Vortex condensates for the SU(3) Chern-Simons theory

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY∗ MARGHERITA NOLASCO AND GABRIELLA TARANTELLO

Abstract. We investigate SU (3)- periodic vortices in the self-dual ChernSimons theory proposed by Dunne in [13],[15]. At the first admissible non-zero energy level E = 2π, and for each (broken and unbroken) vacuum state φ(0) of the system, we find a family of periodic vortices asymptotically gauge equivalent to φ(0) , as the Chern-Simons coupling parameter k → 0. At higher energy levels, we show the existence of multiple gauge distinct periodic vortices with at least one of them asymptotically gauge equivalent to the (broken) principal embedding vacuum, when k → 0.

1. Introduction In recent years several Chern-Simons field theories [14] have been proposed largely motivated by their possible applications to the physics of high critical temperature superconductivity. In fact, the corresponding Chern-Simons vortex theory has revealed a much richer structure compared with that described by the ”classical” Yang-Mills framework [21], [26]. The study of Chern-Simons vortices has been particularly succesfully in the abelian situation, see [31], [30], [4], [34], [29], [37], [8], [9], [6] and [7], where the existence and multiplicity of vortices with different nature (e.g. topological, nontopological, periodically constrained, etc.) has been established for the models proposed in [18], [20] and [23]. Only recently some progress has been made towards the existence of nonabelian CS-vortices, concerning the self-dual Chern-Simons theory proposed by Dunne in [13], [15], see also [22], [25] and [24]. Dunne’s model is defined in the (2 + 1)-Minkowski space R1,2 , with metric tensor gµν = diag(−1, 1, 1). The gauge group is given by a compact Lie group G equipped with a semi-simple Lie algebra (G, [ , ]). The relative Chern-Simons Lagrangean density is given by 2 (1.1) L = −T r{(Dµ φ)† Dµ φ} − kµνα T r{∂µ Aν Aα + Aµ Aν Aα } − V (φ, φ† ), 3 where A = (A0 , A1 , A2 ) is the gauge connection of the principal bundle over R2,1 with structure group G that, together with the Higgs field φ, takes values in G through the adjoint representation of G. The gauge-covariant derivative Dµ = ∂µ + [Aµ , · ] is used to weakly couple the Higgs field φ with the gauge potential A = (A0 , A1 , A2 ). Furthermore, the Levi-Civita antisymmetric tensor µνα is chosen with 012 = 1, k > 0 is the Chern-Simons coupling parameter and “T r{. . . }” refers to the trace in the matrix representation of G. The gauge-invariant scalar potential V (φ, φ† ) is defined by, 1 V (φ, φ† ) = 2 T r{([[φ, φ† ], φ] − v 2 φ)† ([[φ, φ† ], φ] − v 2 φ)}, 4k ∗ Supported by M.U.R.S.T. 40 % Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari .

1

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M. NOLASCO AND G. TARANTELLO

where the constant v 2 plays the role of a mass parameter. In the following we set v 2 = 1. Vortices for L correspond to static solutions (with ”finite” energy) for the EulerLagrange equations corresponding to (1.1). Over R2 , topological and nontopological vortices (see below) have been established in [38] and [36] respectively. Here we look for periodic vortices or condensates, namely for those static solutions which satisfy appropriate periodic boundary conditions to be specified according to the gauge invariance of L. We note immediately that the Euler-Lagrange equations corresponding to L (see (5.31a-b) in [14]) are very difficult to handle directly, even when we restrict to consider “Bogomol’nyi” type vortices, which are obtained by solving the reduced (first order) “relativistic self-dual Chern-Simons equations”: ( D− φ = 0 (1.2) F+− = k12 [φ − [[φ, φ† ], φ], φ† ]. where D− = D1 − iD2 and F+− = ∂+ A− − ∂− A+ + [A+ , A− ], with A± = A1 ± iA2 and ∂± = ∂1 ± i∂2 . Solutions to (1.2) define a sort of ”energy minimizers” for the system as they saturate the lower bound: 1 (1.3) E = 2 tr(φ† (φ − [[φ, φ† ], φ])) 2k (modulo some negligible surface terms) for the energy density E corresponding to L (see [14], [15]). From (1.2) it is easy to determine the zero-energy vortices (vacua states), where the gauge field vanishes (modulo gauge transformations) while the Higgs field φ(0) corresponds to a zero of the potential V , namely, it is gauge equivalent to a solution of the algebraic equation (see [14], [15]): (1.4)

[[φ, φ† ], φ] = φ,

To be able to determine other non-zero energy solutions of (1.2), Dunne in [13] has proposed a simplified form of the self-dual system (1.2) in which the fields are algebrically restricted as follows. Let r be the rank of the Lie algebra G, {Ha } the generators of the Cartan subalgebra and {E±a } the family of the simple root step operators (with E−a = Ea† ), normalized according to a Chevalley basis [5],[19]. Hence, they satisfy the following commutation relations: [Ha , Hb ] = 0 [Ha , E±b ] = ±Kab E±b [Ea , E−b ] = δab Ha a, b = 1, . . . , r , and are subject to the normalization conditions: T r{Ha Hb } = Kab T r{Ea E−b } = δab T r{Ha E±b } = 0 a, b = 1, . . . , r, where K = (Kab ) is the Cartan matrix.

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

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We assume that the fields take the form:

Aµ = −i

(1.5)

r X

Aaµ Ha

a=1

(1.6)

φ=

r X

φa Ea

a=1

Aaµ

with (a = 1, . . . , r; µ = 0, 1, 2) real-valued functions and φa (a = 1, . . . , r) complex-valued functions. In view of (1.5) and (1.6) the gauge invariance of L can be expressed in terms of r the gauge group H = span {ei a=1 wa (x)Ha } (where wa a = 1, . . . , r are real-valued smooth functions) generated by the Cartan subalgebra generators {Ha }. In other words, the gauge transformation laws for the components of the gauge potential and the Higgs field take the following simplified form:

P

Aaµ −→ Aaµ + ∂µ wa ,

(1.7)

φa −→ ei

P

r b=1

Kba wb a

φ ,

a = 1, . . . , r. With the algebraic restriction on the fields (1.5) and (1.6), the Lagrangian density L and the potential V are simplified as well. The Chern-Simons term decomposes into r copies of an abelian Chern-Simons term, and we have 2 r r r X X X (1.8) Lrestricted = − Kba Abµ )φa −k µνρ ∂µ Aaν Aaρ −Vrestricted ∂µ φa − i( a=1

a=1

b=1

where the restricted potential becomes r r 1 X a2 1 X a2 |φ | − 2 |φ | Kab |φb |2 + Vrestricted = 2 4k a=1 2k a,b=1 (1.9) r 1 X + 2 |φa |2 Kab |φb |2 Kbc |φc |2 . 4k a,b,c=1

Furthermore, the “relativistic self-dual Chern-Simons equations” (1.2) (away from the zeroes of φa ) combine into the single set of coupled equations: (1.10) ∂+ ∂− ln|φa |2 = −

r r 1 X 1 X b2 b 2 K |φ | + |φ | Kbc |φc |2 Kac ab k2 k2 b=1

a = 1, . . . r

b,c=1

To recover the Aµ=0 component of the gauge potential, we must supplement (1.2) with the Gauss Law constraint of the system, which componentwise reads as follows: (1.11)

a kF12 = J0a

a = 1, . . . , r

a where (after multiplication by −i) F12 = ∂1 Aa2 − ∂2 Aa1 and J0a define, respectively, the components (in the Cartan subalgebra) of the gauge curvature and the current density J0 = −i([φ† , D0 φ] − [(D0 φ)† , φ]). i ([[φ, φ† ], φ] − φ) and, by direct The selfdual equations (1.2), imply that D0 φ = 2k calculation, for the energy density (1.3) we find the expression

(1.12)

E=

r X

a=1

(modulo negligible surface terms).

a F12

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M. NOLASCO AND G. TARANTELLO

Therefore under the decomposition (1.5) and (1.6), the system may be described in terms of r-(abelian) Chern-Simons fields Aaµ coupled to r complex scalar Higgs fields φa with the couplings determined by the Cartan matrix K = (Kab ). Note that, by means of (1.5) and (1.6), the algebraic equation (1.4) may be solved explicitely in terms of the components φa(0) of φ(0) in the Cartan subalgebra. When φa(0) 6= 0, we find that, |φa(0) |2 =

(1.13)

r X

(K −1 )ab

a = 1, . . . r,

b=1

where K −1 is the inverse of the Cartan matrix K. Notice that, on the base of (1.3), vortex solutions of (1.2) in R2 (with sufficiently fast decay as |x| → +∞) satisfies to the energy relation: Z Z 1 tr(φ† (φ − [[φ, φ† ], φ])). (1.14) E= E= 2 2k R2 R2 Therefore, it is to be expected that non-zero energy vortex solutions in R2 become gauge equivalent to the vacua states as |x| → +∞. Results in this direction have been obtained by Yang [38] and Wang-Zhang [36] when the gauge group G = SU (N ). Yang in [38] shows that there exist solutions for (1.2) satisfying the ansatz (1.5) and (1.6), for every prescribed configuration of zeros for φa , and such that |φa | uniformly converges to |φa(0) | in (1.13) as |x| → +∞, a = 1, . . . , r. Thus, Yang’s vortices are asymptotically equivalent (as |x| → +∞) to the so called principal embedding vacuum. Usually, one refers to those as the topological solutions for (1.2) in R2 . More difficult to derive are instead the nontopological solutions, namely those asymptotically gauge-equivalent to the other vacua states, as |x| → +∞. A class of nontopological solutions has been derived recently in [36]. The solutions in [36] are shown to be asymptotically gauge-equivalent to the unbroken vacuum φ(0) = 0, as |x| → +∞. We observe that, when N = 2, the SU (N )-gauge theory corresponding to L in (1.1) reduces to the abelian ChernSimons-Higgs theory introduced by Hong-Kim-Pac [18] and Jackiw-Weinberg [20]. Thus, the above mentioned results [38] and [36] extend the work of [31], [30] and [6] on topological and nontopological abelian Chern-Simons vortices. In the abelian contest we also mention the work of [7] on nontopological Maxwell-Chern-Simons vortices for the Lee-Lee-Min model [23]. However, to establish condensate-type vortices whose feature more closely resemble those of the mixed states predicted by Abrikosov in superconductivity [1], it is necessary to derive solutions for (1.2) subject to gauge invariant periodic boundary condition. For this purpose note that, in the stationary case, the functions wa (a = 1, . . . , r) in (1.7) expressing the gauge invariance of (1.1), depend only on the space-variables x = (x1 , x2 ) and the gauge transformation laws reduce to: (1.15)

Aa0 → Aa0 a

φ −→ e

P i

Aaj → Aaj + ∂j wa r b=1

Kba wb a

φ

j = 1, 2

a = 1, . . . , r.

Therefore, following ‘t Hooft [33], for each of the r-components of the fields we require appropriate periodic boundary conditions to hold in the periodic cell domain: a a b b Ω = {x = (x1 , x2 ) ∈ R2 | − ≤ x1 ≤ , − ≤ x2 ≤ }, 2 2 2 2 as follows.

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

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Let e1 = (a, 0) and e2 = (0, b) and decompose the boundary of Ω by setting ∂Ω = Γ1 ∪ Γ2 ∪ {e1 + Γ2 } ∪ {e2 + Γ1 } ∪ {0, e1 , e2 , e1 + e2 }, with 1 (se1 − e2 ) |s| < 1} 2 1 Γ2 ={x ∈ R2 | x = (se2 − e1 ) |s| < 1}. 2 Γ1 ={x ∈ R2 | x =

We require that each component Aaµ and φa (a = 1, . . . , r) of the vortex condensates (A, φ) satisfies:  i r K ξb (x+e ) a r b k e b=1 ba k φ (x + ek ) = ei b=1 Kba ξk (x) φa (x) (a)    Aa (x + e ) = Aa (x) (b) k 0 0 (1.16) a a a a  j = 1, 2 (c)  (Aj + ∂j ξk )(x + ek ) = (Aj + ∂j ξk )(x)  x ∈ Γ1 ∪ Γ2 \ Γk k = 1, 2 a = 1, . . . , r

P

P

where ξ1a , ξ2a (a = 1, . . . , r) are smooth functions defined in a neighborhood of Γ2 ∪ {e1 + Γ2 } and Γ1 ∪ {e2 + Γ1 }, respectively. Notice that, in analogy to the abelian case, the set of boundary conditions (1.16) produce a “quantization” effect on the ”charges” (see [4] and [35]). In fact, in view of (1.16)-(a), to any vortex condensate we can associate r-integers Na ∈ Z (a = 1, . . . , r) (vortex numbers), corresponding to the phase shift of φa around ∂Ω. More precisely, for a = 1, . . . , r and k = 1, 2, set r X ξˆka (s1 , s2 ) = Kba ξkb (s1 e1 + s2 e2 ) sj ∈ (0, 1), j = 1, 2, b=1

we have: ξˆ1a (1, 0+ ) − ξˆ1a (0, 0+ ) + ξˆ2a (0+ , 0) − ξˆ2a (0+ , 1)+ + ξˆ1a (0, 1− ) − ξˆ1a (1, 1− ) + ξˆ2a (1− , 1) − ξˆ2a (1− , 0) = 2πNa . Consequently, by means of (1.16)-(c) and (1.11), for the “magnetic flux”-component R R a Φa = Ω F12 and the “electric charge”-component Qa = Ω J0a we obtain the relations, r X Kba Φb = 2πNa , Qa = kΦa . b=1

