Sep 17, 2012 - arXiv:1209.3535v1 [math-ph] 17 Sep 2012 ... mention for example the (2 + 1)-dimensional abelian ChernâSimons model of HongâKimâPac [22].
arXiv:1209.3535v1 [math-ph] 17 Sep 2012
Doubly Periodic Self-Dual Vortices for a Relativistic Non-Abelian Chern–Simons Model Xiaosen Han Institute of Contemporary Mathematics, School of Mathematics Henan University, Kaifeng, Henan 475004, PR China Gabriella Tarantello Dipartimento di Matematica, Unversit`a di Roma “Tor Vergata” Via della Ricerca Scientifica, 00133 Rome, Italy
Abstract. In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions for a 2 × 2 nonlinear elliptic system arising in the study of self-dual non-Abelian Chern–Simons vortices. We show that the given system admits at least two solutions when the Chern–Simons coupling parameter κ > 0 is sufficiently small; while no solutions exist for κ > 0 sufficiently large. As in [36], we use a variational formulation of the problem. Thus, we obtain a first solution via a (local) minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as κ → 0. Then we obtain the second solution by a min-max procedure of “mountain pass” type.
1
Introduction
As well known vortices play an important role in many areas of physics, including superconductivity [1, 19, 27], optics [5], cosmology [21, 28, 50], the quantum Hall effect [40], and quark confinement [23, 24, 33–35]. After the pioneer work of Bogomol’nyi [6] and Prasad–Sommerfield [39], rigorous mathematical results about the existence of vortices have been pursued in various self-dual gauge field theories on the basis of an analytical approach that Taubes introduced in [49] to treat the Abelian–Higgs model. Indeed following [49], one is able to reduce the vortex problem to second order elliptic equations with exponential nonlinearity and Dirac source terms. Within this framework we mention for example the (2 + 1)-dimensional abelian Chern–Simons model of Hong–Kim–Pac [22] and Jackiw–Weinberg [26], for which Taubes’ approach has lead to the existence of topological multivortices (as described in [41,51]), non-topological multivortics (as constructed in [8,9,11,13,42]) and doubly periodic vortices (as given in [7, 14, 15, 30, 37, 46, 48]). In the non-Abelian context, rigorous existence results are established in [4, 10, 43, 44], while a series of sharp existence results have been obtained in [12, 29, 31, 32, 47] for non-Abelian models proposed in connection with the quark confinement phenomenon [23, 24, 33, 34]. For more results about self-dual vortices, we refer the readers to the monographs [45, 54]. Here, we are going to analyze a relativistic (self-dual) non-Abelian Chern–Simons model proposed by Dunne in [16, 17]. For this model, Yang [53] first established the existence of topological solutions in a very general situation. Subsequently, for the gauge group SU (3), Nolasco and Taran1
tello [36] proved a multiplicity result about the existence of doubly periodic vortices. The purpose of this paper is to establish analogous multiplicity results for theories that involve more general gauge groups. More precisely, we focus on gauge groups with a semi-simple Lie algebra of rank 2. From the technical point of view, we need to handle a 2 × 2 nonlinear elliptic system on the flat 2–torus, with coupling matrix given by the Cartan matrix associated to the gauge group. Clearly, this more general situation poses new analytical difficulties compared to the (already nontrivial) case analyzed in [36], where the authors handle a (specific) symmetric 2 × 2 system. Actually, we manage to resolve such difficulties for a larger class of 2 × 2 systems, where our vortex problem is included as a particular case.
2
Derivation of a general 2×2 nonlinear elliptic system and statement of the main results
The non-Abelian Chern–Simons model introduced by Dunne in [16, 17], is formulated over the R1+2 -Minkowski space with metric tensor: diag(1, −1, −1), that will be used in the usual way to rise and lower indices. Using the summation convention over repeated lower and upper indices ( ranging over 0, 1, 2), we consider the Lagrangian density: 2 κ µνα ∂µ Aν Aα + Aµ Aν Aα + Tr [Dµ φ]† [D µ φ] − V (φ, φ† ), L = − Trǫ (2.1) 2 3 where Dµ = ∂µ + [Aµ , ·] is the gauge-covariant derivative applied to the Higgs field φ in the adjoint representation of the gauge group G. The associated semi-simple Lie algebra is denoted by G, with [·, ·] the corresponding Lie bracket. Moreover, (Aµ )µ=0,1,2 denotes the G-valued gauge fields and Tr refers to the trace in the matrix representation of G. As usual, we denote by κ > 0 the Chern– Simons coupling parameter, ǫµνα the Levi–Civita totally skew-symmetric tensor with ε012 = 1 and we let V be the Higgs potential. The Euler-Lagrange equations corresponding to (2.1) are given by ∂V , ∂φ† = ǫµνα J α ,
Dµ D µ φ = κFµν
(2.2) (2.3)
with the strength tensor: Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ],
(2.4)
J µ = [φ† , (D µ φ)] − [(D µ φ)† , φ],
(2.5)
and covariant current density:
which is conserved, by satisfying: Dµ J µ = 0. The system also admits a conserved Abelian current: µ † µ µ † Q = −iTr φ D φ − (D φ) φ ,
µ = 0, 1, 2;
that satisfies: ∂µ Qµ = 0, and it is due to the global U (1)-invariance of the system. 2
Note that the energy density associated to (2.1) is given by: E = Tr([D0 φ]† [D0 φ]) + Tr([Di φ]† [Di φ]) + V (φ, φ† ),
(2.6)
that we consider together with the following Gauss law of the system: κF12 = J 0 = [φ† , (D0 φ)] − [(D0 φ)† , φ]
(2.7)
(corresponding to the α = 0 component of (2.3)). Then, with the choice of the Higgs potential: † 1 † † 2 † 2 V (φ, φ ) = 2 Tr [[φ, φ ], φ] − v φ [[φ, φ ], φ] − v φ , κ ( v > 0 is a constant which measures the scale of the broken symmetry) we see that the energy density E can be shown to satisfy ( [16, 17, 54]) E≥
v2 Q0 κ
(neglecting divergence terms). Moreover, the above lower bound is saturated by field configurations satisfying the following relativistic Chern–Simons self-dual equations: D1 φ ± iD2 φ = 0,
iF12 ∓
2 [[[φ, φ† ], φ] − v 2 φ, φ† ] = 0. κ2
(2.8) (2.9)
See [16, 17, 54] for details. It is not difficult to see that the solutions of (2.8) and (2.9) also satisfy the Euler–Lagrange equations (2.2) and (2.3). To handle the self-dual equations (2.8) and (2.9), we follow [16], and use the following decomposition: Aµ = i
r X
Ajµ Hj ,
φ=
r X
φi Ei ,
(2.10)
i=1
j=1
where Aiµ are real-valued vector fields, φi are complex valued scalar fields (i = 1, . . . , r), r is the rank of the semi-simple Lie algebra G, {Hi }1≤i≤r and {Ei }1≤i≤r (with Ei† = E−i ) are the generators of the Cartan subalgebra and the family of simple ladder operators of the semi-simple Lie algebra G, respectively. The consistency of (2.10) can be checked on the basis of the following commutation and trace relation, [Hi , Hj ] = 0, [Ei , E−j ] = δij Hi , [Hi , E±j ] = ±Kij E±j ,
Tr(Ei E−j ) = δij ,
Tr(Hi Hj ) = Kij , Tr(Hi E±j ) = 0,
i, j = 1, . . . , r,
where K = (Kij )i,j=1,...,r is the Cartan matrix [20] of the semi-simple Lie algebra G. It is well-known that the entries Kij of the Cartan matrix K, satisfy the following properties: 3
i) if i = j ∈ {1, . . . , r} then Kjj = 2,
ii) If i 6= j ∈ {1, . . . , r} then Kij ∈ Z− and Kij = 0 ⇔ Kji = 0.
We also know that for a semisimple Lie algebra, det K > 0,
(2.11)
(in fact all its principal diagonal minors are positive), and so K is non-degenerate. Actually, i) and ii) also imply that the entries of the inverse matrix K −1 are all non-negative, see [20] for details. Going back to (2.10), we observe that it always admits a (trivial) zero-energy configuration for which all the gauge fields vanish, while the Higgs field φ satisfies: [[φ, φ† ], φ] − v 2 φ = 0.
(2.12)
All such vacua configurations correspond to minima for the given potential. In particular, using the decomposition (2.10), we can identify the so-called principal embedding r P vacuum: φ(0) = φj(0) Ej whose components φj(0) satisfy: j=1
r X i 2 (K −1 )ij , φ(0) = v 2
i = 1, . . . , r.
