Chin. Ann. Math. 5B:4(004,493 506. ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING BY INHOMO- GENEITIES IN TYPE II SUPERCONDUCTORS K. I. KIM LIU Zuhan Abstract The effect of an applied magnetic field on an inhomogeneous superconductor is studied and the value of the lower critical magnetic field H c at which superconducting vortices appear is estimated. In addition, the authors locate the vortices of local minimizers, which depends on the inhomogeneous term a(x. Keywords Superconductor, Vortices, Pinning mechanism, Critical magnetic field 000 MR Subject Classification 35J55, 35Q40. Introduction Consider the following functional: J(u, A = A u + ε (a(x u + h h ex h ex, (. which corresponds to the free energy of a superconductor in a prescribed constant magnetic field h ex. Here, R is the smooth, bounded, simply connected section of the superconductor and a(x : R is a given function satisfying 0 < min a(x a(x in. The unknowns are the complex-valued order parameter u H (, C and the U( connection A H (, R. h = curla is the induced magnetic field, A = ia. The order parameter u indicates the local state of the material, viz., u is the density of superconducting electron pairs so that, when u, the material is in its superconducting state, whereas when u 0, it is in its normal state. κ = ε is the Ginzburg-Landau parameter depending on the material. The modified Ginzburg-Landau functional (. was first written down by Likharev []. Then, this model has been used and developed by Aftalion, Sandier, Serfaty, Chapman and Richardson [3, 9]. The minima of a(x corresponds to the impurities in the material. It is well known that a superconductor placed in an applied magnetic field may change its phase when the field varies. There are two critical fields H c and H c for which a phase transition occurs. Above H c, superconductivity is destroyed and material is in the normal Manuscript received January 6, 004. Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784, Korea. Corresponding author. E-mail: zuhanl@yahoo.com Department of Mathematics, Xuzhou Normal University, Xuzhou 6, Jiangsu, China. Department of Mathematics, Yangzhou University, Yangzhou 500, Jiangsu, China. Project supported by the National Natural Science Foundation of China (No.007067, the Excellent Young Teachers Program of the Ministry of Education of China, the Jiangsu Provincial National Science Foundation of China and the Combinatorial and Computational Mathematics Center founded by KOSEF.
494 K. I. KIM & LIU, Z. H. phase. Below H c, the material is superconducting everywhere. When H c h ex H c, the material is in mixed state. In this state, since the works of Abrikosov [], the minimizers of (. exhibit vortices, i.e. points where the order parameter vanishes, if the Ginzburg- Landau parameter kappa is large enough. Actually these vortices are observed in nature experimentally. It is of importance to determine where these points are located and estimate the values of H c and H c. There have been many works toward to dealing with such problems [, 4, 3 ]. We do not attempt to give an exhaustive list of reference, but briefly summarize the advances concerning this problem. In the case a(x in, the value of the first critical field (as a function of kappa was predicted by Abrikosov when the domain is the plane. This computation was extended to arbitrary domain by others. However, these were only formal works. Very recently, S. Serfaty [8] was able to derive a completely rigorous proof of the previous predictions. In the case a(x in, Kim and Liu [4, 5, 6] studied the global minimizers of the functional (.. In this paper, we want to find the first critical magnetic field H c and study the local minimiers of (.. Inspired by [0, 8], we will choose a suitable M and construct the local minimizer of functional (. in where D a M = {(u, A H (, C H (, R : F a (u < M ln ε }, (. F a (u = [ u + ε (a(x u ]. (.3 Recall that the Ginzburg-Landau equation of any critical point (u, A H (, C H (, R associated to the functional (. is with the boundary conditions (G.L. Au = ε (a u u, h = (iu, A u { h = hex on, ( u iau ν = 0 on. (.4 Here denotes ( x, x, ν is the unit outer normal vector to and (z, w = Re(z w where z and w are in C. Let us now state our main results and hypothesis. Notice that J(u, A is invariant under U(-gauge transformation, i.e. of the type v = e iφ u, B = A + φ for any φ H (, R, which makes the problem non-compact. Therefore, throughout this paper, we shall impose the gauge condition { div (a(xa = 0 on, (.5 A ν = 0 on. Moreover, since we assume that is simply connected, we can say that there exists ξ H (, R such that { a(xa = ( ξx, ξ x in, (.6 ξ = 0 on.
ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING 495 Hence h = curla = div a ξ. (.7 Without loss of generality, we make the following hypothesis on the function a(x: 0 < b 0 = min a(x a(x. (.8 Now let ξ 0 be a smooth function satisfying ( div a ξ 0 + ξ 0 = in, ξ 0 = 0 on, (.9 and define Λ = { x : ξ 0(x a(x = max ξ } 0. (.0 a From Lemma 4. in [6], the set Λ is a finite set under the assumption that a(x is analytic. Our main theorem is the following Theorem.. Let R be a bounded, smooth, simply connected domain. Assume that a(x is a analytic function and (.8 holds. Assume that M > π(cardλ max a(x. Λ Then, there exist k = ξ 0 (x, kε = O ε (, k3 ε = o ε ( and ε 0 = ε 0 (M > 0 such that a(x max H c = k ln ε + k ε, (. and the following holds for ε < ε 0 : ( i If h ex H c, a solution of (G.L., which minimizes J(u, A in DM a, exists and satisfies b 0 u. (ii If H c + k3 ε h ex H c + O ε (, there exists a solution of (G.L. that minimizes J(u, A in DM a. It has a bounded positive number of vortices aε i of degree one such that and there exists a constant α > 0 such that dist (a ε i, Λ 0 as ε 0, (. dist (a ε i, a ε j α for i j. (.3 Remark.. It follows from (.0 and (. that the distribution of the location of vortices are governed both by the term a(x and by the function ξ 0 which is related to the magnetic field. This is called the pinning mechanism in superconductivity. Remark.. In Theorem., we only assume the smooth property of the inhomogeneous term a(x. This results can be extended to the model of variable thickness superconducting thin film in [0, 8]. This paper is organized as follows. In the next section we shall give some basic estimate for J(u, A. In Section 3, we pay attention to the splitting of the energy J(u, A. In Section 4, we shall split the magnetic field. In Section 5, we give some estimates for F a. In Section 6, we shall obtain the critical magnetic field. In Section 7, we prove Theorem.. In the discussion of the following sections, we always assume that the Abrikosov estimate H c C ln ε holds and h ex C ln ε. Also, for simplicity of the notation, we denote DM a by D hereafter.
496 K. I. KIM & LIU, Z. H.. Basic Estimates In this section, we give the following lemmas and omit their proofs here. Lemma.. For (ũ, Ã minimizing J(u, A in D with gauge condition (.5, we have where ξ is defined by (.6. J(ũ, Ã Ch ex, (. A eũ Ch ex, (. 4ε (a(x ũ Ch ex, (.3 ξ H (,R Ch ex, (.4 Lemma.. For any (ũ, Ã in D satisfying gauge condition (.5 and J(ũ, Ã Ch ex, there exists (u, A D satisfying (.5 such that In addition, if (ũ, Ã minimizes J in D, then there holds u, (.5 F a (u F a (ũ, (.6 J(u, A J(ũ, Ã + o(, (.7 ξ L (,R Ch ex. (.8 F a (u = F a (ũ + o( as ε 0, (.9 J(u, A = J(ũ, Ã + o( = J(u, Ã + o( as ε 0. (.0 Lemma.3. For (u, A D and u, let u γ H (, C be a minimizer of the following minimization problem { [ min v H (,C v + 4ε (a v ] u v } + ε γ, (. where 0 < γ <, 0 b 0 a(x in and a C (. Then, we have u γ in H 3 (, C and u γ satisfies u γ = ε uγ (a u γ + u uγ ε γ, (. F a (u γ F a (u M ln ε, (.3 u γ, (.4 u γ C ε. (.5 Since u γ C ε, the vortices are well defined in the following lemma (see [5]. Lemma.4. There exists λ > 0 and vortex points {a ε i } i T in such that CardT Ch ex, and u γ (x b 0 in \ i T B(a ε i, λε. (.6
ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING 497 In the remaining part of this section, we shall pay our attentions to the discussions on the minimizer u γ. Although we have put a function a(x in the functional (. on u γ, we found that the proofs of the following four lemmas on the properties of u γ can be obtained directly from adjusting the corresponding ones in [, 8] and [9] by replacing the energy density with [ v + ε a( v ] where v = uγ a. Thus, we omit the proof of the following lemmas in this paper. ( Lemma.5. For any 0 < γ < β <, u γ has no vortex i.e. u γ b 0 on the domain {x : dist (x, ε β }. Lemma.6. Card T is uniformly bounded by a constant N which is independent of ε. Let 0 < γ < β < µ < be such that µ = µ N+ > β. For ε small enough we may choose a subset T T and a radius ρ > 0 with λε ε µ ρ ε µ < ε β such that where e ε (u γ = B(a ε i,ρ e ε (u γ u γ b 0 in \ i T B(a ε i, ρ, (.7 u γ a C ln ε on B(a ε i, ρ for i T, (.8 C(β, µ ρ [ u γ + ε (a uγ ] and for i T, (.9 a ε i a ε j 8ρ for i j T. (.0 Lemma.7. For any u D, denote d i = deg(u γ, B(a ε i, ρ. Then we have d i = O(, i T. (. If the fact u C ε is proved (if u is a solution of (G.L., then this is true, then in the sense of [5] the vortices of u are well defined and has the same uniformly bound on the number as the vortices of u γ has. For u, we may also have bigger vortices (b ε i, q i of size ρ, such that u satisfies the same conclusions as in Lemma.6 for u γ. We may compare (a ε i, d i (the vortices of u γ with (b ε i, q i (the vortices of u by the minimal connection between the vortices as in [8]. Now define positive points p i and negative points n i as follows: ( i The set of p i s consists of all the b i s such that q i > 0 repeated q i times, with the a i s such that d i < 0 repeated d i times. (ii The set of n i s consists of all the b i s such that q i < 0 repeated q i times, with the a i s such that d i > 0 repeated d i times. Also define the positive holes by P i = B(p i, ρ and negative holes by N i = B(n i, ρ and add R \ as a hole of multiplicity d i q i. Then there are k positive holes and k negative holes, and the distance between the configurations ā = (a i, d i and b = (b i, q i is defined by the minimal connection [7], dist (ā, b = L(p i, n i = inf σ S k Lemma.8. For small ε, there holds k p i n σ(i. i= dist (ā, b Cε γ ln ε. (.
498 K. I. KIM & LIU, Z. H. 3. Splitting of the Energy J(u, A The results in this section are closely related to those in Section IV. in [8]. As in [8], we shall decompose the energy J(u, A with the following lemma. Lemma 3.. For any (ũ, Ã D satisfying (.5, (. (.4, let (u, A be associated with (ũ, Ã as in Lemma. and let u be associated with uγ, solving the minimization problem (., which has vortices (a i, d i satisfying Lemma.6. Then we have where V (ξ = a ξ + and ξ is defined by (.6. Let J(u, A = F a (u + V (ξ + o(, (3. div a ξ hex div a ξ + π d i ξ(a i, (3. i T 4. Splitting of the Magnetic Field V (ξ In this section, we will split the magnetic field under the assumptions of Lemma 3.. where ξ 0 is the unique solution of the equation ζ = ξ h ex ξ 0, (4. ( div a div a ξ 0 + div a ξ 0 = 0 in, (4. div a ξ 0 = on, (4.3 ξ 0 = 0 on. (4.4 where By computation, we have the following lemma. Lemma 4.. Under the assumptions of Lemma 3., we have Ṽ (ζ = V (ξ = V (h ex ξ 0 + Ṽ (ζ, (4.5 ζ ( + div a a ζ + π d i ζ(a i. (4.6 i T Now, from Lemmas 3. and 4., we can write J(u, A as J(u, A = J 0 + F a (u + πh ex d i ξ 0 (a i + Ṽ (ζ + o(, (4.7 i T where J 0 = h ex ( div a ξ 0 + a ξ 0. (4.8 By a simple computation we also have the following estimate.
ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING 499 Lemma 4.. J 0 satisfies inf J(u, A = J 0 + a + o(, (4.9 {(u,a D:T = } i.e. J 0 + a is (almost the minimal energy for vortex-less configuration. Lemma 4.3. ξ 0 is the unique solution of the equations (4. (4.4. Then ξ 0 satisfies div a ξ 0 + ξ 0 = in, (4.0 ξ 0 = 0 on, (4. < ξ 0 < 0 in. (4. Proof. By the uniqueness, (4.0 follows directly from (4. (4.4. (4. follows from the maximum principle. Lemma 4.4. J 0 + h ex = h ex Proof. By the definition of J 0 and (4.0, the conclusion follows. ξ 0. (4.3 In the last part of this section, we will pay our attention to the discussions on Ṽ (ζ. Since we found that the proof of the following two lemmas on the properties of ζ can be obtained directly from adjusting the corresponding proof of lemmas in [8], we give the statements of the lemmas. Lemma 4.5. The configurations (a i, d i being set, Ṽ (ζ is minimal for ζ satisfying the following equation ( div a div a ζ + div a ζ = π d i δ ai in, (4.4 i T div a ζ = 0 on, (4.5 ζ = 0 on, (4.6 and Ṽ (ζ = π i T d i ζ(a i. (4.7 Lemma 4.6. Ṽ (ζ = π ( d i d j ζ a i (a j C d i (4.8 i,j remains bounded as ε tends to 0. Furthermore ζ p L ( Cdist (p, k, k <, (4.9 ζ p ζ q H ( C p q k, k <. (4.0 i
500 K. I. KIM & LIU, Z. H. Here ζ p is the solution of ( div a div a ζp + div a ζp = πδ p in, (4. div a ζp = 0 on, (4. ζ p = 0 on. (4.3 5. Estimates for F a Denote e ε (u = [ u + ε (a u ] and ρ as before. The same argument of [0] gives the following lemma. Lemma 5.. u γ π a(a i d i ln ρ + W ((a, d,, (a k, d k + O(, (5. ρ i T where we have assumed T = {,,, k} and W ((a, d,, (a k, d k = π a(a i d i d j ln a i a j π d i R 0 (a i, i j T i T R 0 (x = Φ 0 (x i T a(a i d i ln x a i, where Φ 0 (x solves Lemma 5.. div a Φ 0 = π d i δ ai in, (5. i T Φ 0 = 0 on. (5.3 Proof. First, we have F a (u γ π i T a(a i d i ln ρ + π i T a(a i d i ln ρ ε + W ((a, d,, (a k, d k + O(, (5.4 F a (u γ π a(a i d i ln ρ + O(. ε i T (5.5 F a (u γ But we know from [] that e ε (u γ = Hence B(a i,ρ B(a i,ρ i T ρ u γ + i T B(a i,ρ e ε (u γ. (5.6 [ u γ + ε (a uγ ] πa(a i d i ln ρ ε + O(. B(a i,ρ e ε (u γ i T πa(a i d i ln ρ ε + O(. Now combining the above result with (5., we have the conclusions of this lemma.
ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING 50 6. Estimates of the Critical Magnetic Field H c In this section, we shall obtain the critical magnetic field H c. Lemma 6.. Suppose M > π max a(x. Then there exist k ε = O(, k3 ε = o( as ε 0 such that for ln ε h ex = max ξ + t, (6. 0 a there holds ( i If t < k, ε considering (ũ, Ã a minimizing configuration in D, then T = and J(ũ, Ã = inf J = J 0 + a + o(. (6. {(u,a D:T = } (ii If t = k ε, then there is (u, A D with one vortex of degree one such that J(u, A inf J. (6.3 {(u,a D:T = } (iii If t k ε + k ε 3, there exists (u, A in D having one vortex of degree one such that J(u, A < inf J. {(u,a D:T = } Proof. Recall that, for (u, A obtained from (ũ, Ã with the aid of Lemma., (4.7 yields J(u, A = F a (u + J 0 + πh ex d i ξ 0 (a i + Ṽ (ζ + o(, (6.4 i T and if T, using Lemma 5., we have F a (u F a (u γ π i T a(a i d i ln ε + O(. (6.5 Now substituting (6.5 into (6.4, we get J(u, A J 0 + π a(a i d i ln ε πh ex a(a i d i ξ 0(a i + Ṽ (ζ + O(. a(a i i T i T Hence as long as which is valid if J(u, A > inf J(u, A {(u,a D:T = } πh ex a(a i d i ξ 0(a i π a(a i d i ln ε + O(, a(a i i T i T h ex max ln ε ξ 0(x a(x + O(. (6.6