Hence, they obey to the following “quantization” rules: r r X X (1.17) Φa = 2π (K −1 )ba Nb , Qa = 2πk (K −1 )ba Nb . b=1

b=1

Accordingly, for the energy Z Z 1 (1.18) E= E= 2 tr(φ† (φ − [[φ, φ† ], φ]) 2k Ω Ω we may use (1.14) to derive r Z r r X X X a (1.19) E= F12 = 2π (K −1 )ba Nb = 2π |φb(0) |2 Nb , a=1

φb(0)

Ω

a,b=1

b=1

where expresses the components (in the Cartan subalgebra) of the principal embedding vacuum as given in (1.13). Pr Since D− φ = 0, or equivalently ∂− ln φa = i b=1 Ab− Kab (a = 1, . . . , r), as in [4] and [34], each vortex number Na has the (topological) interpretation of counting

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the number of zeroes (according to their multiplicity) of the Higgs scalar component φa in Ω. By virtue of (1.18), we now expect doubly periodic vortex solutions to become asymptotically gauge-equivalent to the vacua-states, when k → 0+ . Thus, in the same spirit of the results [38] and [36] mentioned above, we are going to establish periodic vortex condensates for L where the asymptotic behavior of the Higgs field, as k → 0+ , is prescribed according to any fixed zero of the gauge potential V . For this purpose, we shall focus our attention to the simplest non-abelian case of physical relevance and take the gauge group G = SU (3). Our results should be compared with those obtained in [4], [34] for the abelian case (i.e. G = SU (2)) and in [29] for the Maxwell-Chern-Simons-Higgs model of Lee-Lee-Min  [23].  2 −1 Note that for the Lie group SU (3) the Cartan matrix is K = . Thus, −1 2 the restricted potential takes the form: (1.20) 1 (4|φ1 |6 + 4|φ2 |6 − 3|φ1 |4 |φ2 |2 − 3|φ1 |2 |φ2 |4 + 4k 2 + 4|φ1 |4 + 4|φ2 |4 − 4|φ1 |2 |φ2 |2 + |φ1 |2 + |φ2 |2 ) = 1 = 2 (|φ1 |2 (2|φ1 |2 − |φ2 |2 − 1)2 + |φ2 |2 (2|φ2 |2 − |φ1 |2 − 1)2 ) 4k whose zeroes coincide with the pairs (|φ1 |2 , |φ2 |2 ) = (0, 0) unbroken vacuum, (|φ1 |2 , |φ2 |2 ) = (1, 1) principal embedding vaccuum, (|φ1 |2 , |φ2 |2 ) = (0, 21 ) and 1 ( 2 , 0). Thus, for k > 0 small, we will be interested in deriving SU (3)- vortex solutions for (1.2) - (1.16), under the ansatz (1.5) and (1.6), such that the components φa (a = 1, 2) of the Higgs field satisfy one of the following: • |φ1 | → 1 and |φ2 | → 1 (type I) 1 2 • |φ ( | → 0 and |φ | → 0 (type II) 1 1 |φ | → √2 and |φ2 | → 0 (a) • (type III) 1 1 2 |φ | → 0 and |φ | → √2 (b) Vrestricted (|φ1 |, |φ2 |) =

in some suitable norm as k → 0+ . In this direction, we prove the existence of two gauge distinct family of solutions for any prescribed pair of ”vortex numbers” Na (a = 1, 2) and relative set of vortex points Za = {pa1 , . . . , paNa } ⊂ Ω corresponding to the zeroes of the component φa of the Higgs field, a = 1, 2. Only one of these two family of solutions we can characterize as having the prescribed asymptotic behavior of type I. The existence of solutions of type II and III is proved only when N1 + N2 = 1 that is, a single vortex point is prescribed. In particular, we may conclude that at the energy level E = 2π (see (1.19)) there exist SU (3)- periodic vortices for each of the prescribed type I, II and III. More precisely, we obtain the following results. Theorem 1.1. Let Na be a nonnegative integer and Za = {pa1 , . . . , paNa } ⊂ Ω be an assigned set of Na -points (not necessarily distinct) in Ω, a = 1, 2. For k > 0 sufficiently small, there exist two gauge distinct SU (3)-periodic vortex solutions of (1.2) - (1.16) satisfying the ansatz (1.5) - (1.6) and such that: (i) the component φa of the Higgs field satisfies: |φa | < 1 in Ω, and φa vanishes exactly at each paj ∈ Za with the multiplicity given by the repetition of paj in Za , a = 1, 2.

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

7

(ii) The induced ”magnetic flux”-component Φa and ”electric charge”-component Qa , satisfy: 2π 1 Φa = Qa = (2Na + Nb ) a 6= b = 1, 2. k 3 (iii) The energy E satisfies: E = 2π(N1 + N2 ). Furthermore, one of these two solutions is always of type I, in the sense that the components φa(1) (a = 1, 2) of the Higgs field satisfy: (1.21)

|φa(1) | → 1,

as k → 0+ ,

pointwise a.e. in Ω and strongly in Lp (Ω), ∀ p ≥ 1. While, when N1 + N2 = 1, then the other solution is of type II, and the corresponding components φa(2) (a = 1, 2) of the Higgs field satisfy: (1.22)

|φa(2) | → 0,

as k → 0+ , uniformly in Ω. 3

Remark 1.2. It is understood that in case Na = 0 for some a = 1, 2 then the corresponding set of vortex points Za is taken to be the empty set. Furthermore, a bootstrap argument shows that, in fact, the convergence in (1.22) holds C m (Ω)uniformly, for every m ∈ N, and in any other relevant norm. Theorem 1.1 may be considered as the complete analogue of the results on abelian Chern-Simons periodic vortices (corresponding to G = SU (2)) obtained by Caffarelli-Yang [4] and Tarantello [34]. Note in fact that, when G = SU (2), then r = 1, and the restricted gaugepotential V (|φ|) = 4k12 |φ|2 (1 − 2|φ|2 )2 . So in this case only type I and II- vortices are allowed. In order to insure type II - vortices, the restriction on the vortex numbers not to exceed the value 1 appears as a technical condition, it is present also in the abelian case [34]. At the moment, it is not clear how to remove it, even for the simpler abelian situation where only recently some progress has been made in this direction, see [28], [12] and [10]. Concerning the type III-vortices, specific of the SU (3)-theory, we have the following: Theorem 1.3. For each fixed point p ∈ Ω and k > 0 sufficiently small, there exists an SU (3)-periodic-vortex solution for (1.2) - (1.16) satisfying the ansatz (1.5) (1.6) such that for the component φa (a = 1, 2) of the Higgs field the following holds: (i) (first component): |φ1 | < 1; φ1 never vanishes in Ω and, (1.23)

1 |φ1 | → √ , 2

as k → 0+ ,

pointwise a.e. in Ω and strongly in H1 (Ω). The corresponding induced first-component of the ”magnetic field” Φ1 and ”electric charge” Q1 satisfy: Φ1 = k1 Q1 = 2π 3 ; (ii) (second component): |φ2 | < 1; φ2 admits a simple zero at p ∈ Ω and, (1.24)

|φ2 | → 0,

as k → 0+ ,

pointwise a.e. in Ω and strongly in H1 (Ω).

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M. NOLASCO AND G. TARANTELLO

The corresponding induced second-component of the ”magnetic field” Φ2 and ”electric charge” Q2 satisfy: Φ2 = k1 Q2 = 4π 3 . (iii) The energy E = 2π. 3 Remark 1.4. In words, Theorem 1.3 states the existence of a type III - (a) vortex with first vortex number N1 = 0 and second vortex N2 = 1. Due to the complete symmetry of (1.10) with respect to the indices a = 1, 2, we can also claim the existence of a SU (3)-periodic vortex of type III - (b) simply by exchanging the role between the indices. Thus, Theorem 1.3 may be completed with the existence of another SU (3)- periodic vortex whose component φa of the Higgs field satisfy: (i) (first component): |φ1 | < 1; φ1 admits a simple zero at p ∈ Ω and, |φ1 | → 0

(1.25)

as k → 0+ ,

pointwise a.e. in Ω and strongly in H1 (Ω). The corresponding induced first component of the ”magnetic field” Φ1 = ”electric charge” Q1 = 4π 3 k;

4π 3

and

(ii) (second component): |φ2 | < 1; φ2 never vanishes in Ω and, 1 |φ2 | → √ , 2

(1.26)

as k → 0+ ,

pointwise a.e. in Ω and strongly in H1 (Ω). The corresponding induced second component of the ”magnetic field” Φ2 and ”electric charge” Q2 satisfy Φ2 = k1 Q2 = 2π 3 . (iii) The energy E = 2π. For such a solution the corresponding vortex numbers are given by N1 = 1 and N2 = 0. Thus, in case N1 + N2 = 1, we can combine the results above and conclude: Corollary 1.5. For given p1 , p2 ∈ Ω and k > 0 sufficiently small, at the energy level E = 2π there exists a SU (3)-periodic vortex, satisfying (1.2) - (1.16) and the ansatz (1.5) - (1.6), for each of the asymptotic behaviors prescribed by the type I,II and III (a)-(b), as k → 0+ . Furthermore, either the first component of the Higgs field φ1 admits a simple zero at p1 and the second component φ2 never vanishes; or φ1 never vanishes and φ2 admits a simple zero at p2 . 3 Note that, at the moment, it is not available any existence result concerning vortices in R2 with the asymptotic behavior of the type III, as |x| → +∞. To establish the results above, we take advantage of the equation D− φ = 0, which we may write componentwise as follows: (1.27)

∂− ln φa = i

r X

Ab− Kba

a = 1, . . . , r.

b=1

In fact, by virtue of (1.27) we can follow an approach introduced by Taubes ([35]) for the study of self-dual Ginzburg-Landau vortices, and derive from (1.10) a system

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

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of nonlinear elliptic equations for the real variable functions ua = ln |φa |2 (a = 1, 2) of the following form (see also [38]): (1.28) ( Pr Pr PNa ∆ua = − k12 ( b=1 Kab eub − b,c=1 eub Kbc euc Kac ) + 4π j=1 δpaj on Ω ua doubly periodic on ∂Ω, a = 1, . . . , r where the points pa1 , . . . , paNa ∈ Ω are the prescribed zeroes of the scalar fields φa (a = 1, . . . , r) repeated according to their multiplicity. In fact, from each solution ua (a = 1, . . . , r) of (1.28) we may recover, under the ansatz (1.5) and (1.6), the whole vortex-solution for (1.2), by setting: 1

φa (x) = e 2 ua (x)+i Aa1 − iAa2 = −i

(1.29)

r X

P

Na j=1

Arg(x−pa j)

∂− ln φb (K −1 )ba

b=1

r

Aa0 = −

X 1 (|φa |2 − (K −1 )ba ) 2k b=1

−1

where K is the inverse of the Cartan matrix K. Clearly, from (1.29) we have that φa vanishes exactly at each paj with the multiplicity corresponding to the repetition of paj in Za . We shall devote the following sections to the analysis of the elliptic system (1.28) in case G = SU (3). 2. Variational formulation and preliminary results We study the system (1.28) when the gauge groupconsidered  is SU (3). Recalling 2 −1 that the Cartan matrix for SU (3) is given by K = , the system (1.28) −1 2 takes the following form:  PN1 2u1 − 2e2u2 − 2eu1 + eu2 − eu1 +u2 ) + 4π j=1 δp1 on Ω  ∆u1 = λ(4e PN2 j 2u 2u u u u +u 2 1 2 1 1 2 (2.1) ∆u2 = λ(4e − 2e − 2e + e − e ) + 4π j=1 δp2j on Ω   u1 , u2 doubly periodic on ∂Ω. where we have set λ = k12 > 0. Concerning problem (2.1) we shall prove the following results: Theorem 2.1. (a) Problem (2.1) admits no solutions for λ > 0 small. (b) Every solutions (u1 , u2 ) for (2.1) satisfies : (2.2)

eua ≤ 1,

in Ω

(a = 1, 2).

(c) There exists λ0 > 0 sufficiently large such that, ∀λ > λ0 problem (2.1) admits, at least, two distinct solutions, one of which always satisfies: (2.3)

eua → 1,

as λ → +∞

pointwise a.e. in Ω and strongly in Lp (Ω), ∀ p ≥ 1. 3 Clearly, (2.3) insures the existence of a periodic vortex solution of type I. Concerning the existence of type II and III vortices, we have to limit our attention to consider the case where the vortex-numbers (N1 , N2 ) satisfy: N1 + N2 = 1.