(2.13)
j=1
To obtain non-trivial (self-dual) vortex configurations, we use the following standard notations [16, 54]: ∂± = ∂1 ± i∂2 , Ai± = Ai1 ± iAi2 , i = 1, 2 and observe that, in the static case, the self-dual equations (2.8)-(2.9) can be expressed componentwise as follows: a
∂± ln φ
a F12
= −i
r X
Ab± Kba ,
(2.14)
b=1
2 = ± 2 κ
r X b=1
!
|φa |2 |φb |2 Kba − v 2 |φa |2 ,
(2.15)
away from the zeros of φa , and with a F12 = ∂1 Aa2 − ∂2 Aa1 ,
a = 1, . . . , r.
Following [49], we can combine equations (2.14)-(2.15) into the following r × r system (so called Master equations): ! r X r r r X X X 4 b |φb |2 |φc |2 Kcb Kba − v 2 |φb |2 Kba , (2.16) ∆ ln |φa |2 = ±2 F12 Kba = 2 κ c=1 b=1
b=1
b=1
(away from the zero points of φa ) a = 1, . . . , r, that we need to solve in combination with the following componentwise expression of the Gauss law (2.7): a κF12 = J0a ,
a = 1, . . . , r,
with J0a the component relative to the Cartan subalgebra of the current J0 in (2.5). 4
(2.17)
The corresponding energy density takes the form: E =v
2
r X
a F12 .
(2.18)
a=1
While, the gauge invariance of the theory is expressed by the following transformation laws: Aaµ
→
Aaµ
+ ∂µ ωa
µ = 0, 1, 2,
r P
i
a
φ →e
Kba ωb
b=1
φa
(2.19)
with ωa a smooth real function, that in the static case depends only on the state variables x = (x1 , x2 ) ∈ R2 , a = 1, . . . , r. In this paper, we are interested in obtaining static solutions of (2.16) subject to suitable ’t Hooft boundary conditions over a doubly periodic domain Ω. To be more precise, we let the periodic cell domain Ω to be generated by two linearly independently vectors e1 , e2 ∈ R2 ,
and set
Ω = x = s 1 e1 + s 2 e2 ∈ R 2
so that
Γ j = x = s j ej ,
0 < sj < 1,
0 < sj < 1 ,
j = 1, 2 ,
j = 1, 2
∂Ω = Γ1 ∪ Γ2 ∪ {e1 + Γ2 } ∪ {e2 + Γ1 } ∪ {0, e1 , e2 , e1 + e2 }. In view of (2.19), we require (Aaµ )µ=0,1,2 and φa to satisfy the boundary conditions ! ! r r P P Kba ωb Kba ωb i i a a φ (x + ek ) = e b=1 φ (x), e b=1 Aaµ + ∂µ ωka (x + ek ) = Aaµ + ∂µ ωka (x), µ = 0, 1, 2, x ∈ Γ1 ∪ Γ2 \ Γk , k = 1, 2, a = 1, . . . , r,
(2.20)
where ωka is a smooth function defined in a neighborhood of Γj ∪ {Γ + ek } with j 6= k ∈ {1, 2}, a = 1, . . . , r. As explicitly derived in [36], solutions of (2.16) and (2.20) carry “quantized” electric and magnetic charges, in the sense that the following hold: Φa : = Qa : =
Z Z
a F12 = 2π
r X (K −1 )ba Nb ,
(2.21)
b=1
J 0 = κΦa = 2πκ
r X
(K −1 )ba Nb ,
a = 1, . . . , r
(2.22)
b=1
with Na a suitable integer, that actually counts the zeros of φa in Ω (with multiplicity) a = 1, . . . , r. In addition, from (2.18), (2.21) and (2.22), we obtain the following “quantization” formula for the total energy: Z r r X X E = 2πv 2 E= (K −1 )ab Nb = 2π |φb(0) |2 Nb , (2.23) Ω
a,b=1
b=1
where the last identity follows by (2.13), with φb(0) the component of the principal embedding vacuum. 5
Here we shall focus on the solvability of (2.16) and (2.20) ! with gauge groups rank 2 −1 Besides the group SU (3), with Cartan matrix K = , examples of this situation −1 2 ! 2 −1 the exceptional gauge group B2 (= C2 ) with Cartan matrix K = and G2 with −2 2 ! 2 −1 matrix K = . −3 2 More generally, in the rank r = 2 case, the Cartan matrix takes the form ! 2 −a12 K= −a21 2
r = 2. include Cartan
(2.24)
with ajk ∈ Z+ for j 6= k ∈ {1, 2} and 4 − a12 a21 > 0. In case a12 = 0 = a21 (i.e. G = A1 × A1 ) then the Cartan matrix diagonalizes, and the system (2.16) decouples into two abelian Chern–Simons vortex problems, for which the existence of (at least) two gauge-distinct periodic static configurations has been established in [46], provided κ > 0 is sufficiently small. Our main goal is to extend such multiplicity result to any gauge group of rank 2. More precisely, we prove: Theorem 2.1 Let the gauge group G admit a semisimple Lie algebra G of rank r = 2 and Cartan matrix K specified in (2.24). For Na ∈ N, let Za = {pa,1 , . . . , pa,Na } ⊂ Ω be a set of Na points (not necessarily distinct) a = 1, 2. For κ > 0 sufficiently small, there exist at least two gauge distinct static solutions of (2.8)-(2.9) subject to the ansatz (2.10) and the boundary condition (2.20) such that: (i) the component φa of the Higgs field satisfies: |φa | < |φa(0) | in Ω, with φa(0) the component of the principal embedding vacuum in (2.13); and φa vanishes exactly at each point pa,j ∈ Za with the same multiplicity, a = 1, 2; (ii) the corresponding magnetic flux Φa , electric charge Qa (a = 1, 2) and total energy E, satisfy the “quantization” identity (2.21), (2.22) and (2.23) respectively; (iii) for at least one of the given solutions the following holds: |φa | → |φa(0) |
as
κ → 0,
pointwise a.e. in Ω and strongly in Lp (Ω), for p ≥ 1. (iv) If v u |Ω| u o, n κ > vt 1 +a12 N2 2N2 +a21 N1 , 4π(det K) max 2N 2 2 2(a12 +2) 2(a21 +2)
with aij ≥ 0, i 6= j ∈ {1, 2}, the off-diagonal entries of the Cartan matrix K in (2.24), then problem (2.8)-(2.10) and (2.20) admits no such solutions . As already mentioned, Theorem 2.1 provides a natural extension of the multiplicity result of Nolasco–Tarantello in [36], concerning the group G = SU (3), for which (2.16) enjoys additional symmetries. In fact, to establish Theorem 2.1 we adopt the same variational viewpoint. However we are able to handle systems of the type (2.16) with a more general coupling matrix.
6
More precisely, we take a 2 × 2 matrix K of the form: ! a −b K= , −c d and assume that a, b, c, d > 0 and ad − bc > 0. Notice that the case a, d > 0 and b = c = 0, is already covered in [46]. We denote the zero set of φi by Zi = {pi,1 , . . . , pi,Ni },
i = 1, 2
(2.25)
(repeated with multiplicity) and set, 1 2 φ = v 2 b + d eu1 , ad − bc
2 2 φ = v 2 a + c eu2 , ad − bc
λ=
4v 4 . κ2
(2.26)
By straightforward calculations, we see that the equations in (2.16) subject to the boundary conditions (2.20) take the form: n 1 u1 + b(a + c)eu2 ] = λ ∆u 1 ad−bc [−a(b + d)e 2 o 1 2 e2u1 − b(b + d)(a2 − c2 )eu1 +u2 − bd(a + c)2 e2u2 + a (b + d) 2 (ad−bc) N1 P δp1,j , x ∈ Ω, +4π n j=1 1 (2.27) [c(b + d)eu1 − d(a + c)eu2 ] ∆u2 = λ ad−bc o 1 2 2u1 − c(a + c)(d2 − b2 )eu1 +u2 + d2 (a + c)2 e2u2 + (ad−bc) 2 −ac(b + d) e N2 P δp2,j x ∈ Ω, +4π j=1 u1 and u2 doubly periodic on ∂Ω. Concerning (2.27), we establish the following:
Theorem 2.2 Assume that a, b, c, d > 0 and ad−bc > 0. Given Nj ∈ N and Zj = {pj,1 , . . . , pj,Nj } ⊂ Ω (a set of Nj -point repeated with multiplicity), j = 1, 2, the following holds: 1. Every solution (u1 , u2 ) of (2.27) satisfies eu1 < 1,
eu2 < 1
in
Ω.
(2.28)
2. If 16π(ad − bc) λ< max |Ω|
dN1 + bN2 cN1 + aN2 , a(b + d)2 d(a + c)2
,
then problem (2.27) admits no solutions. 3. There exist λ0 > 0, such that for λ > λ0 problem (2.27) admits at least two distinct solutions, one of which satisfying: eu1 → 1, eu2 → 1, as λ → +∞ (2.29) pointwise a.e. in Ω and strongly in Lp (Ω) for any p ≥ 1.