50 K. I. KIM & LIU, Z. H. As in [8], set { Z ε = t R : there exist (u, A D with at least one vortex and } J(u, A < inf {(u,a D:T = } J for h ex = k ln ε + t, where k = ξ 0 (x/a(x. In the following, we want to show Zε and then define max k ε = inf Z ε and prove that there exists k3 ε = o( such that [k ε + k3, ε + ] Z ε. To show Z ε, let c such that ξ 0 (c/a(c = max ξ 0 (x/a(x. Then, since ξ 0 = 0 on, we have c. Now consider the following minimization problem ν ε (c = min u, (6.7 W \B(c,ε where W = {u : u = av, v H ( \ B(c, ε, S, deg(v, B(c, ε = }. Then, it follows from discussions in [] that ν ε (c = a(x Φ ε + a = πa(c ln ε + O(, (6.8 where Φ ε satisfies \B(c,ε div a Φ ε = 0 in \ B(c, ε, Φ ε = 0 on, Φ ε = const. on B(c, ε, Φ ε a n = π. B(c,ε (6.9 Let u be a minimizer of the problem (6.7 which is well defined on \ B(c, ε. Now we extend it to the whole domain by defining it on B(c, ε as taking polar system centered at c such that ū(r, θ = f(rw(θ with w(θ B(c,ε = u B(c,ε, f in \ B(c, ε, f 0 in B(c, ε 4 and f (r C ε. Then, since ū has one vortex at c, ūγ has one vortex in B(c, Cε γ ln ε. Thus, we have F a (ū, B(c, ε = [ ū + ε (a ū ] and hence B(c,ε B(c,ε F a (ū, = F a (ū, B(c, ε + f L + C ε w L + C, (6.0 ε \B(c,ε ū K + a(cπ ln ε. (6. Then, taking ξ = h ex ξ 0 + ζ c, where ζ c is defined by (4. (4.3, we know from Section 4 that Now we derive that for V (ξ = V (h ex ξ 0 + Ṽ (ζc = J 0 + πh ex ξ 0 (c + πζ c (c + o(. h ex = max ξ 0 (x/a(x ln ε + t = ln ε + t, ξ 0 (c/a(c
ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING 503 J(ū, A K + a(cπ ln ε + J 0 + πh ex ξ 0 (c + o( [ K + a(cπ ln ε + J 0 + πξ 0 (c t a(c ] ξ 0 (c ln ε + o( This in turn implies t Z ε when = K + πξ 0 (ct + J 0 + o(. t K a + o(. π ξ 0 (c Therefore Z ε and k ε = inf Z ε K a + o(. π ξ 0 (c On the other hand, by (6.6, we have h ex k ln ε +O(. So, we know that k ε O( and thus k ε = O(. Now we prove that there exists k ε 3 = o( such that [k ε + k ε 3, + ] Z ε. In fact, let t Z ε and let (u, A satisfy J(u, A < inf J (6. {(u,a D:T = } for h ex, = k ln ε + t and u γ has vortices (a i, d i. Assume t > t and h ex, = k ln ε + t. Then we have J(u, A = F a (u + V (h ex, ξ 0 + Ṽ (ζ + o( = F a (u + J 0 + h ex, πd i ξ 0 (a i + Ṽ (ζ + o( i T F a (u + J 0 + h ex, πd i ξ 0 (a i + Ṽ (ζ + o( + (t t πd i ξ 0 (a i. i T i T This means that if t t k3 ε = o(, then by (6. J(u, A < inf J(u, A. This {(u,a D:T = } in turn implies [k ε + k3, ε + ] Z ε and the proof of this lemma is complete. In summary, we have deduced that H c = k ln ε +k ε where H c is the first critical field defined as the value of h ex for which the minimal energy among vortex-less configurations is equal to the minimal energy among single-vortex configurations. 7. Proof of Theorem. In order to prove Theorem., we need the following lemmas. Lemma 7.. Let h ex = k ln ε + o( ln ε and (ũ, Ã be a minimizer of J(u, A in D. Let {(a i, d i } k i= be vortices of uγ. Then d i = for i T = {,,, k} and there are ε 0 > 0 and α > 0 such that dist (a i, α > 0. Proof. The conclusion follows by the same argument in [0, 8]. Now to prove next lemmas we need the following assumption H c + k ε 3 h ex k ln ε + O(, (7.