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M. NOLASCO AND G. TARANTELLO

Thus, we consider problem (2.1) in the simpler form:

(2.4)

 2u1  − 2e2u2 − 2eu1 + eu2 − eu1 +u2 ) + 4πN1 δp1 ∆u1 = λ(4e ∆u2 = λ(4e2u2 − 2e2u1 − 2eu2 + eu1 − eu1 +u2 ) + 4πN2 δp2   u1 , u2 doubly periodic on ∂Ω

on Ω on Ω

with assigned points pa ∈ Ω, a = 1, 2. We prove: Theorem 2.2. For N1 + N2 = 1 and λ > 0 sufficiently large, problem (2.9) admits − a (weak) solution (u− 1 , u2 ) satisfying: (2.5)

−

eua → 0

as λ → +∞, uniformly in Ω

(a = 1, 2),

(and in any other relevant norm). Furthermore, there always exists a second solution (u∗1 , u∗2 ) such that, (i) if N1 = 0 and N2 = 1, then ∗ 1 , eu 2 → 0 2 pointwise a.e. in Ω and strongly in H1 (Ω);

(2.6)

∗

eu 1 →

as λ → +∞

(ii) if N1 = 1 and N2 = 0, then 1 as λ → +∞ 2 pointwise a.e. in Ω and strongly in H1 (Ω).

(2.7)

∗

eu1 → 0,

∗

eu 2 →

3 It is clear that, by means of the transformations in (1.29), we obtain Theorem 1.1, 1.3 as well as Corollary 1.5 as an immediate conseguence of Theorem 2.1 and 2.2. To establish Theorem 2.1 and 2.2, it will be convenient to distinguish between the singular and regular part of the solutions in (2.1). For this purpose denote by ua0 (a = 1, 2) the unique solution for the problem (see [3]) ( PNa a a ∆ua0 = − 4πN on Ω j=1 δpj |Ω| + 4π R a (2.8) a u =0 u0 doubly periodic on ∂Ω Ω 0 a=1,2 . As well known, ua0 ∈ C ∞ (Ω \ {pa1 . . . paNi }) and if naj is the multiplicity of paj then a ua0 behaves like: ln |x − paj |2nj , as x → paj . a Setting ua = ua0 + va and ha = eu0 we have that, (u1 , u2 ) is a solution for (2.1) if and only if (v1 , v2 ) is a smooth solution for the following system: (2.9)  2 2v1 2 2v2 v1 v2 v1 +v2 1 ) + 4πN on Ω  ∆v1 = λ(4h1 e − 2h2 e − 2h1 e + h2 e − h1 h2 e |Ω| ∆v2 = λ(4h22 e2v2 − 2h21 e2v1 − 2h2 ev2 + h1 ev1 − h1 h2 ev1 +v2 ) +   v1 , v2 doubly periodic on ∂Ω.

4πN2 |Ω|

on Ω

As a preliminary result, we start to derive part (b) of Theorem 2.1.

Proof of (2.2). First of all notice that, in view of (2.8) and (2.9), limx→paj ua = −∞ for a = 1, 2 and j = 1, . . . , Na . Therefore ua attains its maximum value at some point x ¯a ∈ Ω \ {pa1 , . . . , paNa }. Set u ¯a = maxΩ ua = ua (¯ xa ) (a = 1, 2). By symmetry,

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

11

we can assume without loss of generality, that u ¯1 ≥ u ¯2 . Since (u1 , u2 ) is a solution of (2.1), we derive 0 ≥4e2¯u1 − 2e2u2 (¯x1 ) − 2eu¯1 + eu2 (¯x1 ) − eu¯1 +u2 (¯x1 ) ≥ (2.10)

≥2e2¯u1 − 2eu¯1 + eu2 (¯x1 ) − eu¯1 +u2 (¯x1 ) = = 2eu¯1 (eu¯1 − 1) − eu2 (¯x1 ) (eu¯1 − 1) = (2eu¯1 − eu2 (¯x1 ) )(eu¯1 − 1).

Thus, eu¯1 − 1 ≤ 0 and consequently (2.11)

eua (x) ≤ 1

for any x ∈ Ω,

and a = 1, 2. 2 1

1

We notice that (2.9) admits a variational formulation on H (Ω) × H (Ω). Here 1 H1 (Ω) denotes the space of doubly periodic functions v ∈ Hloc (R2 ) with periodic cell domain Ω. It defines an Hilbert space equipped with the standard H 1 (Ω)-scalar product. 1 2 2 We shall R denote Rby k 2· k the usual norm on H (Ω) as given by kvk = k∇vk2 + 2 2 kvk2 = Ω |∇v| + Ω |v| . It is easy to check that (weak) solutions for (2.9) correspond to critical points in H1 (Ω) × H1 (Ω) for the (unbounded) functional (2.12) Z Z 1 ∇v1 · ∇v2 ) + λ W (v1 , v2 )+ Iλ (v1 , v2 ) = (k∇v1 k2 + k∇v2 k2 + 3 Ω Ω Z Z 4π 4π + (2N1 + N2 ) v1 + (2N2 + N1 ) v2 , v1 , v2 ∈ H1 (Ω) 3 3 Ω Ω with W (v1 , v2 ) given by

(2.13)

W (v1 , v2 ) =h21 e2v1 + h22 e2v2 − h1 ev1 − h2 ev2 − h1 h2 ev1 +v2 + 1 = 1 3 = (2h1 ev1 − h2 ev2 − 1)2 + (h2 ev2 − 1)2 = 4 4 1 3 = (2h2 ev2 − h1 ev1 − 1)2 + (h1 ev1 − 1)2 ≥ 0. 4 4

To simplify notations, from now on we shall assume, without loss of generality, that |Ω| = 1. Integrating (2.9) over Ω, we find that any solution (v1 , v2 ) for (2.9) satisfies the following constraint conditions: (2.14) ( R R R R R 1 =0 4 Ω h21 e2v1 − 2 Ω h22 e2v2 − 2 Ω h1 ev1 + Ω h2 ev2 − Ω h1 h2 ev1 +v2 + 4πN λ R 2 2v R 2 2v R R R 4πN2 v2 v1 v1 +v2 2 1 4 Ω h2 e − 2 Ω h1 e − 2 Ω h2 e + Ω h1 e − Ω h1 h2 e + λ = 0. Considering (2.14) together with (2.2), we easly see that (2.1) (or equivalently (2.9)) cannot admit solutions for λ > 0 small. This establishes part (a) of Theorem 2.1. Furthermore R conditions (2.14) R may be more clearly interpreted if we set vi = ci + wi with Ω wi = 0 and ci = Ω vi (i = 1, 2). Indeed, after some simple algebraic manipulation, from (2.14) we get a quadratic system for the variables eci (i = 1, 2) as follows: (2.15) ( R R R 2e2c1 Ω h21 e2w1 − ec1 ( Ω h1 ew1 + ec2 Ω h1 h2 ew1 +w2 ) + 4π 3λ (2N1 + N2 ) = 0 R 2 2w R R 4π 2c2 c2 w2 c1 w1 +w2 2 2e h e − e ( Ω h2 e + e Ω h1 h2 e ) + 3λ (2N2 + N1 ) = 0. Ω 2

12

M. NOLASCO AND G. TARANTELLO

Set E = {w ∈ H1 (Ω) :

Z

w = 0}.

Ω

We shall see that, for each fixed pair (w1 , w2 ) ∈ E × E satisfying: Z Z 32π (2Ni + Nj ) h2i e2wi i 6= j i, j = 1, 2. ( hi ewi )2 ≥ 3λ Ω Ω the system (2.15) admits four distinct solutions (c1 , c2 ). For this purpose, from now on, we take R 2 R 2 h h 32π Ω 1 R R λ> max{(2N1 + N2 ) ; (2N2 + N1 ) Ω 2 2 } 3 ( Ω h1 )2 ( Ω h2 ) (2.16)

and define the set

Aλ = {(w1 , w2 ) ∈ E × E : wi satisfies

(2.17)

(2.16),

i = 1, 2}.

Note that (0, 0) ∈ Aλ . For given (w1 , w2 ) ∈ Aλ , we introduce the smooth functions gi± : [0, +∞) → R (i = 1, 2) defined as follows R R h ew1 + X Ω h1 h2 ew1 +w2 ± Ω 1 R g1 (X) ≡ ± 4 Ω h21 e2w1 qR R R 2 2w1 ( Ω h1 ew1 + X Ω h1 h2 ew1 +w2 )2 − 32π 3λ (2N1 + N2 ) Ω h1 e R 2 ± 4 Ω h1 e2w1 R R (2.18) h2 ew2 + X Ω h1 h2 ew1 +w2 R g2± (X) ≡ Ω ± 4 Ω h22 e2w2 qR R 2 R 2w2 ( Ω h2 ew2 + X Ω h1 h2 ew1 +w2 )2 − 32π 3λ (2N2 + N1 ) Ω h2 e R 2 . ± 4 Ω h2 e2w2 and set

( F + (X) ≡ X − g1+ (g2+ (X)), F ± (X) ≡ X − g1+ (g2− (X)),

F − (X) ≡ X − g1− (g2− (X)) F ∓ (X) ≡ X − g1− (g2+ (X))

It is easy to check that solutions of (2.15) correspond to the zeroes of the smooth functions F ∗ : [0, +∞) → R, with ∗ = +, −, ±, ∓. Notice that gi± (X) > 0 for any X ≥ 0 (i = 1, 2) and so F ∗ (0) < 0 (∗ = +, −, ±, ∓). Moreover, lim gi− (X) = 0

(2.19)

X→+∞

(i = 1, 2).

while gi+ (X) = X

(2.20) Therefore,

R

h1 h2 ew1 +w2 R + o(1), 2 Ω h2i e2wi

Ω

as X → +∞

R ( Ω h1 h2 ew1 +w2 )2 F + (X) = (1 − R 2 2w R 2 2w ) + o(1), X 4 Ω h1 e 1 Ω h2 e 2

For the remaining F ∗ , (∗ = −, ±, ∓) we have

F ∗ (X) = 1 + o(1), as X → +∞. X Therefore, for any choise ∗ = +, −, ±, ∓, it follows: lim F ∗ (X) = +∞,

X→+∞

(i = 1, 2).

as X → +∞.

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

13

and hence, by continuity, F ∗ (X ∗ ) = 0, for some X ∗ > 0 and ∗ = +, −, ±, ∓. Furthermore, for i, j = 1, 2, i 6= j (2.21) R h h ew1 +w2 dgi± ± Ω 1 2 (X) = ±gi (X) q R , R R 2 dX 2wi ( Ω hi ewi + X Ω h1 h2 ew1 +w2 )2 − 32π (2N + N ) h e i j i 3λ Ω and so, F ± and F ∓ are strictly increasing. Moreover, for (w1 , w2 ) ∈ Aλ (i = 1, 2) and X > 0, we have R h h ew1 +w2 g1+ (g2+ (X)) dF + Ω 1 2 (X) = − q R · R R 2 dX 2w1 ( Ω h1 ew1 + g2+ (X) Ω h1 h2 ew1 +w2 )2 − 32π 3λ (2N1 + N2 ) Ω h1 e R h h ew1 +w2 g2+ (X) Ω 1 2 ·q R +1> R R 2 2w2 ( Ω h2 ew2 + X Ω h1 h2 ew1 +w2 )2 − 32π (2N + N ) h e 2 1 2 λ Ω g1+ (g2+ (X)) F + (X) = , X X and, analougously, >1 −

dF − g − (g − (X)) F − (X) (X) > 1 − 1 2 = . dX X X ∗

So, FX is strictly increasing, for ∗ = +, −, and X > 0. In conclusion, for any (w1 , w2 ) ∈ Aλ and ∗ = +, −, ±, ∓ there exists a unique ∗ X ∗ > 0 such that F ∗ (X ∗ ) = 0. Set ec1 (w1 ,w2 ) = X ∗ , and observe that, by the + − strict monotonicity of gi and gi , i = 1, 2 (see (2.21)), there exists a unique c∗i = c∗i (w1 , w2 ) (i = 1, 2; ∗ = +, −, ±, ∓) satisfying, +

+

ec2 = g2+ (ec1 );

±

±

ec2 = g2− (ec1 );

∓

∓

ec2 = g2+ (ec1 );

−

−

ec2 = g2− (ec1 ).

ec1 = g1+ (ec2 ), (2.22)

ec1 = g1+ (ec2 ), ec1 = g1− (ec2 ), ec1 = g1− (ec2 ),

+

+

±

±

∓

∓

−

−

Consequently, for given (w1 , w2 ) ∈ Aλ , setting vi∗ = wi + c∗i (w1 , w2 ), i = 1, 2, ∗ = +, −, ±, ∓ we have that the pair (v1∗ , v2∗ ) satisfy (2.14). Our goal will R be to deriveRsolutions (v1 , v2 ) for (2.9) which decompose as vi = wi + ci with Ω wi = 0, ci = Ω vi and ci = c∗i (w1 , w2 ) (i = 1, 2) with prescribed ∗ to coincide with either +, −, ± or ∓. This will yield to solutions of (2.9) with specific asymptotic behavior as λ → +∞. Note that, by the complete symmetry of the problem, the case ∗ = ± and ∗ = ∓ are similar in nature. So, we shall limit our attention to the case ∗ = ± with the understanting that, by changing the role between the indices, analogous considerations hold also when ∗ = ∓. We start with the following: Lemma 2.3. For every (w1 , w2 ) ∈ Aλ we have + R • (i) eci Ω hi ewi ≤ 1 for i = 1, 2; − R • (ii) eci Ω hi ewi ≤ 8π j; λ (2Ni + Nj ) for i = 1, 2, i 6= R ± R c± 2 • (iii) ec1 Ω h1 ew1 ≤ 21 + 16π (2N + N ) and e h ew2 ≤ 8π 2 1 3λ λ (2N2 + N1 ) . Ω 2 Proof. (i) In view of (2.18), from (2.22) for i, j = 1, 2 i 6= j we have R + R h ewi + ecj Ω h1 h2 ew1 +w2 c+ Ω i i R e ≤ . 2 Ω h2i e2wi

14

M. NOLASCO AND G. TARANTELLO

Iterating such an inequality, by means of H¨older inequality, we get, e

c+ 1

R R Z Z h e w1 h h ew1 +w2 w2 c+ Ω 1 Ω 1 2 R 1 R R ≤ + ( h2 e + e h1 h2 ew1 +w2 ) ≤ 2 Ω h21 e2w1 4 Ω h21 e2w1 Ω h22 e2w2 Ω Ω R R R h1 h2 ew1 +w2 Ω h2 ew2 h1 e w 1 1 + Ω Ω R 2 R 2 + ec1 . ≤ R 2 2w + 2w 2w 1 1 2 4 2 Ω h1 e 4 Ω h1 e h e Ω 2 +

By symmetry, an analogous estimate holds for ec2 . Hence, (2.23) R R R  R  Z h h ew1 +w2 Ω h1 ew1 Ω h2 ew2 + 4 ( Ω hi ewi )2 Ω 1 2R R 2 R eci hi e w i ≤ + , 3 2 Ω hi e2wi 4 Ω h21 e2w1 Ω h22 e2w2 Ω

i = 1, 2,

and, using H¨ older inequality, we derive the desired estimate.