7
Remark 2.1 As already noticed, when b = c = 0, problem (2.27) decouples in two abelian self-dual Chern-Simons equations: N ∆u = λeui (eui − 1) + 4π Pi δ , x ∈ Ω, pi,j i j=1 ui doubly periodic on ∂Ω, i = 1, 2 for which the existence and multiplicity results claimed above have been established in [46].
Thus in view of (2.26), [46] and Theorem 2.2, we deduce (by standard arguments [49]) the statement of Theorem 2.1. Hence we devote the following section to the proof of Theorem 2.2.
3
Existence of doubly periodic solutions
In this section we ∆u1 ∆u2 u 1
analyze problem (2.27), and for convenience we rewrite it as follows: =
λ ad−bc
[a(b + d)eu1 (eu1 − 1) + b(a + c)eu2 (1 − eu2 )] N1 P u1 + ceu2 ) (eu1 − eu2 ) + 4π (ae δp1,j , x ∈ Ω, + λ(a+c)(b+d)b 2 (ad−bc) j=1
=
λ ad−bc
[d(a +
c)eu2
(eu2
(beu1 + deu2 ) (eu2 + λ(a+c)(b+d)c (ad−bc)2 and u2
(1 − eu1 )] N2 P − eu1 ) + 4π δp2,j , x ∈ Ω,
− 1) + c(b +
doubly periodic on
∂Ω.
d)eu1
(3.1)
j=1
We start to establish the following:
Proposition 3.1 Let (u1 , u2 ) satisfy (3.1). Then ui < 0 in Ω, i = 1, 2. Proof.
˜i ≡ max ui = Notice that ui attains its maximum value at a point x ˜i ∈ Ω \ Zi , so that u Ω
ui (˜ xi ), i = 1, 2. We start by showing u ˜i ≤ 0, for i = 1, 2. Indeed, in case u ˜1 ≥ u ˜2 , then we use the first equation in (3.1) to obtain: i h λ 0 ≥ ∆u1 (˜ x1 ) = a(b + d)eu˜1 eu˜1 − 1 + b(a + c)eu2 (˜x1 ) 1 − eu2 (˜x1 ) ad − bc λ(a + c)(b + d)b u˜1 u ˜1 u2 (˜ x1 ) u2 (˜ x1 ) e ae − e + ce + (ad − bc)2 i h λ ≥ a(b + d)eu˜1 − b(a + c)eu2 (˜x1 ) eu˜1 − 1 . ad − bc
Since
a(b + d)eu˜1 − b(a + c)eu2 (˜x1 ) ≥ (ad − bc)eu˜1 > 0 we find that necessarily, u ˜1 ≤ 0, and the desired conclusion follows in this case. On the other hand if u ˜2 ≥ u ˜1 , then we can use a similar argument for the second equation in (3.1) to deduce that u ˜2 ≤ 0. Thus, in any case, we have: u ˜i ≤ 0, i = 1, 2. To obtain that actually the strict inequality holds, we use the strong maximum principle. It can be applied, since for example we see that u1 satisfies: (a + c)(b + d)b λ u1 u2 u2 (ae + ce ) (1 − eu2 ) b(a + c)e + ∆u1 + c1 (x)u1 = ad − bc ad − bc ≥ 0 in Ω 8
with λ 1 − eu1 (a + c)(b + d)b u1 u2 u1 c1 (x) = (ae + ce ) . a(b + d)e + ad − bc ad − bc u1 Similarly for u2 . Therefore we conclude that ui < 0 in Ω, i = 1, 2. In particular we have established the first conclusion of Theorem 2.2. To proceed further, we let ui = ui0 + vi , i = 1, 2 with ui0 being the unique solution of the problem (see [3]) Ni P i ∆ui0 = 4π δpi,s − 4πN |Ω| , s=1 R i ui0 doubly periodic on ∂Ω i = 1, 2. Ω u0 dx = 0,
Consequently, problem (2.27) (or (3.1)) can be formulated in terms of the unknown (v1 , v2 ) as follows: h i h 1 λ λ u10 +v1 + b(a + c)eu20 +v2 + ∆v = a2 (b + d)2 e2u0 +2v1 −a(b + d)e 1 ad−bc (ad−bc)2 i 2 − c2 )eu10 +u20 +v1 +v2 − bd(a + c)2 e2u20 +2v2 + 4πN1 , −b(b + d)(a |Ω| i h h 1 2 λ λ 2 2u10 +2v1 (3.2) c(b + d)eu0 +v1 − d(a + c)eu0 +v2 + (ad−bc) ∆v2 = ad−bc 2 −ac(b + d) e i 2 − b2 )eu10 +u20 +v1 +v2 + d2 (a + c)2 e2u20 +2v2 + 4πN2 , −c(a + c)(d |Ω| v , v doubly periodic on ∂Ω. 1 2
Actually, to emphasize the variational structure of (3.2), we shall use the following equivalent formulation: i h ad−bc 2u10 +2v1 − (ad − bc)eu10 +v1 − b(a + c)eu10 +u20 +v1 +v2 ∆(dv + bv ) = λ a(b + d)e 1 2 b+d 1 +bN2 ) + 4π(ad−bc)(dN , (b+d)|Ω| i h ad−bc 2u20 +2v2 − (ad − bc)eu20 +v2 − c(b + d)eu10 +u20 +v1 +v2 (3.3) ∆(cv + av ) = λ d(a + c)e 1 2 a+c 1 +aN2 ) + 4π(ad−bc)(cN , (a+c)|Ω| v1 , v2 doubly periodic on ∂Ω. 2 We introduce the Hilbert space: H(Ω) ≡ W 1,2 Ze1R+Ze2 of Ω-periodic L2 -functions whose R derivatives also belong to L2 (Ω), equipped with the usual norm: kwk2 = kwk22 +k∇wk22 = Ω w2 dx+ R 2 Ω |∇w| dx, w ∈ H(Ω). It is not difficult to check that weak solutions to (3.3) are critical points in H(Ω) × H(Ω) of the functional: Z Z a d 2 2 ∇v1 · ∇v2 dx + λ Q(v1 , v2 )dx k∇v1 k2 + k∇v2 k2 + Iλ (v1 , v2 ) = 2b 2c Ω Ω Z Z α1 α2 + v1 dx + v2 dx, (3.4) |Ω| Ω |Ω| Ω 9
where 2 1 Q(v1 , v2 ) = 2ab(ad−bc) Q22 (v2 ), Q21 (v1 , v2 ) + (a+c) 2ac h i Q1 (v1 , v2 ) = a(b + d)eu10 +v1 − b(a + c)eu20 +v2 − (ad − bc) , 2 +v u 2 0 −1 , Q2 (v2 ) = e α = 4π d N + N , α = 4π N + a N . 1 2 1 2 b 1 c 2
(3.5)
In view of our assumption, notice that the quadratic part of Iλ is positive definite. In fact we obtain a first critical point for Iλ via (local) minimization.
