Therefore, we get u1,nk (yk ) → +∞ and yk → (2) (2) yk ∈ Dk be such that u2,nk (yk ) = maxDk u2,nk .
β . 2πrk2
0, as k → +∞. Moreover, let
We define 1 (1) δ1,k = exp[− u1,nk (yk )] 2 (5.41) 1 (2) δ2,k = exp[− u2,nk (yk )]. 2 Following essentially the same blow-up argument as above, since Z β = lim eu1,nk < 4π, k→+∞
Dk
˜ k = {x ∈ R2 : δ2,k x+y (2) ∈ Dk } we may assume that δ2,k < δ1,k . Then, we define D k ˜ k , we set and, for x ∈ D (5.42)
(2)
(2)
(2)
(2)
u ˜1,k (x) =u1,nk (δ2,k x + yk ) − u2,nk (yk ) u ˜2,k (x) =u2,nk (δ2,k x + yk ) − u2,nk (yk ).
R Now, again by the same blow-up argument, we get σ2 (0) ≥ limk→+∞ Dk eu2,nk ≥ 4π, and hence, recalling that we are in the case σ1 (0) > 4π, by lemma 5.6 (see fig.2) we can conclude, as above, that max{σ1 (0), σ2 (0)} ≥ 8π . Now, as a direct consequence of corollary 5.3, corollary 5.7 and lemma 5.8, since for (µ1 , µ2 ) ∈ Λ we have max{µ1 , µ2 } < 8π, we may conclude that each sequence ui,n (i = 1, 2) may have at most one blow-up point pi and, if both sequences do have a blow-up point, then p1 6= p2 . More precisely, we have Corollary 5.9. For (µ1 , µ2 ) ∈ Λ we have S1 ∩ S2 = ∅ and card (Si ) ≤ 1, for i = 1, 2. Moreover, if min{µ1 , µ2 } = 6 4π we have S1 ∪ S2 = ∅. Now, we can conclude: Proof of Theorem 2: Let (µ1,n , µ2,n ) ∈ A and µi,n → µi , with (µ1 , µ2 ) satisfying (i) and (ii), and let (w1,n (x), w2,n (x)) be a solution of (P )µ1,n ,µ2,n , as established in Proposition 3.5. Then, let Z (5.43) ui,n (x) = wi,n (x) − ln ewi,n + ln µi,n + zi,n (x), i = 1, 2, Ω
where zi,n is defined in (5.2). Then, by corollary 5.9, we have that S1 ∪ S2 = ∅ and hence ui,n (i = 1, 2) are bounded sequences in L∞ (Ω). Since zi,n (R i = 1, 2) are bounded sequences in L∞ (Ω), we conclude also that wi,n (x) − ln Ω ewi,n are bounded sequences in L∞ (Ω), for i = 1, 2. Then, by standard elliptic estimates, we have that wi,n → w ¯i ,
22
M. LUCIA AND M. NOLASCO
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