To obtain (ii), we use again (2.22) together with (2.18). Thus, for i = 1, 2, i 6= j we have (2.24)

8π(2Ni + Nj ) 1 R − R c 3λ w j h e i +e h h ew1 +w2 Ω i Ω 1 2

−

eci ≤

and (ii) easly follows. As above, to obtain (iii), note that from (2.22) and (2.18), we have (2.25) and so,

±

ec2 ≤

8π(2N2 + N1 ) 1 , R ± R c 3λ h ew2 + e 1 Ω h1 h2 ew1 +w2 Ω 2 ±

ec2 ≤

(2.26) On the other hand, (2.27)

e

c± 1

≤

R

Ω

8π(2Ni + Nj ) R . λ Ω h2 e w 2

± R h1 ew1 + ec2 Ω h1 h2 ew1 +w2 R , 2 Ω h21 e2w1

while from (2.25) we also have, (2.28)

±

ec2 ≤

8π(2N2 + N1 ) 1 . ± R c 3λ e 1 Ω h1 h2 ew1 +w2

Furthermore, from (2.22) and (2.18) it is easy to check that (2.29)

±

ec1 ≥

R h e w1 RΩ 12 . 4 Ω h1 e2w1

Combining (2.28) with (2.29), from (2.27) we get R R ± R Z ( Ω h1 ew1 )2 ec2 Ω h1 h2 ew1 +w2 Ω h1 ew1 c± w1 1 R e h1 e ≤ R 2 2w + ≤ 2 Ω h1 e 1 2 Ω h21 e2w1 Ω R R ( h1 ew1 )2 4π(2N2 + N1 ) Ω h1 ew1 (2.30) ≤ RΩ 2 2w + ≤ ± R 2 Ω h1 e 1 3λec1 Ω h21 e2w1 R ( Ω h1 ew1 )2 16π(2N2 + N1 ) ≤ R 2 2w + , 3λ 2 Ω h1 e 1 and the desired estimate follows by H¨older inequality.



VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

15

R Remark 2.4. Note that, by Jensen’s inequality, Ω hi ewi ≥ 1 (i = 1, 2), which combined with the estimates in lemma 2.3, gives in particular that, − 1 eci ≤ O( ), i = 1, 2; λ (2.31) ± ± 1 1 1 ec1 ≤ + O( ) and ec2 ≤ O( ). 2 λ λ This suggests that solutions of (2.9) with the prescribed asymptotic behavior of type I, II and III (see the Introduction) should correspond to those with mean-values as − ± given by c+ i , ci and ci , respectively. With this aim, we consider the functionals Jλ+ , Jλ− , Jλ± and Jλ∓ defined on Aλ and obtained by inserting the contraint (2.22) into Iλ . More precisely, for (w1 , w2 ) ∈ Aλ , define +

eci ≤ 1,

Jλ∗ (w1 , w2 ) = Iλ (w1 + c∗1 (w1 , w2 ), w2 + c∗2 (w1 , w2 )),

with ∗ = +, −, ±, ∓.

Using (2.15) we find: (2.32) Z 1 2 2 ∇w1 · ∇w2 )+ = (k∇w1 k + k∇w2 k + 3 Ω Z Z ∗ ∗ λ λ + (1 − ec1 h1 ew1 ) + (1 − ec2 h2 ew2 ) − 4π(N1 + N2 )+ 2 Ω 2 Ω 4π 4π (2N1 + N2 )c∗1 + (2N2 + N1 )c∗2 , + 3 3 with ∗ = +, −, ± and ∓. Remark 2.5. It is easy to check that Jλ∗ is Frech´et differentiable in the interior of Aλ . Moreover if (w1 , w2 ) (in the interior of Aλ ) is a critical point for Jλ∗ , then (w1 + c∗1 , w2 + c∗2 ) defines a critical point for Iλ . Concerning the existence of critical points for Jλ∗ , we will show that such a functional is bounded below and attains its infimum on Aλ . However, only when ∗ = +, we can prove that the corresponding minimum point belongs to the interior of Aλ , for λ > 0 large (see Proposition 3.4 below). In the other cases ∗ = −, ± and ∓, we can prove that this occurs only when N1 + N2 = 1 (see section 5). It is an interesting open question to know what happens for the remaining cases. Jλ∗ (w1 , w2 )

3. Multivortex solutions of the I type In this section we analyze the functional Jλ+ , and the corresponding minimization problem in Aλ . We start with a preliminary lemma already derived in [27]: Lemma 3.1. If (w1 , w2 ) ∈ Aλ then ∀τ ∈ (0, 1] we have Z Z 1−τ 1 3λ wi τ (3.1) hi e ≤ ( ) ( hτi eτ wi ) τ 32π(2N + N ) i j Ω Ω

i, j = 1, 2

i 6= j.

1 Proof. Let τ ∈ (0, 1) and let a = 2−τ so that τ a + 2(1 − a) = 1. By interpolation inequality we have Z Z Z hi ewi ≤( hτi eτ wi )a ( h2i e2wi )1−a ≤ Ω Ω Z ZΩ 3λ τ τ wi a ≤( hi e ) ( )1−a ( hi ewi )2(1−a) i 6= j. 32π(2Ni + Nj ) Ω Ω

16

M. NOLASCO AND G. TARANTELLO

Consequently, for i 6= j, i, j = 1, 2 Z ( hi ewi )2a−1 ≤ ( Ω

Z 3λ )1−a ( hτi eτ wi )a , 32π(2Ni + Nj ) Ω

that is, Z

hi e

wi

Ω

Z 1−τ 1 3λ τ ≤( ) ( hτi eτ wi ) τ . 32π(2Ni + Nj ) Ω 

In view of lemma 3.1 we derive the coerciveness of Jλ+ as follows: Proposition 3.2. For λ > 0 sufficiemtly large, there exist constants α, C > 0 (independent of λ) such that Jλ+ (w1 , w2 ) ≥ α(k∇w1 k2 + k∇w2 k2 ) − C(ln λ + 1)

(3.2)

for all (w1 , w2 ) ∈ Aλ . Moreover, Jλ+ attains its infimum on Aλ . Proof. From (2.18) and (2.22) it follows immediately that R hi e w i c+ i e ≥ RΩ 2 2w i = 1, 2. 4 Ω hi e i Hence, from (2.16) we find a suitable constant C > 0, independent of λ, such that for every (w1 , w2 ) ∈ Aλ , we have Z + (3.3) ci (w1 , w2 ) ≥ −ln λ − ln hi ewi − C, i = 1, 2. Ω

Whence, using lemma 3.1, for fixed τ ∈ (0, 1) and any (w1 , w2 ) ∈ Aλ , we obtain: Z 2 X 4π 1X + 2 k∇wi k2 − (2Ni + Nj ) ln hi e w i − Jλ (w1 , w2 ) ≥ 6 i=1 3 Ω i6=j=1,2

2

4π 1X (2Ni + Nj ) ln λ − C ≥ k∇wi k22 − 3 6 i=1 i6=j=1,2 Z X 4π 1−τ i 1 3λ − (2Ni + Nj ) ln[( ) τ ( eτ (wi +u0 ) ) τ ]− 3 16π(2Ni + Nj ) Ω

−

X

i6=j=1,2

−

X

i6=j=1,2

≥

2 1X

6

k∇wi k22

i=1

−

4π (2Ni + Nj ) ln λ − C ≥ 3 −

X

i6=j=1,2

X

i6=j=1,2

4π (2Ni + Nj ) (ln 3τ

Z

eτ wi + max ui0 )−

Ω

Ω

4π 1 ( )(2Ni + Nj )ln λ − C, 3 τ

for some constant C > 0 independent of λ. Recall that, by Moser-Trudinger’s inequality [3](see [16], [11] and [27], for alternative proofs) we have that (3.4)

Z Ω

ew ≤ C exp(

1 k∇wk22 ), 16π

∀w ∈ E,

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

17

with C a positive constant depending only on Ω. Thus, for any (w1 , w2 ) ∈ Aλ , we obtain 1 X τ (2Ni + Nj ) Jλ+ (w1 , w2 ) ≥ (1 − )k∇wi k22 − 6 2 i6=j=1,2 (3.5) 4π X (2Ni + Nj ) − ln λ − Cτ , 3 τ i6=j=1,2

with Cτ > 0 a suitable constant independent of λ. Hence, it sufficies to take 0 < τ < maxi6=j=1,22 (2Ni +Nj ) in (3.5), to derive (3.2) and conclude that Jλ+ is coercive on Aλ . Since Jλ+ is weakly lower semicontinuous on the weakly closed set Aλ , we immediately conclude that the infimum of Jλ+ is attained on Aλ .  Our next goal is to prove that, for λ sufficiently large, such a minimum point lies in the interior of Aλ . To this purpose we will estimate the functional Jλ+ on the boundary ∂Aλ of Aλ . Lemma 3.3. For λ > 0 sufficiently large, (3.6)

inf (w1 ,w2 )∈∂Aλ

Jλ+ (w1 , w2 ) ≥

√ λ − C( λ + ln λ + 1), 2

with C > 0 a suitable constant independent of λ. Proof. For (w1 , w2 ) ∈ ∂Aλ , we have that the identity Z Z 32π (3.7) ( hi ewi )2 = (2Ni + Nj ) h2i e2wi 3λ Ω Ω necessarily holds for i = 1 for i = 2. Using (3.7) into (2.23), Z + ec2 h2 e w 2 ≤ Ω (3.8)

i 6= j

or 2. Without loss of generality assume that (3.7) holds by means of H¨older’s inequality, we find ! R R R h1 e w 1 Ω h2 e w 2 4 ( Ω h2 ew2 )2 Ω R + R 2 ≤ 1 R 1 3 2 Ω h22 e2w2 4( Ω h1 e2w1 ) 2 ( Ω h22 e2w2 ) 2

≤ C(

1 1 + √ ), λ λ

for some constant C > 0 independent of λ. Hence, for λ > 0 sufficiently large, we have Z + 2C (3.9) ec2 h2 e w 2 ≤ √ . λ Ω Using (2.32) with ∗ = +, the arguments of Proposition 3.2 with τ = maxi6=j=1,22 (2Ni +Nj ) , lemma 2.3-(i) and (3.9), we get Z Z + + λ λ Jλ+ (w1 , w2 ) ≥ (1 − ec2 h2 ew2 )− (1 − ec1 h1 ew1 ) + 2 Ω 2 Ω 2π − max (2Ni + Nj )2 ln λ − C ≥ 3 i,j=1,2;i6=j √ λ ≥ − C( λ + ln λ + 1), 2 for any (w1 , w2 ) ∈ ∂Aλ , with C > 0 a suitable constant independent of λ.