3.1
Constrained minimization
For a solution (v1 , v2 ) of (3.3), after integration over Ω, we find the following natural constraints: Z Z Z 1 2 u10 +v1 2u10 +2v1 e dx − b(a + c) eu0 +u0 +v1 +v2 dx dx − (ad − bc) a(b + d) e Ω
Ω
Ω
4π(ad − bc)(dN1 + bN2 ) + = 0, λ(b + d) Z Z Z 1 2 2 2 eu0 +u0 +v1 +v2 dx eu0 +v2 dx − c(b + d) e2u0 +2v2 dx − (ad − bc) d(a + c) Ω
Ω
Ω
(3.6)
4π(ad − bc)(cN1 + aN2 ) + = 0. λ(a + c)
(3.7)
From (3.5)-(3.7), we obtain Z Z Z 1 a d u20 +v2 u10 +v1 Q(v1 , v2 )dx = 1−e dx 1−e dx + 1 + 1+ 2 b c Ω Ω Ω α1 + α2 . − 2λ Therefore, if we decompose v1 , v2 as follows: Z Z 1 wi dx = 0, ci = vi = wi + ci , vi dx, |Ω| Ω Ω
(3.8)
i = 1, 2,
then form (3.6) and (3.7) we find: Z 4π(ad − bc)(dN1 + bN2 ) 1 2c1 e2u0 +2w1 dx − ec1 R1 (w1 , w2 , ec2 ) + e = 0, λa(b + d)2 Ω Z 4π(ad − bc)(cN1 + aN2 ) 2 e2u0 +2w2 dx − ec2 R2 (w1 , w2 , ec1 ) + e2c2 = 0, λd(a + c)2 Ω
(3.9) (3.10)
with c2
R1 (w1 , w2 , e ) = R2 (w1 , w2 , ec1 ) =
Z ad − bc 1 eu0 +w1 dx + a(b + d) Ω Z ad − bc 2 eu0 +w2 dx + d(a + c) Ω
10
Z b(a + c) c2 1 2 eu0 +u0 +w1 +w2 dx, e a(b + d) ZΩ c(b + d) c1 1 2 eu0 +u0 +w1 +w2 dx. e d(a + c) Ω
(3.11) (3.12)
A necessary condition for the solvability of (3.9) and (3.10) with respect to c1 and c2 is that, Z 16π(ad − bc)(dN1 + bN2 ) 1 e2u0 +2w1 dx, (3.13) (R1 (w1 , w2 , ec2 ))2 ≥ 2 λa(b + d) Ω Z 16π(ad − bc)(cN1 + aN2 ) 2 c1 2 (R2 (w1 , w2 , e )) ≥ e2u0 +2w2 dx. (3.14) 2 λd(a + c) Ω On the other hand, from Proposition 3.1, we see that u10 + v1 + c1 < 0 and u20 + v2 + c2 < 0 in Ω. Therefore, as a consequence of (3.13)-(3.14), we obtain: Z Z 16π(ad − bc)(dN1 + bN2 ) 1 2u10 +2w1 e dx ≤ |Ω| e2u0 +2w1 dx, 2 λa(b + d) ZΩ ZΩ 2 2 16π(ad − bc)(cN1 + aN2 ) e2u0 +2w2 dx ≤ |Ω| e2u0 +2w2 dx. 2 λd(a + c) Ω Ω Thus, we obtain the following necessary condition for the solvability of (3.3), 16π(ad − bc) dN1 + bN2 cN1 + aN2 λ≥ max , , |Ω| a(b + d)2 d(a + c)2
(3.15)
and deduce part 2. of Theorem 2.2. Conditions (3.13) and (3.14) suggest to focus only with pairs (v1 , v2 ) that, under the decomposition: vi = wi + ci , i = 1, 2, satisfy: Z
w1 dx = 0 and
Ω
Z
w2 dx = 0 and
Ω
Z
Z
u10 +w1
e
dx
Ω u20 +w2
e
dx
Ω
2
≥
2
≥
16πa(dN1 + bN2 ) λ(ad − bc)
16πd(cN1 + aN2 ) λ(ad − bc)
Z Z
1
e2u0 +2w1 dx,
(3.16)
Ω 2
e2u0 +2w2 dx;
(3.17)
Ω
and where (c1 , c2 ) satisfy (3.9) and (3.10). Hence we define the admissible set: A = (w1 , w2 ) ∈ H(Ω) × H(Ω) such that
(3.16) and
(3.17) hold .
(3.18)
On the basis of (3.9) and (3.10), we aim to obtain (c1 , c2 ) from the equations: q R 2u1 +2w 1 +bN2 ) c 2 1 dx 0 R1 (w1 , w2 , e ) ± [R1 (w1 , w2 , ec2 )]2 − 16π(ad−bc)(dN 2 Ωe λa(b+d) ec1 = R 2u1 +2w 1 2 Ωe 0 ≡ g1± (ec2 ),
c2
e
=
R2 (w1 , w2 , ec1 ) ±
≡ g2± (ec1 ).
q
16π(ad−bc)(cN1 +aN2 ) R
[R2 (w1 , w2 , ec1 )]2 − R 2 2 Ω e2u0 +2w2
λd(a+c)2
To this end, we follow [36] 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)),
11
2u20 +2w2
Ωe
(3.19)
dx (3.20)
so that the solutions of (3.19) and (3.20), with all possible choices of signs: ∗ = +, −, ±, ∓ corresponds to the zeros of the function: F ∗ : [0, +∞) 7→ R. At this point, as in [36], it suffices to check the following claims. Claim 1. The functions gi± (X) is strictly monotonic with respect to X > 0, i = 1, 2. In fact, by direct computation we have: dg1± (X) dX dg2± (X) dX
= ±g1± (X) q = ±g2± (X) q
b(a+c) a(b+d)
R
u10 +u20 +w1 +w2 dx Ωe
1 +bN2 ) [R1 (w1 , w2 , X)]2 − 16π(ad−bc)(dN λa(b+d)2 c(b+d) R u10 +u20 +w1 +w2 dx d(a+c) Ω e
[R2 (w1 , w2 , X)]2 −
R
16π(ad−bc)(cN1 +aN2 ) λd(a+c)2
and by definition (see (3.19) and (3.20)) gi± (X) > 0,
1
,
(3.21)
;
(3.22)
2u0 +2w1 dx Ωe
R
2
2u0 +2w2 dx Ωe
∀ X > 0, i = 1, 2.
(3.23)
Claim 2. For any (w1 , w2 ) ∈ A, there exits a unique X ∗ (w1 , w2 ) > 0 such that F ∗ (X ∗ (w1 , w2 )) = 0; with ∗ = +, −, ±, ∓. To prove Claim 2, observe that F ∗ (0) < 0, with ∗ = +, −, ±, ∓. Next, we check easily that, lim g− (X) X→+∞ i
= 0,
g1+ (X) X→+∞ X
=
g2+ (X) X→+∞ X
=
lim
lim
and consequently:
lim
X→+∞
F + (X) X
F ∗ (X) X→+∞ X lim
= 1− = 1,
i = 1, 2, R 1 2 b(a + c) Ω eu0 +u0 +w1 +w2 dx , R 2u1 +2w 1 dx a(b + d) 0 e Ω R 1 2 c(b + d) Ω eu0 +u0 +w1 +w2 dx , R 2u2 +2w 2 dx d(a + c) e 0 Ω
R
1 2 eu0 +u0 +w1 +w2 dx
2
Ω bc ad − bc ≥ > 0, R 2u1 +2w R 2u2 +2w 1 dx 2 dx ad Ω e 0 ad 0 e Ω
∗ = −, ±, ∓.
In particular, lim F ∗ (X) = +∞,
X→+∞
∗ = +, −, ±, ∓,
and from (3.21) and (3.22), we see that dF ∗ (X) > 0, dX
∗ = ±, ∓,
hence we deduce the statement of Claim 2 for ∗ = ±, ∓.
12
On the other hand, from (3.21)-(3.22) and (3.16)-(3.17) we obtain R 2 + + + u10 +u20 +w1 +w2 dx + 2 2 g (g (X))g (X) e 1 2 2 Ω dF (X) b (a + c) q = 1− 2 dX a (b + d)2 [R (w , w , g + (X))]2 − 16π(ad−bc)(dN1 +bN2 ) R e2u10 +2w1 dx 1
×q
> 1−
1
2
2
λa(b+d)2
Ω
1
[R2 (w1 , w2 , X)]2 −
16π(ad−bc)(cN1 +aN2 ) λd(a+c)2
g1+ (g2+ (X)) F + (X) = . X X
R
2u20 +2w2 dx Ωe
Similarly, for ∗ = −, we have: dF − (X) g− (g− (X)) F − (X) >1− 1 2 = . dX X X Thus, for X > 0 the function this case as well.
F ∗ (X) X
is strictly increasing, with ∗ = +, −, and Claim 2 follows in
From the above discussion we see that, for any (w1 , w2 ) ∈ A, there exists a unique c∗j = c∗j (w1 , w2 ) for j = 1, 2 and ∗ = +, −, ±, ∓ such that v1∗ = w1 + c∗1 (w1 , w2 ),
v2∗ = w2 + c∗2 (w1 , w2 ),
∗ = +, −, ±, ∓
satisfy (3.6)-(3.7). Notice also that, by the above property, c∗1 and c∗2 depend smoothly on (w1 , w2 ) ∈ A. We shall use those properties only for ∗ = +, although it is reasonable to expect that other choices may lead to stronger multiplicity results, as in [36]. Thus, in what follows we consider the functional + Jλ+ (w1 , w2 ) = Iλ (w1 + c+ 1 (w1 , w2 ), w2 + c2 (w1 , w2 )),
(w1 , w2 ) ∈ A.
From (3.4) and (3.8), we see that Jλ+ (w1 , w2 )
Z a d 2 2 ∇w1 · ∇w2 dx k∇w1 k2 + k∇w2 k2 + = 2b 2c Ω Z Z + λ a d 2 +w 1 +w u c u c+ 2 1 + dx 1−e 2e 0 dx + 1 + 1−e 1e 0 1+ 2 b c Ω Ω α1 + α2 + (3.24) +α1 c+ 1 + α2 c2 − 2
with α1 , α2 defined in (3.5). It is easy to check that the functional Jλ+ is Frech´et differentiable in the interior of A. Moreover, + if (w1 , w2 ) is a critical point of Jλ+ and lies in the interior of A, then (w1 +c+ 1 (w1 , w2 ), w2 +c2 (w1 , w2 )) gives a critical point of Iλ . In the sequel, we show that Jλ+ is bounded from below and admits an interior minimum. Lemma 3.1 For any (w1 , w2 ) ∈ A, there holds: Z i c+ i eu0 +wi dx ≤ |Ω|, e Ω
13
i = 1, 2.