504 K. I. KIM & LIU, Z. H. where with H c = k ln ε + k ε (7. k = max ξ 0(x/a(x, kε = O(, k3 ε = o(. (7.3 Lemma 7.. Let (u, A be obtained from a minimizer (ũ, Ã of J in D, and (aε i, dε i be the vortices of u γ. Denote Then we have Λ = { x : ξ 0(x a(x = max y ξ } 0(y. (7.4 a(y dist (a ε i, Λ 0 as ε 0, i T, (7.5 dist (a ε i, a ε j α > 0 for i j T. (7.6 The first result (7.5 remains true if we only assume h ex k ln ε + o( ln ε. Proof. If T, we have from Lemma 7. that d ε i = for all i T. Denote d = i T d ε i = CardT = deg(u γ,. Then it follows from Lemma 5., as in [0, 8], that W (a,, a k + π i T a(a ε i ln ε + O( F a (u γ πh ex where W (a,, a k = W ((a,,, (a k,. Then i T ξ 0 (a ε i + o( ln ε, π a(a ε i ln ε πh ex a(a ε i max ξ 0(x a(x i T i T ( πh ex a(a ε ξ0 (a ε i i a(a ε i T i + max ξ 0(x + o( ln ε, a(x which in turn gives hence ( ξ0 (a ε i a(a ε i + max ξ 0(x a(x i T Moreover, since W (a ε,, a ε k O( and o( ln ε h ex 0, dist (a ε i, Λ 0 as ε 0, i T. W (a ε,, a ε k + π i T a(a ε i ln ε + O( πh ex i T ξ 0 (a ε i + O(, we have W (a ε,, a ε k O( if h ex k ln ε + O(. Therefore we conclude that a ε i aε j remains bounded from below uniformly since as in [5] we could prove that W + if a ε i aε j 0 for some i j. Lemma 7. is proved. Lemma 7.3. Assume that M > π(cardλ max a(x. Then, for ε sufficiently small, a Λ minimizer (ũ, Ã of J in D is a solution of (G.L., and (u, A = (ũ, Ã, where (u, A comes from (ũ, Ã in Lemma..
ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING 505 Proof. It suffices to show that (ũ, Ã is not on the boundary of D. Since we have proved that W is a bounded quantity in Lemma 7. and dist (a ε i, aε j α > 0, we have π a(a i ln ε + O( π a(a i ln ε + O(. i T i T Hence, this inequality together with Lemma. yields that F a (u = π i T a(a i ln ε + O(. (7.7 Now it follows from Lemma 7. that as ε 0, a ε i c i Λ and thus we get from (7.6 that c i c j for i j. Hence d = CardT CardΛ. (7.8 Thus π a(a i πcardλ max a(x + δ < M for ε ε 0. (7.9 Λ i T ( Here, δ M πcardλ max a(x. Thus Λ F a (ũ π i T a(a i ln ε + O( (M δ ln ε, (7.0 which implies that (ũ, Ã is not on D. Thus Lemma 7.3 is proved. Now since (u, A is a solution of (G.L., we can show that u C/ε. Then u has bigger vortices of size ρ : {(b i, q i } i T. The following lemma compares the vortices of u γ with the vortices of u. Lemma 7.4. For sufficiently small ε, we have ( i If (u, A is a solution of (G.L. such that J(u, A Ch ex, then u, and u C ε. ( (ii If (u, A is a solution of (G.L. such that u γ has no vortex i.e. u γ b 0 and J(u, A J 0 + ( a + o(, then u has not vortex on u b 0. (iii If (u, A is a solution of (G.L. given by Theorem., then its vortices (of size ρ satisfy the same conclusions as those of u γ. (iv In addition, if (a i i T are the vortices of u γ of degree one, then the vortices {b i } i T of u are of degree one and Lemma.6 (on u γ is satisfied by u. Proof. The conclusions follow by the same argument in [9]. Proof of Theorem.. Let (ũ, Ã be a minimizer of J(u, A in D. Then, from Lemma 7.3, we have that (ũ, Ã is a solution of (G.L., and (u, A = (ũ, Ã, where (u, A comes from (ũ, Ã by Lemma.. Now, in the case that h ex H c, we know that u γ has not a vortex from Lemma 6.. Combining this with the conclusion (ii of Lemma 7.4, we obtain that u has no vortex and u b 0 in. Moreover, from the conclusion (i of Lemma 7.4, we have u in. Thus the conclusion (i of Theorem. is proved. In the case that H c + k3 h ex H c + O(, we know that u γ has vortices from Lemma 6.. Then by the conclusions of Lemma 7., Lemma 7., and the conclusions (iii and (iv in Lemma 7.4, we obtain the conclusion (ii of Theorem.. The proof of Theorem. is complete.
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