18

M. NOLASCO AND G. TARANTELLO

In order to find suitable test-functions in the interior of Aλ where the reverse estimate in (3.6) holds, we recall here some results obtained in [34] concerning the Abelian Chern-Simons-Higgs equation. In [34] (Proposition 3.1) for µ > 0 sufficiently large there exist R is proved R that v¯µi = c¯iµ + w ¯µi , with c¯iµ = Ω v¯µi and Ω w ¯µi = 0 (i = 1, 2) solution of ( i i ∆v = µev+u0 (ev+u0 − 1) + 4πNi (3.10) v ∈ H 1 (Ω), such that ui0 + v¯µi < 1 in Ω, c¯iµ → 0 and w ¯µi → −ui0 pointwise a.e., as µ → +∞ i (i = 1, 2). Since hi = eu0 ∈ L∞ (Ω), by dominated convergence, we have that, i hi ew¯µ → 1 strongly in Lp (Ω) for any p ≥ 1. In particular, Z 1 2 (3.11) h1 h2 ew¯µ +w¯µ → 1, as µ → +∞. Ω

Hence, for fixed λ0 > 0 large and  ∈ (0, 1), we can find µ > 0 sufficiently large to insure that, setting w ¯i, = w ¯µi  (i = 1, 2), we have (w ¯1, , w ¯2, ) ∈ Aλ for every λ ≥ λ0 and R 2 Ω h2j e2w¯j, + 1 R R (3.12) max > 1 − . j=1,2 4 h2 e2w¯1, Ω h22 e2w¯2, − 1 Ω 1 R R Recalling that, by Jensen’s inequality, Ω h1 h2 ew¯1, +w¯2, ≥ 1 and Ω h2j e2w¯j, ≥ R ( Ω hj e2w¯j, )2 ≥ 1, j = 1, 2, by means of (2.22) and (2.18) we obtain, (3.13) e

c+ ¯1, ,w ¯2, ) i (w

R R + ( Ω hi ew¯1, + ecj (w¯1, ,w¯2, ) Ω h1 h2 ew¯1, +w¯1, ) R ≥ × 4 Ω h2i e2w¯i, s ! R 2 2w ¯i, h e 32π i × 1+ 1− (2Ni + Nj ) RΩ ≥ 3λ ( Ω hi ew¯i, )2 R + h ew¯i, 1 + ecj (w¯1, ,w¯2, ) R 2 RΩ i2 ≥ + × 2 Ω hi e2w¯i, 4 Ω hi e2w¯i, s ! R 2 hi e2w¯i, 32π Ω × 1− (2Ni + Nj ) R −1 ≥ 3λ ( Ω hi ew¯i, )2 +

1 + ecj (w¯1, ,w¯2, ) 8π(2Ni + Nj ) R 2 ≥ − , 2 w ¯ i, 3λ 2 Ω hi e

for i, j = 1, 2 and i 6= j.

+ Thus, setting c+ ¯1, , w ¯2, ) i = 1, 2, an iteration of the estimate above i, = ci (w yields to + + 1 1 eci, ≥ R 2 2w¯i, + R 2 2w¯1, R 2 2w¯2, (1 + eci, )+ 2 Ω hi e 4 Ω h1 e h e Ω 2

1 8π(2Nj + Ni ) 8π(2Ni + Nj ) − ≥ R 2 2w¯i, − ≥ 3λ 3λ 2 Ω hi e R + 2 Ω h2j e2w¯j, + 1 eci, 4π(4Nj + 5Ni ) ≥ R 2 2w¯1, R 2 2w¯2, + R 2 2w¯1, R 2 2w¯2, − , 3λ 4 Ω h1 e h e 4 Ω h1 e h e Ω 2 Ω 2

for i 6= j = 1, 2. Consequently, for i 6= j = 1, 2,

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

e

c+ i,

19

R 2 Ω h2j e2w¯j, + 1 4π(4Nj + 5Ni ) ≥ R 2 2w¯1, R 2 2w¯2, − . 9λ 4 Ω h1 e h e −1 Ω 2

In view of (3.12), we conclude that +

eci, ≥ 1 −  −

4π max Nj λ j=1,2

i = 1, 2,

that gives +

(1 − eci,

(3.14)

Z

hi ew¯i, ) ≤  +

Ω

4π max Nj , λ j=1,2

i = 1, 2

for all λ ≥ λ0 . Now, we are ready to prove: Proposition 3.4. For λ > 0 sufficiently large, (3.15)

inf (w1 ,w2 )∈∂Aλ

Jλ+ (w1 , w2 ) >

inf (w1 ,w2 )∈Aλ

Jλ+ (w1 , w2 ).

Proof. Fix  ∈ (0, 21 ) and consider (w ¯1, , w ¯2, ) satisfying (3.14) for λ ≥ λ0 . + Since ci ≤ 0, i = 1, 2 (see remark 2.4), by (2.32) (with ∗ = +) and (3.14), we have: Jλ+ (w ¯1, , w ¯2, ) ≤ k∇w ¯1, k2 + k∇w ¯2, k2 + λ ≤ C + λ,

(3.16)

with C > 0 a suitable constant depending on  only. Comparing with lemma 3.3, it follows inf (w1 ,w2 )∈∂Aλ

Jλ+ (w1 , w2 ) −

inf (w1 ,w2 )∈Aλ

Jλ+ (w1 , w2 ) ≥

inf (w1 ,w2 )∈∂Aλ

Jλ+ (w1 , w2 )−

√ 1 − Jλ+ (w ¯1, , w ¯2, ) ≥ λ( − ) − C( λ + lnλ) − C → +∞, 2 as λ → +∞, and the proposition is proved.  ¿From proposition 3.2 and 3.4 we may conclude that, for λ > 0 sufficiently large and (N1 , N2 ) ∈ N × N, there exists (w1,λ , w2,λ ) in the interior of Aλ where Jλ+ attains its infimum. Consequently, (Jλ+ )0 (w1,λ , w2,λ ) = 0, and ( + v1,λ = w1,λ + c+ 1 (w1,λ , w2,λ ) (3.17) + v2,λ = w2,λ + c+ 2 (w1,λ , w2,λ ) defines a critical point for Iλ , namely a (weak) solution of (2.9). Next we prove that, as in the Abelian Chern-Simons-Higgs theory (see [34]), the + + solution characterized by the choice of the “plus” sign in (2.18), namely (v1,λ , v2,λ ), give rise to a periodic vortex of type I. More precisely, we prove + + Proposition 3.5. For λ sufficiently large, let (v1,λ , v2,λ ) be the solution of (2.9) as given by (3.17), then, i

+

• eu0 +vi,λ → 1 as λ → +∞, pointwise a.e. in Ω and in Lp (Ω), ∀ p ≥ 1 (i = 1, 2). In order to prove proposition 3.5 we start with the following lemma: + + Lemma 3.6. Let (v1,λ , v2,λ ) be given by (3.17), then, Z + + (3.18) W (v1,λ , v2,λ ) → 0, as λ → +∞. Ω

20

M. NOLASCO AND G. TARANTELLO

Proof. In view of (3.16) we may conclude that ∀ > 0 ∃λ > 0 and C > 0 such that ∀λ ≥ λ we have inf Jλ+ ≤ λ + C .

(3.19)

Aλ

On the other hand, following the argument of Proposition 3.2 (with τ = 2 (maxi6=j=1,2 (2Nj + Ni ))−1 ) we obtain Z 4π + + + + (3.20) inf Jλ+ = Jλ+ (w1,λ , w2,λ )≥λ W (v1,λ , v2,λ )− max(2Ni + Nj )2 ln λ − C, Aλ 3 i6=j Ω for C > 0 a suitable constant independent of λ. So putting together (3.19) and (3.20), we obtain that Z + + lim sup W (v1,λ , v2,λ ) ≤ , ∀ > 0 λ→+∞

Ω

and the conclusion follows.  Proof of Proposition 3.5. +

Recalling (2.13), by lemma 3.6 we have hi evi,λ → 1 in L2 (Ω) as λ → +∞ and + + i = 1, 2. Since (v1,λ + u10 , v2,λ + u20 ), is a solution of (2.1), by (2.2), we have that +

i

evi,λ +u0 ≤ 1,

in Ω

i = 1, 2.

vi+ +ui0

Hence, e → 1 pointwise a.e., and, by dominated convergence, strongly in Lp (Ω), ∀ p ≥ 1, as λ → +∞. 2 We conclude this section by observing that the type I multivortex solution + + (v1,λ , v2,λ ) can be characterized variationally as follows: + + + + Lemma 3.7. Let (v1,λ , v2,λ ) be given by (3.17). Then (v1,λ , v2,λ ) is a local minimum for the functional Iλ .

Proof. For any (w1 , w2 ) ∈ Aλ observe that (3.21)

+ ∂ci Iλ (w1 + c+ 1 (w1 , w2 ), w2 + c2 (w1 , w2 )) = 0,

i = 1, 2.

Moreover, for i = 1, 2, i 6= j we have Z ∂c22 Iλ (w1 + c1 , w2 + c2 ) =λ (4h2i e2(wi +ci ) − hi ewi +ci − h1 h2 ec1 +c2 ew1 +w2 ) i Ω Z 2 ∂c1 c2 Iλ (w1 + c1 , w2 + c2 ) = − λ h1 h2 ec1 +c2 ew1 +w2 . Ω

In view of (2.18) we have (3.22) Z Z + + + + 1 32π vi,λ v1,λ +v2,λ + + 2 2 (2Ni + Nj ) h2i e2vi,λ ) 2 ∂c2 Iλ (v1,λ , v2,λ ) = λ(( hi e + h1 h2 e ) − i 3λ Ω Ω and, since (w1,λ , w2,λ ) lies in the interior of Aλ , Z + + + + 2 ∂c2 Iλ (v1,λ , v2,λ ) > λ h1 h2 ev1,λ +v2,λ i

i = 1, 2.

Ω

+ + Therefore, at the point (v1,λ , v2,λ ), the Hessian matrix of Iλ (w1 + c1 , w2 + c2 ) w.r.t. the variables (c1 , c2 ) is strictly positive defined. P Let vi = wi + ci (i = 1, 2), by + continuity, there exists δ > 0 such that for any i=1,2 kvi − vi,λ k ≤ δ, we have (w1 , w2 ) ∈ Aλ and

(3.23)

+ + Iλ (v1 , v2 ) ≥ Iλ (w1 + c+ 1 (w1 , w2 ), w2 + c2 (w1 , w2 )) = Jλ (w1 , w2 ).

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

21

Therefore, (3.24)

Iλ (v1 , v2 ) ≥ Jλ+ (w1 , w2 ) ≥

inf (w1 ,w2 )∈Aλ

+ + Jλ+ (w1 , w2 ) = Iλ (v1,λ , v2,λ ),

+ + , v2,λ ) defines a local minimum for Iλ . and so (v1,λ

 4. The Mountain Pass solution In this section we prove the existence of a second solution of “mountain pass” type and obtain the proof of Theorem 2.1. We start to show that the functional Iλ satisfies the compactness condition of Palais-Smale. Lemma 4.1. Let {(v1,n , v2,n )} be a sequence in H1 (Ω) × H1 (Ω) satisfying: (1) Iλ (v1,n , v2,n ) → α as n → +∞, (2) kIλ0 (v1,n , v2,n )k → 0 as n → +∞, then {(v1,n , v2,n )} admits a convergent subsequence in H1 (Ω) × H1 (Ω). R R Proof. Set vi,n = wi,n + ci,n (i = 1, 2), where Ω wi,n = 0 and ci,n = Ω vi,n . For any (ψ1 , ψ2 ) ∈ H1 (Ω) × H1 (Ω), (4.1) Z Z 2 ∇w2,n ∇ψ2 )+ Iλ0 (v1,n , v2,n )[ψ1 , ψ2 ] = ( ∇w1,n ∇ψ1 + 3 Ω Ω Z Z 1 + ( ∇w1,n ∇ψ2 + ∇w2,n ∇ψ1 )+ 3 Ω Ω Z 4π + λ (2h21 e2v1,n − h1 ev1,n − h1 h2 ev1,n +v2,n + (2N1 + N2 ))ψ1 + 3 Ω Z 4π + λ (2h22 e2v2,n − h2 ev2,n − h1 h2 ev1,n +v2,n + (2N2 + N1 ))ψ2 . 3 Ω Choosing ψi = 1 and ψj = 0 (i, j = 1, 2; i 6= j) in (4.1) we get, as n → +∞, Z 4π (4.2) |λ (2h2i e2vi,n − hi evi,n − h1 h2 ev1,n +v2,n ) + (2Ni + Nj )| ≤ o(1). 3 Ω By (2.13), it implies: Z Z Z v1,n 2λ W (v1,n , v2,n ) + λ h1 e +λ h2 ev2,n ≤ 2λ + o(1). Ω

Ω

Ω

Hence, as n → +∞, we have Z W (v1,n , v2,n ) ≤ 1 + o(1); Ω Z (4.3) hi evi,n ≤ 2 + o(1) i = 1, 2. Ω

¿From (4.3), and Jensen’s inequality, we also get eci,n ≤ 2 + o(1),

(4.4)

as n → +∞.