(3.25)
Remark 3.1 Using Jensen’s inequality, from (3.25) follows that +
eci ≤ 1, Proof.
i = 1, 2.
Using (3.19)-(3.20), we have R u1 +w ad−bc 1 dx + 0 a(b+d) Ω e c+ e1 ≤ R Ω
c+ 2
e
ad−bc d(a+c)
≤
R
2
u0 +w2 dx + Ωe R Ω
R u1 +u2 +w +w b(a+c) c+ 1 2 dx 0 0 2 a(b+d) e Ωe , 1 e2u0 +2w1 dx
(3.26)
R u1 +u2 +w +w c(b+d) c+ 1 2 dx 1 0 0 Ωe d(a+c) e ; 2 e2u0 +2w2 dx
(3.27)
and so, by using (3.26)-(3.27) and H¨older inequality, we find R u1 +w R 2 1 2 b(ad−bc) R ad−bc e 0 1 dx eu0 +w2 dx Ω eu0 +u0 +w1 +w2 dx + Ω Ω a(b+d) ad(b+d) c1 + ≤ e R 2u1 +2w R 2u1 +2w R 2u2 +2w 1 dx 1 dx 2 dx 0 0 0 Ωe Ωe Ωe R 2 + bc u10 +u20 +w1 +w2 dx e ec1 ad Ω + R 2u1 +2w R 2u2 +2w 2 dx 1 dx 0 0 Ωe Ωe R R u1 +u2 +w +w b(ad−bc) R ad−bc u10 +w1 dx u20 +w2 dx 1 2 dx 0 0 e e bc + Ωe a(b+d) Ω ad(b+d) Ω ≤ + + ec1 . R 2u1 +2w R 2u1 +2w R 2u2 +2w 1 dx 1 dx 2 dx ad 0 0 0 Ωe Ωe Ωe Consequently,
c+ 1
e
Similarly, we obtain c+ 2
e
d b+d
≤ R
Ω
a a+c
≤ R
Ω
u10 +w1 dx Ωe 1 e2u0 +2w1 dx
R
u20 +w2 dx Ωe 2 e2u0 +2w2 dx
R
+
+
b b+d
u20 +w2 dx u10 +u20 +w1 +w2 dx Ωe Ωe . R 2u1 +2w R 2u2 +2w 1 dx 2 dx 0 0 Ωe Ωe
c a+c
u10 +u20 +w1 +w2 dx u10 +w1 dx Ωe Ωe . R 2u1 +2w R 2u2 +2w 1 dx 2 dx 0 0 e e Ω Ω
R
R
R
R
(3.28)
(3.29)
Next, we can use (3.28)-(3.29) and H¨older inequality to deduce that R 2 R u1 +w R R 2 1 2 d u10 +w1 dx Z b e e 0 1 dx Ω eu0 +w2 dx Ω eu0 +u0 +w1 +w2 dx b+d Ω 1 +w b+d Ω u c+ 1 + e 0 dx ≤ e1 R 2u1 +2w R 2u1 +2w R 2u2 +2w 1 dx 1 dx 2 dx 0 0 0 e e Ω Ω Ω Ωe ≤ |Ω|, R 2 R u1 +w R u2 +w R u1 +u2 +w +w a u20 +w2 dx Z c 1 dx 2 dx e 0 0 e e e 0 0 1 2 dx + Ω a+c 2 a+c Ω Ω ec2 eu0 +w2 dx ≤ + R 2u2 +2w R 2u1 +2w R 2u2Ω+2w 2 dx 1 dx 2 dx 0 0 0 Ω Ωe Ωe Ωe ≤ |Ω|, and (3.25) is established. The following property of functions in A was pointed out first in [38] and used in [36]. In our context, it takes the following form: Lemma 3.2 For any (w1 , w2 ) ∈ A and s ∈ (0, 1), we have 1 1−s Z Z s s λ(ad − bc) su10 +sw1 u10 +w1 e dx , e dx ≤ 16πa(dN1 + bN2 ) Ω Ω 1 1−s Z Z s s λ(ad − bc) su20 +sw2 u20 +w2 e dx . e dx ≤ 16πd(cN1 + aN2 ) Ω Ω 14
(3.30) (3.31)
Proof. Although the proof of (3.30), (3.31) follows exactly as in [36, 38], here we give the proof 1 such that sγ + 2(1 − γ) = 1. Then using H¨older inequality for completeness. Let s ∈ (0, 1), γ = 2−s and (3.16) we have 1−γ γ Z Z Z 2u10 +2w1 su10 +sw1 u10 +w1 e dx e dx e dx ≤ Ω
Ω
Ω
≤
λ(ad − bc) 16πa(dN1 + bN2 )
1−γ Z
1
esu0 +sw1 dx Ω
γ Z
1
eu0 +w1 dx Ω
2(1−γ)
,
which implies Z
u10 +w1
e Ω
dx ≤
λ(ad − bc) 16πa(dN1 + bN2 )
=
λ(ad − bc) 16πa(dN1 + bN2 )
1−γ Z 2γ−1
1−s Z
su10 +sw1
e
dx
Ω
s
su10 +sw1
e
1
γ 2γ−1
s
dx
Ω
.
Analogously, using H¨older inequality and (3.17), we can get (3.31). Lemma 3.2 will allow us to show that the functional Jλ+ is coercive on A. To this purpose, we need the following well-known Moser–Trudinger inequality (see [3]): Z Z 1 2 w k∇wk2 , ∀ w ∈ H(Ω) : wdx = 0, (3.32) e dx ≤ C1 exp 16π Ω Ω where C1 is a positive constant depending on Ω only. Lemma 3.3 There exist suitable constants C2 > 0 and C3 > 0 independent of λ such that, for every (w1 , w2 ) ∈ A there holds: (3.33) Jλ+ (w1 , w2 ) ≥ C2 k∇w1 k22 + k∇w2 k22 − C3 (ln λ + 1). Proof.
From (3.19)-(3.20), we see that: R u1 +w ad−bc 1 dx 0 2a(b+d) Ω e c+ , e1 ≥ R 2u1 +2w 1 dx 0 e Ω
c+ 2
e
Thus, by the constraints (3.16)-(3.17), we find: +
ec1 ≥ that is,
8π(dN1 + bN2 ) , R 1 λ(b + d) Ω eu0 +w1 dx c+ 1 c+ 2
+
ec2 ≥
≥
u20 +w2 dx Ωe . 2 e2u0 +2w2 dx
ad−bc 2d(a+c)
R
Ω
R
8π(cN1 + aN2 ) , R 2 λ(a + c) Ω eu0 +w2 dx
Z 1 8π(bN1 + dN2 ) − ln λ − ln eu0 +w1 dx, ≥ ln b+d ZΩ 8π(cN1 + aN2 ) 2 ≥ ln − ln λ − ln eu0 +w2 dx. a+c Ω
For any s ∈ (0, 1), using Lemma 3.2 and Moser–Trudinger inequality (3.32), we have Z 1 eu0 +w1 dx ln Ω Z 1−s 1 ad − bc 1 ≤ esu0 +sw1 dx ln λ + ln + ln s 16πa(dN1 + bN2 ) s Ω s 1 − s ad − bc 1 2 ≤ k∇w1 k2 + ln λ + ln + max u10 + ln C1 ; x∈Ω 16π s 16πa(dN1 + bN2 ) s 15
(3.34) (3.35)
(3.36)
and similarly: Z s 1−s ad − bc 1 2 eu0 +w2 dx ≤ ln k∇w2 k22 + ln λ + + max u20 + ln C1 . x∈Ω 16π s 16πd(cN + aN ) s 1 2 Ω Therefore from (3.24), (3.8), and (3.25), for any given ε > 0, we obtain: d a ε 1 + + 2 Jλ (w1 , w2 ) ≥ − − k∇w1 k2 + k∇w2 k22 + α1 c+ 1 + α2 c2 , 2b 2 2c 2ε
(3.37)
(3.38)
where α1 and α2 are given in (3.5). So, with the optimal choice 1 d c ε= + , 2 b a we deduce: Jλ+ (w1 , w2 ) ≥
a(ad − bc) ad − bc + k∇w1 k22 + k∇w2 k22 + α1 c+ 1 + α2 c2 . 4ab 2c(ad + bc)
Then, from (3.39) and (3.34)-(3.37) for s ∈ (0, 1) we conclude a(ad − bc) sα2 ad − bc sα1 2 + − − k∇w1 k2 + k∇w2 k22 Jλ (w1 , w2 ) ≥ 4a 16π 2c(ad + bc) 16π α1 + α2 − ln λ − C, s
(3.39)
(3.40)
with C a positive constant independent of λ and α1 and α2 given in (3.5). At this point, by choosing s > 0 sufficiently small, the statement of Lemma 3.3 follows. Since Jλ+ (w1 , w2 ) is weakly lower semicontinuous in A, by lemma 3.3 we conclude that Jλ+ (w1 , w2 ) attains the infimum in A. Next we show that, for λ is sufficiently large, the minimizer of Jλ+ belongs to the interior of A. To this end, we observe the following: Lemma 3.4 There exists a positive constant C4 , independent of λ, such that, √ d a |Ω| + min 1 + , 1 + λ − C4 (ln λ + λ + 1). inf Jλ (w1 , w2 ) ≥ 2 b c (w1 ,w2 )∈∂A Proof.
or
On the boundary of A, we have 2 Z Z 16πa(dN1 + bN2 ) 1 u10 +w1 = e dx e2u0 +2w1 dx λ(ad − bc) Ω Ω Z
Ω
u20 +w2
e
dx
2
=
16πd(cN1 + aN2 ) λ(ad − bc)
Z
2
e2u0 +2w2 dx.