Furthermore, using (2.13) together with (4.3) we derive, as n → +∞, that Z (4.5) (hi evi,n − 1)2 ≤ 2 + o(1), i = 1, 2 Ω

(4.6)

Z

h2i e2vi,n ≤ C,

Ω

with C > 0 a suitable constant.

i = 1, 2,

22

M. NOLASCO AND G. TARANTELLO

Set (w1,n + w2,n )+ = max{w1,n + w2,n , 0} and take (4.1) with ψ1 = ψ2 = (w1,n + w2,n )+ then (4.7) Iλ0 (v1,n ,v2,n )[ψ1 , ψ2 ] = k∇(w1,n + w2,n )+ k2 + Z + λ (2h21 e2v1,n + 2h22 e2v2,n − 4h1 h2 ev1,n +v2,n )(w1,n + w2,n )+ + Ω Z X Z + 2λ h1 h2 ev1,n +v2,n (w1,n + w2,n )+ − λ hi evi,n (w1,n + w2,n )+ + Ω

i=1,2

Z

Ω

Z

+ 4π(N1 + N2 ) (w1,n + w2,n )+ ≥ 2λ h1 h2 ev1,n +v2,n (w1,n + w2,n )+ − Ω Ω Z X Z 1 + 2 12 − λ( ((w1,n + w2,n ) ) ) ( h2i e2vi,n ) 2 . Ω

i=1,2

Ω

Therefore, by (4.6) and assumption (2) we get, (4.8) Z h1 h2 ev1,n +v2,n (w1,n + w2,n )+ ≤ Ck(w1,n + w2,n )+ k2 + n k(w1,n + w2,n )+ k ≤ Ω

≤ C(k∇w1,n k2 + k∇w2,n k2 ), with n → 0 as n → +∞ and C > 0 a suitable constant independent of n ∈ N. Note that in (4.8) we have used the well known estimate: k(w1,n + w2,n )+ k2 ≤k(w1,n + w2,n )+ k ≤ kw1,n + w2,n k2 ≤ ≤C0 (k∇w1,n k2 + k∇w2,n k2 ), with a suitable constant C0 > 0 (independent of n ∈ N). Now, take ψ1 = w1,n , ψ2 = w2,n in (4.1) and use (4.6), (4.8), and Poincar´e inequality to derive 1 Iλ0 (v1,n , v2,n )[w1,n , w2,n ] ≥ (k∇w1,n k2 + k∇w2,n k2 )+ 3 Z X 2 +λ (2h2i e2vi,n − hi evi,n )wi,n − Ω i=1

−λ

Z

h1 h2 ev1,n +v2,n (w1,n + w2,n )+ ≥

Ω 2 Z X 1 2 2 ≥ (k∇w1,n k +k∇w2,n k ) + 2λ h2i e2ci,n (e2wi,n − 1)wi,n + 3 Ω i=1 2 Z 2 Z X X 1 +2λ h2i e2ci,n wi,n − λ ( h2i e2vi,n ) 2 kwi,n k2 − i=1

Ω

i=1

Ω

X 1 X −C(k∇w1,n k2 +k∇w2,n k2 ) ≥ k∇wi,n k22 − C k∇wi,n k2 , 3 i=1,2 i=1,2 where we used that (e2wi,n − 1)wi,n ≥ 0, i = 1, 2 a.e. in Ω. Hence, by assumption (2), we conclude that k∇w1,n k and k∇w2,n k are bounded sequences. Moreover, from assumption (1), Iλ (v1,n , v2,n ) is bounded below uniformly on n ∈ N and we get a constant C > 0, independent of n ∈ N, such that

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

23

1 4π(N1 + N2 ) min{c1,n , c2,n } ≥ − (k∇w1,n k2 + k∇w2,n k2 )− 6Z −λ W (v1,n , v2,n ) + Iλ (v1,n , v2,n ) ≥ −C,

(4.9)

Ω

that is, the sequence ci,n (i = 1, 2) is also bounded from below. Therefore, after passing to a subsequence, we get (4.10)

vi,n * v¯i ,

(i = 1, 2)

as n → +∞,

weakly in H1 (Ω), strongly in Lp (Ω), p ≥ 1 and R pointwise a.e. in Ω. Moreover, evi,n → ev¯i strongly in Lp (Ω) p ≥ 1, and ci,n → Ω v¯i = c¯i . Consequently, for any (ψ1 , ψ2 ) ∈ H1 (Ω) × H1 (Ω) we derive Iλ0 (v1,n , v2,n )[ψ1 , ψ2 ] → Iλ0 (¯ v1 , v¯2 )[ψ1 , ψ2 ] = 0,

(4.11)

namely (¯ v1 , v¯2 ) defines a critical point for Iλ . In order to obtain strong convergence in H1 (Ω)×H1 (Ω) we choose ψ1 = v1,n − v¯1 and ψ2 = v2,n − v¯2 into (4.1). By assumption (2) and (4.11), we obtain |(Iλ0 (v1,n , v2,n )−Iλ0 (¯ v1 , v¯2 ))[v1,n − v¯1 , v2,n − v¯2 ]| ≤

(4.12)

≤n (kv1,n − v¯1 k + kv2,n − v¯2 k) = o(1),

as n → +∞. Consequently, 2 2 Z X 1X 2 k∇(wi,n − w ¯i )k ≤ − 2λ h2i (e2vi,n − e2¯vi )(vi,n − v¯i )+ 3 i=1 Ω i=1 2 Z X +λ hi (evi,n − ev¯i )(vi,n − v¯i )+ i=1

+λ

Z

Ω

h1 h2 (ev1,n +v2,n − ev¯1 +¯v2 )(v1,n + v2,n − (¯ v1 + v¯2 )) + o(1) = o(1),

Ω

as n → +∞, and the desired conclusion follows.



Proof of Theorem 2.1-(c) In section 3 we proved, for λ > 0 sufficiently large, the existence of a solution of problem (2.9) with the desired asymptotic behavior (2.3) as λ → +∞ (see Propo+ + sition 3.5). Moreover, in lemma 3.7, we have shown that such a solution (v1,λ , v2,λ ) defines a local minimum for Iλ , namely X + + + (4.13) ∃ δ0 > 0 : Iλ (v1 , v2 ) ≥ Iλ (v1,λ , v2,λ ), provided kvi − vi,λ k ≤ δ0 . i=1,2

In order to find a second solution for (2.9), we observe that Iλ admits a “mountain pass” structure. In fact, there exists a constant Cλ > 0 (depending only on λ) such that for c ∈ R we have: 4π + + + (4.14) Iλ (v1,λ , w2,λ + c) − Iλ (v1,λ , v2,λ ) ≤ Cλ + (2N2 + N1 )c. 3 We distinguish two cases. + + (i) If (v1,λ , v2,λ ) is not a strict local minimum for Iλ , namely (4.15)

∀ 0 < δ < δ0

P

inf

+ i=1,2 kvi −vi,λ k=δ

+ + Iλ = Iλ (v1,λ , v2,λ ),

then, by an application of Ekeland’s lemma (see 1.6 ), we obtain P [17], corollary + δ δ δ a local minimum (v1,λ , v2,λ ) for Iλ , such that i=1,2 kvi,λ − vi,λ k = δ for every

24

M. NOLASCO AND G. TARANTELLO

δ ∈ (0, δ0 ). Therefore, in this case, we find a one-parameter family of (weak) solutions for (2.9). Otherwise, + + (ii) if (v1,λ , v2,λ ) is a strict local minimum for Iλ , then ∃ δ1 ∈ (0, δ0 ) :

(4.16)

P

inf

+ i=1,2 kvi −vi,λ k=δ1

+ + Iλ (v1 , v2 ) > Iλ (v1,λ , v2,λ ).

Moreover, in view of (4.14), there exists c¯ < 0 such that |¯ c − c+ 2 (w1,λ , w2,λ )| > δ1 and + + + Iλ (v1,λ , w2,λ + c¯) < Iλ (v1,λ , v2,λ ).

(4.17)

We introduce the class of paths + + + Γλ = {γ ∈ C([0, 1], H1 (Ω) × H1 (Ω)) : γ(0) = (v1,λ , v2,λ ); γ(1) = (v1,λ , w2,λ + c¯)}

and define + + αλ = inf max Iλ (γ(t)) > Iλ (v1,λ , v2,λ ). γ∈Γ t∈[0,1]

In view of lemma 4.1, (4.16) and (4.17), we can apply the Mountain Pass theorem of Ambrosetti-Rabinowitz [2] to conclude that αλ defines a critical level for Iλ . Namely, there exists (¯ v1,λ , v¯2,λ ) ∈ H1 (Ω) × H1 (Ω) such that Iλ0 (¯ v1,λ , v¯2,λ ) = 0

+ + Iλ (¯ v1,λ , v¯2,λ ) = αλ > Iλ (v1,λ , v2,λ ).

and

Hence (¯ v1,λ , v¯2,λ ) defines a (weak) solution of (2.9) distinct from the local minimum + + (v1,λ , v2,λ ). 2 5. Vortex solutions of the II & III-type In this section we are going to establish Theorem 2.2, by proving the existence of one-vortex solutions from minima on Aλ of the functionals Jλ− , Jλ± and Jλ∓ (see remark 2.5). We start to discuss the minimization problem for Jλ− . To this purpose we prove a preliminary lemma: Lemma 5.1. There exists a constant C > 0, independent of λ, such that for any (w1 , w2 ) ∈ Aλ −

eci ≥

(5.1)

C λ Ω h i ew i R

i = 1, 2.

Proof. By symmetry, it sufficies to show (5.1) for i = 1. ¿From (2.22) and (2.18), we have

(5.2)

−

ec1 ≥

4π(2N1 + N2 ) . R − R w 3λ( Ω h1 e 1 + ec2 Ω h1 h2 ew1 +w2 )

On the other hand, by H¨ older inequality, (2.16) and lemma 2.3-(ii) we have Z Z Z − − 1 1 ec2 h1 h2 ew1 +w2 ≤ec2 ( h21 e2w1 ) 2 ( h22 e2w2 ) 2 ≤ Ω Ω Ω Z Z 3λ c− w2 2 p ≤ e h e h1 e w 1 ≤ 2 (5.3) 32π (2N1 + N2 )(2N2 + N1 ) Ω Ω r Z 3 2N2 + N1 ≤ h1 e w 1 . 4 2N1 + N2 Ω

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

25

Combining (5.2) and (5.3), we obtain −

ec1 ≥

4π(2N1 + N2 ) 1 R q , λ Ω h1 e w 1 2 +N1 3(1 + 34 2N ) 2N1 +N2

and the desired estimate is established.  Lemma 5.1 permits to obtain the following: Proposition 5.2. If N1 + N2 = 1, then there exists a constant C > 0, independent of λ, such that for all (w1 , w2 ) ∈ Aλ we have: (5.4)

1 (k∇w1 k2 + k∇w2 k2 ) + λ − 4πln λ − C. 30 attains its infimum on Aλ .

Jλ− (w1 , w2 ) ≥

Moreover, Jλ−

Proof. Recalling (2.32), from lemma 5.1 we get (5.5) Jλ− (w1 , w2 )

Z 1 2 2 ≥ (k∇w1 k + k∇w2 k + ∇w1 ∇w2 ) + λ 3 Ω 4π 1 4π 1 (2N1 + N2 )ln R + (2N2 + N1 )ln R − C, + 3 3 λ Ω h1 e w 1 λ Ω h2 e w 2

for any (w1 , w2 ) ∈ Aλ Rand for some constant C > 0 independent of λ. 1 Using the estimate Ω |∇w1 ||∇w2 | ≤ 2 k∇w1 k2 + 2 k∇w2 k2 , valid for any  > 0, and Moser-Trudinger inequality (3.4), for some constant C > 0 (independent of λ), we obtain 1 1 2N1 + N2 1  2N2 + N1 Jλ− (w1 , w2 ) ≥ (1 − − )k∇w1 k2 + (1 − − )k∇w2 k2 + 3 2 4 3 2 4 + λ − 4π(N1 + N2 )ln λ − C ≥ 2 1 1 (3 − − N1 )k∇w1 k2 + (3 − 2 − N2 )k∇w2 k2 + λ − 4πln λ − C, ≥ 12  12 provided N1 + N2 = 1. Thus (5.4) follows by choosing  = 54 (1 − N2 ) + 45 (1 − N1 ). Therefore Jλ− is coercive on Aλ . By its weak lower semicontinuity on the weakly closed set Aλ , we immediately conclude that Jλ− attains its infimum on Aλ .  Let (w1,λ , w2,λ ) ∈ Aλ satisfy Jλ− (w1,λ , w2,λ ) = inf Aλ Jλ− , in order to prove that ( − v1,λ = w1,λ + c− 1 (w1,λ , w2,λ ) (5.6) − v2,λ = w2,λ + c− 2 (w1,λ , w2,λ ) defines a (weak) solution of (2.9), it suffices to show that (w1,λ , w2,λ ) lies in the interior of Aλ . Indeed, we have Proposition 5.3. For N1 + N2 = 1 and λ > 0 sufficiently large, we have (5.7)

inf (w1 ,w2 )∈∂Aλ

Jλ− (w1 , w2 ) >

inf (w1 ,w2 )∈Aλ

Jλ− (w1 , w2 ).

Proof. For (w1 , w2 ) ∈ ∂Aλ the identity Z Z 32π(2Ni + Nj ) wi 2 (5.8) ( hi e ) = h2i e2wi λ Ω Ω holds for i = 1 or 2.

i 6= j

26

M. NOLASCO AND G. TARANTELLO

Now let (w1,λ , w2,λ ) satisfies Jλ− (w1,λ , w2,λ ) = inf Aλ Jλ− and by contradiction assume that (w1,λ , w2,λ ) ∈ ∂Aλ . W.l.o.g. we may suppose that (5.8) holds for i = 2. By Jensen’s inequality it follows that, Z h22 e2w2,λ → +∞, as λ → +∞, Ω

and, by Moser-Trudinger’s inequality (3.4), necessarily k∇w2,λ k → +∞,

(5.9)

as λ → +∞.

Furthermore, by proposition 5.2 we get (5.10)

Jλ− (w1,λ , w2,λ ) ≥

1 (k∇w1,λ k2 + k∇w2,λ k2 ) + λ − 4πln λ − C, 30

with C > 0 a suitable constant independent of λ. On the other hand, by lemma 2.3-(ii) and lemma 5.1 there exist constants C1 , C2 > 0, independent of λ, such that

(5.11)

C1 λ C1 4π C2 λ ) + (1 − )+ (2N1 + N2 )ln + Jλ− (0, 0) ≤ (1 − 2 λ 2 λ 3 λ C2 4π (2N2 + N1 )ln ≤ λ − 4π(N1 + N2 )lnλ + C, + 3 λ

with C > 0 a suitable constant independent of λ. Thus, if N1 + N2 = 1, from (5.10) and (5.11) we obtain a contradiction since, 0 ≥ Jλ− (w1,λ , w2,λ ) − Jλ− (0, 0) ≥

1 k∇w2,λ k2 − C → +∞, 30

as λ → +∞.

Hence, (w1,λ , w2,λ ) must belong to the interior of Aλ .



Remark 5.4. Note that, if we use estimate (3.1) into (5.5) we can derive, as in Proposition 3.2, that the functional Jλ− is bounded below and attains its infimum on Aλ for all integers N1 and N2 . However, in the general situation, the estimate (5.4) gets worst with respect to λ and becomes: (5.12)

Jλ− (w1 , w2 ) ≥ α(k∇w1 k22 + k∇w2 k22 ) + λ − 4πcα ln λ − C,

for some α > 0 and some constant cα = cα (N1 , N2 ) ≥ 0, depending on N1 and N2 α → +∞, as N1 + N2 → +∞. in such a way that N1c+N 2 Thus, the estimate (5.12) is no longer sufficient for the arguments in the proof of proposition 5.3 to yield (5.7). So we have established that (5.6) defines a solution for (2.9) provided λ > 0 is − − sufficiently large. Next, we show that (v1,λ , v2,λ ) exhibits a different asymptotic + + behavior as λ → +∞ w.r.t. the family (v1,λ , v2,λ ) obtained in Proposition 3.5 above. Indeed, we have − − Proposition 5.5. Let N1 + N2 = 1 and (v1,λ , v2,λ ) be given by (5.6). Then,

• (i) c− i (w1,λ , w2,λ ) → −∞, as λ → +∞. • (ii) there exists a constant C > 0 (independent of λ) such that k∇wi,λ k ≤ C (i = 1, 2). Furthermore, any sequence λn → +∞ admits a subsequence (still denoted by λn ) such that for wi,n = wi,λn (i = 1, 2) we have wi,n → w ¯i

strongly in H1 (Ω),

(i = 1, 2)

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

and (w ¯1 , w ¯2 ) satisfies: (5.13)  4   −∆w1 = 4πN1 ( 3

R h he e −∆w2 = 4πN2 ( 43 R h he e   R w = 0, Ω i

1

Ω

2

w1 w1 1 w2 2

Ω

w1

− −

1

wi ∈ H (Ω)

1 3 1 3

R h he e R h he e 2

Ω

1

Ω

w2 w2 2 w1 1

w1

− 1) + − 1) +

R R

h1 ew1 8π 3 N2 ( Ω h1 ew1 h2 ew2 8π 3 N1 ( Ω h2 ew2

R h he e − R h he e −

2

Ω

1

Ω

27

w2 w2 2 w1 1

w1

) )

i = 1, 2.

Proof. By remark 2.4 and lemma 5.1 we have − 1 eci (w1,λ ,w2,λ ) = O( ) λ and (i) immediately follows . (ii) Recalling (5.10) and (5.11) we have

i = 1, 2.

1 (k∇w1,λ k2 + k∇w2,λ k2 ) − C, 30 hence, we immediately derive k∇wi,λ k ≤ C (i = 1, 2) for some suitable constant C > 0 independent of λ. Therefore, passing to subsequences if necessary, we derive wi,n = wi,λn → w ¯i weakly in H1 (Ω), strongly in Lp (Ω) ∀p ≥ 1 and pointwise a.e. in Ω. Furthermore, by dominated convergence, we also have Z Z i i (5.14) eα(u0 +wi,n ) → eα(u0 +w¯i ) i = 1, 2 and α > 0. 0 ≥ Jλ− (w1,λ , w2,λ ) − Jλ− (0, 0) ≥

Ω

Ω

− Consequently, taking into account (i), and the definition of c− i,n = ci,n (w1,n , w2,n ) in (2.22), as n → +∞, we get −

λn eci,n →

(5.15)

4π 2Ni + Nj R 3 Ω hi ew¯i

i, j = 1, 2 i 6= j.

The weak convergence in H1 (Ω), together with (5.14) and (5.15), yield to the conclusion that (w ¯1 , w ¯2 ) is a weak solution for (5.13). Finally, to prove that wi,n → w ¯i (i = 1, 2) strongly in H1 (Ω), notice that Z Z Z 2 |∇wi,n − ∇w ¯i | = − (∆wi,n − ∆w ¯i )(wi,n − w ¯i ) = hi,n (wi,n − w ¯i ), Ω

Ω

Ω

where −

−

4π 2N1 + N2 R )− 3 Ω h1 ew¯1 4π 2N2 + N1 R − )− 3 Ω h2 ew¯2

h1,n =λn ec1,n (2h1 ew1,n − 2h1 ew¯1 ) + 2h1 ew¯1 (λn ec1,n − −

(5.16)

−

− λn ec2,n (h2 ew2,n − h2 ew¯2 ) − h2 ew¯2 (λn ec2,n −

−

−

−

− λn (4e2c1,n h21 e2w1,n − 2e2c2,n h22 e2w2,n − ec1,n ec1,n h1 ew1,n h2 ew2,n ), and h2,n is given by Hence, khn kp is uniformly bounded R the symmetric expression. for any p ≥ 1 and Ω |∇wi,n − ∇w ¯i |2 ≤ khn k2 kwi,n − w ¯i k2 → 0.  − − Corollary 5.6. Let (v1,λ , v2,λ ) be given by (5.6). Then as λ → +∞, −

• evi,λ → 0 uniformly in C k (Ω), ∀k > 0, i=1,2. Proof. In view of (i) in Proposition 5.5 it is enough to prove that any sequence λn → +∞ admits a subsequence (which we still denote by λn ) such that wi,n = wi,λn → w ¯i in C k (Ω) for any k ≥ 0. This is readily established since, −∆(wi,n − w ¯i ) = hi,n , with hi,n given in (5.16), and khi,n kp → 0 as n → +∞ for any p ≥ 1. Consequently, kwi,n − w ¯i kC 1,α → 0 as

28

M. NOLASCO AND G. TARANTELLO

n → +∞ and α ∈ (0, 1). A bootstrap argument then gives kwi,n − w ¯i kC k → 0, for any k ∈ N.  To conclude the proof of Theorem 2.2, we consider the analogous minimization problem for Jλ± and Jλ∓ on Aλ . We start with the following: Proposition 5.7. Let N1 + N2 = 1 and ∗ = ± or ∓, there exists a constant C > 0, independent of λ, such that 3 1 (5.17) Jλ∗ (w1 , w2 ) ≥ (k∇w1 k2 + k∇w2 k2 ) + λ − 4πln λ − C, 30 4 for all (w1 , w2 ) ∈ Aλ . Moreover, Jλ± attains its infimum on Aλ . Proof. It suffices to prove (5.17) with ∗ = ±, the other case ∗ = ∓ follows analogously by exchanging the role between the indices. In view of (2.18), from (2.22) it follows immediately that R h1 e w 1 8π(2N1 + N2 ) 1 c± R (5.18) e 1 ≥ RΩ 2 2w , ≥ . 1 3λ 4 Ω h1 e h e w1 Ω 1 While, (5.19) ±

ec2 ≥

8π 4π R R ≥ , R ± R w 3λ max{ Ω h2 e 2 , Ω h1 h2 ew1 +w2 } 3λ( Ω h2 ew2 + ec1 Ω h1 h2 ew1 +w2 ) ±

c1 where we have used that (see remark 2.4). R R e ≤w1 +w w2 In case Ω h2 e ≥ Ω h1 h2 e 1 2 , by (2.32) we can use lemma 2.3-(iii), (5.18), (5.19) and Moser-Trudinger inequality (3.4), to derive

(5.20) Z 3 1 ∇w1 ∇w2 ) + λ− Jλ± (w1 , w2 ) ≥ (k∇w1 k2 + k∇w2 k2 + 3 4 Ω Z Z 4π 4π w1 − (2N1 + N2 )ln h1 e − (2N2 + N1 )ln h2 e w 2 − 3 3 Ω Ω − 4π(N1 + N2 )ln λ − C ≥ 1 1 2N1 + N2 1  2N2 + N1 ≥ (1 − − )k∇w1 k2 + (1 − − )k∇w2 k2 + 3 2 4 3 2 4 3 + λ − 4π(N1 + N2 )ln λ − C 4 for every  > 0 and C > 0 independent of λ. Since N1 + N2 = 1, as in Proposition 5.2, we can certainly make a choise R R of  > 0 in (5.20) in order to insure (5.17). Now suppose that Ω h2 ew2 < Ω h1 h2 ew1 +w2 , proceeding as above, in this case we get (5.21) Z 1 3 2 2 ≥ (k∇w1 k + k∇w2 k + ∇w1 ∇w2 ) + λ − 4π(N1 + N2 )ln λ− 3 4 Ω Z Z 4π 4π − (2N1 + N2 )ln h1 e w1 − (2N2 + N1 )ln h1 h2 ew1 +w2 − C ≥ 3 3 Ω Ω 1 (2N1 + N2 ) 2 2 2 ≥ (k∇w1 k + k∇w2 k + k∇w1 + ∇w2 k ) − k∇w1 k2 − 6 12 (2N2 + N1 ) 3 − k∇w1 + ∇w2 k2 + λ − 4π(N1 + N2 )ln λ − C ≥ 12 4 1 3 2 2 ≥ (k∇w1 k + k∇w2 k ) + λ − 4πln λ − C, 24 4

Jλ± (w1 , w2 )

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

29

provided N1 + N2 = 1. In any case we get the desired estimate (5.17). Thus Jλ± is coercive on Aλ . Since it is weakly lower semicontinuous on the weakly closed set Aλ , we immediately conclude that Jλ± attains its infimum on Aλ .  Remark 5.8. By similar considerations to those of remark 5.4, we can assert that, in fact, the functional Jλ∗ , ∗ = ± or ∓, is bounded below and attains its infimum on Aλ for all integers N1 and N2 . However, we need to restrict to the case N1 +N2 = 1 in order to insure a sharp form of estimate (5.17) (see (5.36) below), which is crucial for the existence of a minimizer in the interior of Aλ . ∗ ∗ Let ∗ = ± or ∓ and denote by (w1,λ , w2,λ ) ∈ Aλ a minimum of Jλ∗ in Aλ , namely Jλ∗ (w1,λ , w2,λ ) = inf Aλ Jλ∗ . Define ( ∗ ∗ ∗ ∗ v1,λ = w1,λ + c∗1 (w1,λ , w2,λ ) (5.22) ∗ ∗ ∗ ∗ ∗ v2,λ = w2,λ + c2 (w1,λ , w2,λ ), ∗ = ± or ∓. ∗ ∗ ∗ ∗ To show that (w1,λ , w2,λ ) lies in the interior of Aλ , and hence that (v1,λ , v1,λ ) defines a (weak) solution of (2.9), we prove the following preliminary result which holds for any choise of N1 , N2 ∈ N. ± ± Lemma 5.9. (i) Let (v1,λ , v2,λ ) be given by (5.22), then Z ± 1 h1 ev1,λ → . 2 Ω ∓ ∓ (ii) Let (v1,λ , v2,λ ) be given by (5.22), then Z ∓ 1 h2 ev2,λ → . 2 Ω

Proof. By symmetry, we only need to establish (i). R R As in the proof of lemma 3.3, let v¯µ1 = c¯1µ + w ¯µ1 , with c¯1µ = Ω v¯µ1 and Ω w ¯µ1 = 0 be the solution of ( 1 1 ∆v = µev+u0 (ev+u0 − 1) + 4πN1 v ∈ H1 (Ω) satisfying w ¯µ1 → −u10 pointwise a.e. in Ω and (5.23)

1

hi ew¯µ → 1,

in Lp (Ω), ∀p ≥ 1,

as µ → +∞ (i = 1, 2) (see [34]- Proposition 3.1). Observe that, from (2.22) and (2.18), we have s R R 2 Z w1 2 ( h e ) h e2w1 32π(2N + N ) 1 1 2 c± w Ω RΩ 1 e1 h1 e 1 ≥ R 2 2w (1 + 1 − )≥ 3λ 4 Ω h1 e 1 ( Ω h1 ew1 )2 Ω (5.24) R 1 ( Ω h1 ew1 )2 8π ≥ R 2 2w − (2N1 + N2 ). 1 2 Ω h1 e 3λ Recalling (5.23) and lemma 2.3-(iii), we find λ0 > 0 sufficiently large and c0 > 0 such that for any  > 0 there exist µ > 0 with the property that (w ¯µ1  , 0) ∈ Aλ and Z ± 1 1 1 8π ec1 (w¯µ ,0) h1 ew¯µ ≥ −  − (2N1 + N2 ), 2 3λ ZΩ (5.25) ± 1 c0 ec2 (w¯µ ,0) h2 ≥ , λ Ω for every λ ≥ λ0 .