(3.41)
(3.42)
(3.43)
Ω
Suppose for example that (3.42) holds. Then using (3.28) and H¨older inequality we obtain R 2 R u1 +w R u2 +w R u1 +u2 +w +w d u10 +w1 dx Z b 1 dx 2 dx e 0 0 e 0 0 1 2 dx Ω b+d b+d Ω e Ωe c+ u10 +w1 1 + e e dx ≤ R 2u1 +2w R 2u1 +2w R 2u2Ω+2w 1 dx 1 dx 2 dx 0 0 0 Ω Ωe Ωe Ωe p 16πad(dN1 + bN2 ) 4b πa(dN1 + bN2 )|Ω| + p ≤ λ(b + d)(ad − bc) λ(ad − bc)(b + d) 16
which implies: Z Z a d u20 +w2 c+ u10 +w1 c+ 2 1 dx 1−e e dx + 1 + 1−e e 1+ b c Ω Ω √ |Ω| d ≥ 1+ λ − C5 λ − C6 2 b λ 2
with C5 > 0 and C6 > 0 suitable constants independent of λ. + Now, estimating c+ 1 , c2 as in Lemma 3.3, we arrive at the estimate (3.41). At this point, we need to test Jλ+ over a suitable function in the interior of A, for which the opposite inequality in (3.41) holds. To this end, we follow [36] and recall that, for µ > 0 sufficiently large, there exist periodic solutions vµi (i = 1, 2) for the problem: 4πNi i i in Ω (3.44) ∆v = µeu0 +v eu0 +v − 1 + |Ω| R i 1 i i i i such that ui0 + vµi < 0 in Ω, ciµ := |Ω| Ω vµ dx → 0 and wµ := vµ − cµ → −u0 pointwise a.e. as µ → +∞. Those facts were proved in [46]. i Since eu0 ∈ L∞ (Ω) (i = 1, 2), by the dominated convergence theorem, we have i
i
eu0 +wµ → 1 strongly in as µ → +∞. In particular,
Z
i
Ω
i
Lp (Ω)
e2u0 +2wµ dx → |Ω|,
for any p ≥ 1
i = 1, 2
as µ → +∞. Therefore, for λ0 large and for fixed ε ∈ (0, 1), we can find µε ≫ 1, so that (wµ1 ε , wµ2 ε ) lies in the interior of A for every λ > λ0 , and the following holds: i h R 1 (ad−bc)|Ω| 2u10 +2wµ ε dx + c|Ω| a e Ω a+c ≥ 1 − ε, (3.45) R 2u2 +2w2 R 2u1 +2w1 µε dx − bc|Ω|2 µε dx 0 ad Ω e 0 Ωe i h R 2 2 (ad−bc)|Ω| d Ω e2u0 +2wµε dx + b|Ω| b+d ≥ 1 − ε. (3.46) R R 2 1 2 1 ad Ω e2u0 +2wµε dx Ω e2u0 +2wµε dx − bc|Ω|2
Using Jensen’s inequality, and Remark 3.1, in view of (3.19)-(3.20) by a straightforward calculation we get R R u1 +w1 2 1 2 1 2 1 b(a+c) c+ ad−bc e 0 µε dx + a(b+d) e 2 (wµε ,wµε ) Ω eu0 +u0 +wµε +wµε dx 1 ,w 2 ) a(b+d) Ω c+ (w e 1 µε µε ≥ R 1 1 2 Ω e2u0 +2wµε dx v R 2u1 +2w1 u µ u ε dx 16πa(dN1 + bN2 ) Ω e 0 × 1 + u R 2 t1 − 1 λ(ad − bc) u10 +wµ ε dx Ωe ≥
ad−bc a(b+d) |Ω|
+ R
Ωe
and similarly 2 1 c+ 2 (wµε ,wµε )
e
2 1 b(a+c) c+ 2 (wµε ,wµε ) a(b+d) |Ω|e 1 2u10 +2wµ ε
≥
dx
−
8πa(dN1 + bN2 ) λ(ad − bc)|Ω|
2 1 c(b+d) c+ ad−bc 1 (wµε ,wµε ) d(a+c) |Ω| + d(a+c) |Ω|e R 2u2 +2w2 µε dx 0 Ωe
17
−
8πd(cN1 + aN2 ) . λ(ad − bc)|Ω|
(3.47)
(3.48)
Then inserting (3.48) into (3.47) we find, 2 1 c+ 1 (wµε ,wµε )
8πa(dN1 + bN2 ) λ(ad − bc)|Ω| dx Ωe 2 1 b(a+c)|Ω| c(b+d) c+ ad−bc 1 (wµε ,wµε ) |Ω| + |Ω|e 8πd(cN1 + aN2 ) a(b+d) d(a+c) R d(a+c) + R 2u1 +2w1 − 2 +2w 2 2u λ(ad − bc)|Ω| µε dx µε dx 0 0 Ωe Ωe
≥
e
ad−bc a(b+d) |Ω| 1 2u10 +2wµ ε
R
≥
R
ad−bc a(b+d) |Ω| 1 2u10 +2wµ ε
Ωe
−
dx
+
2 1 b(ad−bc) bc 2 c+ 2 1 (wµε ,wµε ) ad(b+d) |Ω| + ad |Ω| e R 2u1 +2w1 R 2u2 +2w2 µε dx µε dx 0 0 Ωe Ωe
bd(a + c)(cN1 + aN2 ) 8π a(dN1 + bN2 ) + , − λ(ad − bc)|Ω| a(b + d) which implies +
1
2
ec1 (wµε ,wµε )
i h R 2 2 d Ω e2u0 +2wµε dx + b|Ω| ≥ R R 2 1 2 1 ad Ω e2u0 +2wµε dx Ω e2u0 +2wµε dx − bc|Ω|2 8πad bd(a + c)(cN1 + aN2 ) − a(dN1 + bN2 ) + . λ(ad − bc)2 |Ω| a(b + d) (ad−bc)|Ω| b+d
(3.49)
Similarly, we get +
1
2
ec2 (wµε ,wµε )
i h R 1 1 a Ω e2u0 +2wµε dx + c|Ω| ≥ R R 2 1 2 1 ad Ω e2u0 +2wµε dx Ω e2u0 +2wµε dx − bc|Ω|2 8πad ac(b + d)(dN1 + bN2 ) − d(cN1 + aN2 ) + . λ(ad − bc)2 |Ω| d(a + c) (ad−bc)|Ω| a+c
(3.50)
Then, by combining (3.49)-(3.50) and (3.45)-(3.46) we conclude that, bd(a + c)(cN1 + aN2 ) 8πad 1 ,w 2 ) c+ (wµ µ 1 ε ε a(dN1 + bN2 ) + , ≥ 1−ε− e λ(ad − bc)2 |Ω| a(b + d) + 8πad 2 1 ac(b + d)(dN1 + bN2 ) ec2 (wµε ,wµε ) ≥ 1 − ε − d(cN1 + aN2 ) + ; 2 λ(ad − bc) |Ω| d(a + c) for all λ > λ0 . As a consequence, for all λ > λ0 , we obtain that, Z + 1 1 2 1 1 − ec1 (wµε ,wµε ) eu0 +wµε dx Ω bd(a + c)(cN1 + aN2 ) 8πad a(dN + bN ) + , ≤ |Ω|ε + 1 2 λ(ad − bc)2 a(b + d) Z + 2 2 2 1 1 − ec2 (wµε ,wµε ) eu0 +wµε dx Ω 8πad ac(b + d)(dN1 + bN2 ) ≤ |Ω|ε + d(cN1 + aN2 ) + . λ(ad − bc)2 d(a + c)
(3.51)
(3.52)
Lemma 3.5 For λ > 0 sufficiently large, there holds: Jλ+ (wµ1 ε , wµ2 ε ) −
inf
(w1 ,w2 )∈∂A
18
Jλ+ (w1 , w2 ) < −1.