30

M. NOLASCO AND G. TARANTELLO

Consequently, 1  1 k∇w ¯µ1  k2 + λ(1 − − ) + O(ln λ), 3 4 2 On the other hand, we have Z ± λ λ (5.27) Jλ± (w1,λ , w2,λ ) ≥ (1 − ec1 h1 ew1,λ ) + + O(ln λ), 2 2 Ω (5.26)

Jλ± (w ¯µ1  , 0) ≤

as λ → +∞.

as λ → +∞.

Therefore, Z ± λ λ (1 − ec1 h1 ew1,λ ) + − 2 2 Ω 1  1 1 2 − ( k∇w ¯µ k + λ(1 − − )) + O(ln λ), 3 4 2

0 ≥ Jλ± (w1,λ , w2,λ ) − Jλ± (w ¯µ1  , 0) ≥

from which we derive Z ± 1 lim sup ( − ec1 h1 ew1,λ ) ≤ , ∀  > 0. λ→+∞ 2 Ω At this point, taking into account lemma 2.3-(iii), we conclude Z ± 1 h1 ev1,λ → , as λ → +∞. 2 Ω  Remark 5.10. Putting together (2.30) and (5.24), we have that necessarily R ( Ω h1 ew1,λ )2 R 2 2w (5.28) → 1, as λ → +∞. h e 1,λ Ω 1 Using lemma 5.9, we derive: Proposition 5.11. (i) If N1 = 0 and N2 = 1 then, for λ > 0 sufficiently large, (5.29)

inf (w1 ,w2 )∈∂Aλ

Jλ± (w1 , w2 ) >

inf (w1 ,w2 )∈Aλ

Jλ± (w1 , w2 ).

(ii) If N1 = 1 and N2 = 0 then, for λ > 0 sufficiently large, (5.30)

inf (w1 ,w2 )∈∂Aλ

Jλ∓ (w1 , w2 ) >

inf (w1 ,w2 )∈Aλ

Jλ∓ (w1 , w2 ).

Proof. Again by symmetry we only need to establish (i). Let us suppose that (w1,λ , w2,λ ) satisfies J ± (w1,λ , w2,λ ) = inf Aλ J ± and (w1,λ , w2,λ ) ∈ ∂Aλ . In view of remark 5.10 necessarily Z Z 32π(2N2 + N1 ) (5.31) ( h2 ew2,λ )2 = h22 e2w2,λ . λ Ω Ω As a consequence of (5.31) we get Z h22 e2w2,λ → +∞

as λ → +∞

Ω

and, by Moser- Trudinger inequality (3.4), necessarily (5.32)

k∇w2,λ k → +∞

as λ → +∞. 1

Now, note that, if N1 = 0 then u10 = 0, and in particular h1 = eu0 = 1. By explicit calculation we see that, ± 1 1 ec1 (0,0) = + O( ), 2 λ (5.33) ± 1 ec2 (0,0) =O( ), as λ → +∞. λ

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

31

Therefore, in view of (2.13) and (5.33) we get Jλ± (0, 0) (5.34)

Z λ λ 8π c± (0,0) c± (0,0) 1 2 ≤ (1 − e ) + (1 − e h2 ) − ln λ + C ≤ 2 2 3 Ω 3 8π ≤ λ− ln λ + C as λ → +∞ 4 3 .

for a suitable constant C > 0, independent of λ. On the other hand, by lemma 5.9, for λ sufficiently large, we can insure that, Z (5.35) c± (w , w ) ≥ −ln ew1,λ − ln 4. 1,λ 2,λ 1 Ω

Thus, using the same arguments of Proposition 5.7, by lemma 2.3-(iii) we find constants α, C > 0 (independent of λ) such that Z ± λ Jλ± (w1,λ , w2,λ ) ≥α(k∇w1,λ k2 + k∇w2,λ k2 ) + (1 − ec1 ew1,λ )+ 2 Ω Z ± 8π λ (5.36) (1 − ec2 h2 ew2,λ ) − ln λ − C ≥ + 2 Ω 3 8π 3 ≥α(k∇w1,λ k2 + k∇w2,λ k2 ) + λ − ln λ − C. 4 3 Combining (5.34) and (5.36) we conclude that, (5.37)

0 ≥ Jλ± (w1,λ , w2,λ ) − Jλ± (0, 0) ≥ α(k∇w1,λ k22 + k∇w2,λ k22 ) − C,

and, in view of (5.32), we reach a contradiction.  To conclude we determine the asymptotic behavior, as λ → +∞, of the family of solutions given by (5.22). ∗ ∗ Proposition 5.12. Let (v1,λ , v2,λ ) be given by (5.22) with ∗ = ± or ∓. We have ± ± ± ± ± 1 • (case ∗ = ±): c± 1 (w1,λ , w2,λ ) → ln 2 , c2 (w1,λ , w2,λ ) → −∞ ± w1,λ → 0 strongly in H1 (Ω) as λ → +∞;

along any sequences λn → +∞, there exists a subsequence (still denoted by ± λn ) such that w2,λ → w± strongly in H1 (Ω) with w± satisfying: n    w 1 −∆w = 4π R h2 e − in Ω |Ω| h 2 ew (5.38) Ω R w = 0. Ω (5.39)

In particular, ± 1 ev1,λ → and 2

±

ev2,λ → 0,

strongly in H1 (Ω), as λ → +∞.

∓ ∓ ∓ ∓ ∓ 1 • (case ∗ = ∓): c∓ 1 (w1,λ , w2,λ ) → −∞, c2 (w1,λ , w2,λ ) → ln 2 , ∓ w2,λ → 0 strongly in H1 (Ω) as λ → +∞;

along any sequences λn → +∞, there exists a subsequence (still denoted by ∓ λn ) such that w1,λ → w∓ strongly in H1 (Ω) with w∓ satisfying: n    w 1 −∆w = 4π R h1 e − in Ω |Ω| h ew (5.40) Ω 1 R  w = 0. Ω

32

M. NOLASCO AND G. TARANTELLO

In particular, (5.41)

∓

ev1,λ → 0

and

∓

ev2,λ →

1 , 2

strongly in H1 (Ω),

as λ → +∞.

Proof. As usual we only need to prove the result in case ∗ = ±, the other case follows by exchanging the role between the indices. Recall that in this case N1 = 0, and hence h1 = 1, N2 = 1. By (5.37) we have ± 2 k∇wi,λ k2 ≤ C,

(5.42)

for suitable C > 0 (independent of λ) and i = 1, 2. Consequently, Z ± (5.43) 1≤ hi ewi,λ ≤ C (i = 1, 2) Ω

with C > 0 independent of λ. ± ± ± Thus, by setting c± i,λ = ci (w1,λ , w2,λ ), we have: c± 2,λ =

(5.44)

8π (2N2 + N1 ) 1 + o( ) R ± w 9λ λ h2 e 2,λ Ω

and (5.45)

c± 1,λ =

R w± 4π 1 Ω e 1,λ 1 + (N1 + N2 )(N2 − 9N1 ) + o( ) ± 2 R e2w1,λ 9λ λ Ω

as λ → +∞. ¿From (5.43) we derive immediately that c± 2,λ → −∞, as λ → +∞. Furthermore, in view of (5.28) and (5.42), along any sequence λn → +∞, we find a subsequence (still denoted by λn ) such that, ± w1,n := w1,λ → 0 weakly in H1 (Ω), strongly in Lp (Ω), p ≥ 1 and pointwise a.e. n in Ω.

Analogously, for suitable w± ∈ H1 (Ω), we may claim that, ± w2,n := w2,λ → w± weakly in H1 (Ω), strongly in Lp (Ω), p ≥ 1 and pointwise n a.e. in Ω.

Note in particular that, ew1,n → 1, ew2,n → ew± in Lp (Ω), ∀ p ≥ 1. Thus, using (2.9) (with |Ω| = 1 ) together with (5.44) - (5.45), we find: ±

±

±

−∆(w1,n + 2w2,n ) =3λn h2 ec2,n ew2,n (1 + ec1,n ew1,n − 2ec2,n ew2,n ) − 4π(2N2 + N1 )   h2 ew2,n 1 R =4π(2N2 + N1 ) − + φn in Ω |Ω| h ew2,n Ω 2 ± p with c± i,n := ci,λn (i = 1, 2) and φn → 0 strongly in L (Ω), ∀ p ≥ 1. Consequently, by elliptic regularity theory, we obtain (after taking a subsequence if necessarily)

(5.46) and w± (5.47)

1 w1,n + w2,n → w± strongly in C 1,α (Ω), α ∈ (0, 1) 2 satisfies:    w 1 −∆w = 2π(2N + N ) R h2 e − in Ω 2 1 |Ω| h ew Ω 2 R  w = 0. Ω

with N2 = 1 and N1 = 0, namely (5.40).

VORTEX CONDENSATES FOR THE SU(3) CHERN-SIMONS THEORY

33

On the other hand, if we insert (5.44)-(5.45) into the first equation in (2.9) (with N1 = 0 and N2 = 1) we get: R  2w1,n  ( ew1,n )2 e ew1,n 8π h2 ew2,n R R R ∆w1,n =λn RΩ 2w1,n − + (2 − ew1,n )+ 2w w w2,n 1,n 1,n 9 e e e h e 2 Ω Ω Ω Ω 8π w1,n w1,n + e (2e − 1) + ψn in Ω 9 with ψn → 0 strongly in Lp (Ω), ∀p ≥ 1. Therefore, R Z Z ( ew1,n )2 e2w1,n ew1,n ( R 2w1,n − R w1,n )w1,n − fn w1,n , (5.48) k∇w1,n k22 = −λn RΩ 2w1,n e e Ω Ωe Ω Ω Ω 8π 9 (1

R h he e

w±

) in Lp (Ω), ∀ p ≥ 1. R tw1,n Note that, the function h(t) := Ω e etw1,n w1,n is increasing in t ∈ R, since Ω Z Z etw1,n etw1,n 2 R R h0 (t) = w − ( w )2 = 1,n tw tw1,n 1,n 1,n e e Ω Ω Ω Ω Z Z etw1,n etw1,n R R = (w − w )2 ≥ 0 ∀ t ∈ R. 1,n tw1,n tw1,n 1,n Ω Ωe Ω Ωe

with fn →

+

2

w±

R

Ω

2

(R

e2w1,n ew1,n R − )w1,n = h(2) − h(1) ≥ 0, 2w e 1,n ew1,n Ω Ω

Thus, Z Ω

and from (5.48) we derive

k∇w1,n k2 → 0,

(5.49)

(5.50)

λn

Z Ω

(R

as n → +∞;

e2w1,n ew1,n − R w1,n )w1,n → 0 2w 1,n e e Ω Ω

as n → +∞.

Taking into account (5.46), we can also assert that, w2,n → w±

strongly in H1 (Ω).

± →0 Since (5.49) holds along any sequence λn → +∞, we may conclude that, w1,λ strongly in H1 (Ω) as λ → +∞. 1 Finally, from (5.45) we get c± 1,λ → ln 2 , as λ → +∞. This concludes the proof. 

Clearly, Theorem 2.2 it is now an immediate consequence of Corollary 5.6 and Proposition 5.12. Final remarks: It is an interesting open problem to know if Theorem 2.2 remains valid without the restriction N1 + N2 = 1. To test whether or not our approach could be generalized for a more general choise of N1 , N2 ∈ N, we could start by investigating the existence question for problems (5.13) and (5.47). While problem (5.47) has appeared already in the abelian theory, see [34], and it has been studied in [32] and [11], the elliptic system (5.13) is a novelty of the SU (3)-theory and it is certainly worthwhile investigating. Acknowledgments: The authors wish to express their gratitude to G. Dunne for useful comments.

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[34] G. Tarantello, Multiple condensates solutions for the Chern-Simons-Higgs theory, J. Math. Phys. 37 (1996), 3769–3796. [35] C. Taubes, Arbitrary n-vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys. 72 (1980), 277–292. [36] G. Wang and L. Zhang, Non-Topological solutions of relativistic SU(3) Chern-Simons Higgs model, Comm. Math. Phys. 202 (1999), 501–515. [37] R. Wang, The existence of Chern-Simons vortices, Comm. Math. Phys. 137 (1991), 587–597. [38] Y. Yang, The Relativistic Non-Abelian Chern-Simons Equations, Comm. Math. Phys. 186 (1997), 199–218. ` di L’Aquila, Via Vetoio, Cop(M. Nolasco) Dipartimento di Matematica, Universita pito, 67010 L’Aquila, Italy E-mail address: [email protected] ` di Roma “Tor Vergata”, Via (G. Tarantello) Dipartimento di Matematica, Universita della Ricerca Scientifica, 00133 Roma, Italy E-mail address: [email protected]