(3.53)
+ Proof. Using (3.51)-(3.52) and the fact that c+ 1 ≤ 0, c2 ≤ 0, we conclude that, for any small ε ∈ (0, 1), there exists a constant Cε > 0 such that, |Ω| d a 2 1 + Jλ (wµε , wµε ) ≤ 2+ + ελ + Cε . (3.54) 2 b c
So by virtue of Lemma 3.4 we get inf Jλ+ (w1 , w2 ) Jλ+ (wµ1 ε , wµ2 ε ) − (w1 ,w2 )∈∂A √ |Ω| d a d a ≤ min 1 + , 1 + −1 + 2 + + ε λ + C(ln λ + λ + 1), 2 b c b c
(3.55)
with C > 0 independent of λ. Clearly, (3.53) easily follows from (3.55) by choosing ε > 0 sufficiently small and λ > 0 sufficiently large. From Lemma 3.3 and 3.5 we easily conclude: ¯ > 0 such that for every λ > λ, ¯ the functional J + attains it minimum Corollary 3.1 There exists λ λ at a point (w1,λ , w2,λ ), which lies in the interior of A. Furthermore, + v1,λ = w1,λ + c+ 1 (w1,λ , w2,λ ),
+ v2,λ = w2,λ + c+ 2 (w1,λ , w2,λ )
(3.56)
defines a critical point for the functional Iλ in H(Ω) × H(Ω), namely a (weak) solution for (3.2). Concerning such a solution we prove: + + ) be the solution of (3.2) found above and defined by (3.56). We , v2,λ Proposition 3.2 Let (v1,λ have i) + i eu0 +vi,λ → 1, as λ → +∞ (i = 1, 2) (3.57)
pointwise a.e. in Ω and in Lp (Ω) for any p ≥ 1. + + ii) (v1,λ , v2,λ ) defines a local minimum for Iλ in H(Ω) × H(Ω).
Proof.
If we use (3.33) together with (3.54) we readily find that, Z 2 + i eu0 +vi,λ − 1 dx → 0, as λ → +∞, i = 1, 2.
(3.58)
Ω
1
+
2
+
2
+
Since, by Proposition 3.1, we know that eu0 +v1,λ < 1, eu0 +v2,λ < 1 in Ω, so by the dominated convergence theorem we conclude: 1
+
eu0 +v1,λ → 1,
eu0 +v2,λ → 1 as
λ → +∞
pointwise a.e. in Ω and in Lp (Ω), ∀ p ≥ 1 . To establish ii), we check that for any (w1 , w2 ) ∈ A and corresponding (c1 , c2 ) given by (3.9),(3.10), we have: ∂c1 Iλ (w1 + c1 (w1 , w2 ), w2 + c2 (w1 , w2 )) = 0 = ∂c2 Iλ (w1 + c1 (w1 , w2 ), w2 + c2 (w1 , w2 ))
19
and ∂c22 Iλ (w1 + c1 (w1 , w2 ), w2 + c2 (w1 , w2 )) 1 Z a(b + d)2 λ 2u10 +2w1 c1 c2 2c1 = e dx − e R1 (w1 , w2 , e ) , 2e b(ad − bc) Ω
(3.59)
∂c22 Iλ (w1 + c1 (w1 , w2 ), w2 + c2 (w1 , w2 )) 2 Z d(a + c)2 λ 2u20 +2w2 2c2 c2 c1 = e 2e dx − e R2 (w1 , w2 , e ) , c(ad − bc) Ω
(3.60)
∂c21 c2 Iλ (w1 + c1 (w1 , w2 ), w2 + c2 (w1 , w2 )) Z (a + c)(b + d)λ c1 c2 1 2 eu0 +u0 +w1 +w2 dx. e e =− (ad − bc) Ω
(3.61)
+ Next we use (3.59)-(3.61) with (c1 , c2 ) = (c+ 1 , c2 ), so that (3.19) and (3.20) hold with + sign. Thus, for vi+ = wi + c+ i , i = 1, 2, after straightforward calculation we find:
∂c22 Iλ (v1+ , v2+ ) 1
a(b + d)2 λ = b(ad − bc)
(
ad − bc a(b + d)
(
ad − bc d(a + c)
b(a + c) a(b + d) Ω 1 Z 2 16π(ad − bc)(dN1 + bN2 ) 2u10 +2v1+ − , dx e λa(b + d)2 Ω Z
1
+
eu0 +v1 dx +
Z
1
2
+
+
eu0 +u0 +v1 +v2 dx Ω
2
∂c22 Iλ (v1+ , v2+ ) 2
d(a + c)2 λ = c(ad − bc)
c(b + d) dx + e d(a + c) Ω 1 Z 2 4π(ad − bc)(cN1 + aN2 ) 2u20 +2v2+ dx e − . λd(a + c)2 Ω Z
u20 +v2+
Z
u10 +u20 +v1+ +v2+
e Ω
dx
2
In case (w1 , w2 ) lies in the interior of A, then we can use the strict inequality in (3.16), (3.17) and obtain Z + + 1 2 (a + c)(b + d)λ + + 2 eu0 +u0 +v1 +v2 dx, ∂c2 Iλ (v1 , v2 ) > 1 (ad − bc) ZΩ + + 1 2 (a + c)(b + d)λ ∂c22 Iλ (v1+ , v2+ ) > eu0 +u0 +v1 +v2 dx. 2 (ad − bc) Ω Therefore we have checked that, if (w1 , w2 ) is an interior point of A then the Hessian matrix of + Iλ (w1 +c1 , w2 +c2 ) with respect to (c1 , c2 ) is strictly positive definite at (c+ 1 (w1 , w2 ), c2 (w1 , w2 )). We + + apply such property, near the critical point (v1,λ , v2,λ ). Indeed, by continuity, for δ > 0 sufficiently small, we can ensure that, if (v1 , v2 ) = (w1 + c1 , w2 + c2 ) satisfies: + + kv1 − v1,λ k + kv2 − v2,λ k ≤ δ,
then (w1 , w2 ) belongs to the interior of A and + Iλ (v1 , v2 ) = Iλ (w1 + c1 , w2 + c2 ) ≥ Iλ (w1 + c+ 1 (w1 , w2 ), w2 + c2 (w1 , w2 )) + + = Jλ+ (w1 , w2 ) ≥ Iλ (v1,λ , v2,λ ).
+ + Consequently, (v1,λ , v2,λ ) defines a local minimizer for Iλ in H(Ω) × H(Ω), as desired.
20
3.2
Mountain-Pass solution
To complete the proof of Theorem 2.2, it remains to establish the existence of a second solution. Again we use the variational approach and show that the functional Iλ admits also a “saddle” critical point of “mountain-pass” type. We start to establish the following: Lemma 3.6 The functional Iλ satisfies the (P.S.) condition in H(Ω) × H(Ω). Namely, every sequence (v1,n , v2,n ) ∈ H(Ω) × H(Ω) satisfying: Iλ (v1,n , v2,n ) → m0
kIλ′ (v1,n , v2,n )k∗
→0
as as
n → +∞,
n → +∞,
(3.62) (3.63)
admits a strongly convergent subsequence in H(Ω)×H(Ω), where m0 is a constant and k·k∗ denotes the norm of dual space of H(Ω) × H(Ω). Proof. we have:
Let εn = kIλ′ (v1,n , v2,n )k∗ → 0, n → +∞, and observe that ∀ (ψ1 , ψ2 ) ∈ H(Ω)×H(Ω),
Iλ′ (v1,n , v2,n )[(ψ1 , ψ2 )] Z Z Z Z d a = ∇v2,n · ∇ψ1 dx ∇v1,n · ∇ψ2 dx + ∇v1,n · ∇ψ1 dx + ∇v2,n · ∇ψ2 dx + b Ω c Ω Ω Ω Z i h λ 0 0 + Q1 (v1,n , v2,n ) a(b + d)eu1 +v1,n ψ1 − b(a + c)eu2 +v2,n ψ2 dx ab(ad − bc) Ω Z Z Z α2 α1 λ(a + c)2 u20 +v2,n Q2 (v2,n )e ψ1 dx + ψ2 dx (3.64) ψ2 dx + + ac |Ω| Ω |Ω| Ω Ω and |Iλ′ (v1,n , v2,n )(ψ1 , ψ2 )| ≤ εn (kψ1 k + kψ2 k). In particular, if we take ψ1 = ψ2 = ψ ∈ H(Ω) in (3.64) we find: Z b+d a+c ′ ∇ Iλ (v1,n , v2,n )[(ψ, ψ)] = v1,n + v2,n · ∇ψ b c Ω Z Z λ Q1 (v1,n , v2,n )ψdx +2λ Q(v1,n , v2,n )ψdx + ab Ω Ω Z Z α1 + α2 λ(a + c)2 Q2 (v2,n )ψdx + ψdx, + ac |Ω| Ω Ω
(3.65)
(3.66)
where we recall that Q, Q1 , Q2 and α1 , α2 are defined in (3.5). As a consequence for ψ ≡ 1, we deduce that, Z Z 2 Q22 (v2,n )dx ≤ C Q1 (v1,n , v2,n )dx + Ω
Ω
for some suitable constant C > 0. In particular, Z Z 2 2(u10 +v1,n ) e2(u0 +v2,n ) dx ≤ C e dx + Ω
Ω
with a (possible different) C > 0.
21
(3.67)
R R 1 Decompose vj,n = wj,n + cj,n with Ω wj,n dx = 0, cj,n = |Ω| Ω vj,n dx (j = 1, 2) and observe that, (by assumption) " r
r 2 #
d c 1 c a − v1,n + v2,n Iλ (v1,n , v2,n ) = k∇v1,n k22 + ∇
2 b a a c 2 Z +λ Q(v1,n , v2,n )dx + α1 c1,n + α2 c2,n → m0 as n → +∞. (3.68) Ω
Moreover, from (3.67) and Jensen’s inequality, we find cj,n ≤ c0 ,
∀ n ∈ N,
j = 1, 2
(3.69)
with suitable c0 > 0. Next, let
b+d a+c w1,n + w2,n b c R so that Ω zn dx = 0. If we take in (3.66) ψ = zn+ = max{zn , 0}, from (3.67) we find #2 r Z "r λ a d 1 2 k∇zn+ k22 + (b + d)eu0 +v1,n − (a + c)eu0 +v2,n zn+ dx ad − bc Ω b c !Z r ad 2λ(a + c)(b + d) 1 2 + −1 eu0 +v1,n eu0 +v2,n zn+ dx ad − bc bc Ω zn =
≤ C(kzn+ k2 + εn kzn+ k).
(3.70)
(3.71)
Then from (3.71) and Poincar´e inequality we obtain Z 1 2 eu0 +v1,n eu0 +v2,n zn+ dx ≤ C(k∇w1,n k2 + k∇w2,n k2 ).
(3.72)
Ω
To proceed further we choose ψ1 = w1,n and ψ2 = w2,n in (3.64) and after straightforward calculations we obtain: r
r 2
c a d c 2 ′
− w1,n + w2,n k∇w1,n k2 + ∇ I (v1,n , v2,n )[(w1,n , w2,n )] =
b a a c 2 Z λ 2 2 2(u10 +v1,n ) 2 2 2(u20 +v2,n ) a (b + d) e w1,n + b (a + c) e w2,n + ab(ad − bc) Ω i λ(a + c)2 Z 1 2 2 −ab(a + c)(b + d)eu0 +v1,n eu0 +v2,n (w1,n + w2,n ) + e2(u0 +v2,n ) w2,n dx ac Ω Z Z a + c b+d 1 2 eu0 +v1,n w1,n dx + eu0 +v2,n w2,n dx . (3.73) −λ b c Ω Ω Clearly, in view of (3.67) we can estimate Z j eu0 +vj,n wj,n dx ≤ Ckwj,n k2 . Ω
While from (3.69) we get Z Z Z j j j e2(u0 +vj,n ) wj,n dx = e2(u0 +cj,n ) (ewj,n − 1) wj,n dx + e2(u0 +cj,n ) wj,n dx Ω
Ω
Ω
c0
2uj0
≥ −e ke
k2 kwj,n k2
≥ −Ck∇wj,n k2 ,
22
j = 1, 2.
Furthermore, we see that Z 1 2 eu0 +v1,n eu0 +v2,n (w1,n + w2,n )dx ΩZ 1 2 eu0 +v1,n eu0 +v2,n (w1,n + w2,n )+ dx ≤ Ω Z 1 2 eu0 +c1,n (ew1,n − 1) eu0 +v2,n (w1,n + w2,n )+ dx = {w1,n ≤0≤w2,n }
+
Z
eu0 +c1,n eu0 +v2,n (w1,n + w2,n )+ dx
Z
eu0 +c2,n (ew2,n − 1) eu0 +v1,n (w1,n + w2,n )+ dx
Z
eu0 +c2,n eu0 +v1,n (w1,n + w2,n )+ dx
1
2
{w1,n ≤0≤w2,n }
+
2
1
{w2,n ≤0≤w1,n }
+
2
1
{w2,n ≤0≤w1,n }
Z
+
1
2
eu0 +v1,n eu0 +v2,n (w1,n + w2,n )+ dx
{w1,n >0}∩{w2,n >0}
≤ C (k∇w1,n k2 + k∇w2,n k2 ) +
c b + b+d a+c
Z
Ω
1
2
eu0 +v1,n eu0 +v2,n zn+ dx.
Thus from (3.72), we conclude: Z 1 2 eu0 +v1,n eu0 +v2,n (w1,n + w2,n )dx ≤ C (k∇w1,n k2 + k∇w2,n k2 ) . Ω
Using the estimates above, together with (3.73) and (3.63), we conclude that
d c − b a
r
r 2
c a
w1,n + w2,n k∇w1,n k2 + ∇
≤ C. a c 2
(3.74)
Now from (3.68), (3.69) and (3.74), we deduce that {cj,n } is also uniformly bounded from below, for j = 1, 2. Consequently, {vj,n } is a uniformly bounded sequence in H(Ω), for j = 1, 2. So, along a subsequence, (denoted the same way), and for suitable vj ∈ H(Ω) (j = 1, 2) we have: vj,n → vj as n → +∞ weakly in H(Ω), and strongly in Lp (Ω), p ≥ 1 and pointwise a.e. in Ω; j
j
eu0 +vj,n → eu0 +vj as n → +∞ in Lp (Ω), p ≥ 1; j = 1, 2.
In particular, (v1 , v2 ) is a critical point for Iλ and by the above convergence properties we have: r
r 2
c a
(v1,n − v1 ) + (v2,n − v2 ) k∇(v1,n − + ∇
a c 2 ′ ′ = Iλ (v1,n , v2,n ) − Iλ (v1 , v2 ) [(v1,n − v1 , v2,n − v2 )] + o(1) → 0, as n → +∞.
d c − b a
v1 )k22
Thus, vj,n → vj strongly in H(Ω) as n → +∞, j = 1, 2; and the proof of Lemma 3.6 is completed.
23
+ + To proceed further, we need to use the minimization property of (v1,λ , v2,λ ) as given in Proposition 3.2-(ii). In case it defines a degenerate (local) minimum for Iλ , in the sense that for every δ > 0 sufficiently small,
inf
+ + kv1 −v1,λ k+kv2 −v2,λ k=δ
{
}
+ + Iλ (v1 , v2 ) = Iλ (v1,λ , v2,λ ).
then we obtain a 1-parameter family of (degenerate) local minima of Iλ , (see Corollary 1.6 of [18]), and the conclusion of Theorem 2.2 is obviously established in this case. + + Hence, we suppose that (v1,λ , v2,λ ) defines a strict local minimum for Iλ . In particular, for δ > 0 sufficiently small, the following holds: + + )< , v2,λ Iλ (v1,λ
inf Iλ (v1 , v2 ) := γ0 . + + k+kv2 −v2,λ k=δ} {kv1 −v1,λ
(3.75)
In addition, we observe that, + + − ξ) → −∞, − ξ, v2,λ Iλ (v1,λ
as
ξ → +∞.
+ ¯ i = 1, 2, we find Hence, for fixed ξ¯ > 1 sufficiently large and v¯i = vi,λ − ξ, + + − v¯2 k > δ − v¯1 k + kv2,λ kv1,λ
and
+ + ). , v2,λ Iλ (¯ v1 , v¯2 ) < Iλ (v1,λ
(3.76)
Lemma 3.6 together with (3.75) and (3.76), allow us to use the “mountain-pass” lemma of Ambrosetti-Rabinowitz [2] and conclude the existence of a second critical point (˜ v1,λ , v˜2,λ ) for Iλ satisfying: + + Iλ (˜ v1,λ , v˜2,λ ) ≥ γ0 > Iλ (v1,λ , v2,λ ). (3.77) + + By virtue of (3.77), such critical point yields to a solution for (3.2) distinct from (v1,λ , v2,λ ). This completes the proof of Theorem 2.2.
It would be interesting to see whether, as for the gauge group SU (3), a stronger multiplicity result holds, in relation to each vacua state of the system. For example, it is natural to expect that the “mountain-pass” solution is asymptotically gauge equivalent to the unbroken vacuum for λ → +∞; as it occurs in the Abelian case, see [15]. Acknowledgement The research of X. Han was supported by the by the National Natural Science Foundation of China under grant 11201118, Henan Basic Science and Frontier Technology Program Funds under grant 112300410054, and the Key Youth Teacher Foundation of Department Education of Henan Province under grant 2011GGJS-210. The research of G. Tarantello has been supported by FIRBIdeas project: Analysis and Beyond, and by PRIN-project: Nonlinear elliptic problems in the study of vortices and related topics.
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