Asymptotics for Maxwell-Chern-Simons multivortices

Asymptotics for Maxwell-Chern-Simons multivortices Tonia Ricciardi

∗

Dipartimento di Matematica e Applicazioni Universit` a di Napoli Federico II Via Cintia, Complesso Monte S. Angelo 80126 Napoli, Italy fax: +39–081–675665 E-mail: [email protected]

Abstract We consider a system of two elliptic PDE’s with exponential nonlinearities, defined on a compact surface, whose solutions correspond to multivortices for a self-dual Maxwell-Chern-Simons gauge theory. We analyze the limit behavior of its solutions as the Chern-Simons action dominates the Maxwell action and we provide a rigorous justification in the C q -norm, for all q ≥ 0, of the asymptotics formally derived by the physicists.

key words: Chern-Simons vortex theory, nonlinear elliptic PDE’s MSC 2000 subject classification: 35J60

1

Introduction

Chern-Simons gauge theories are of interest in several areas of physics, including high-Tc superconductivity, as they can describe particles which are ∗

Partially supported by CNR Fellowship no. 203.01.69 (19/01/98) and by MURST Project “Variational Methods and Nonlinear Differential Equations”.

1

both electrically and magnetically “charged”. On the other hand, the introduction of the Chern-Simons term into the Lagrangian results in serious difficulties for a rigorous treatment of the corresponding equations of motion. Therefore a considerable effort has been devoted in recent years to the study of models with a self-dual structure. Such a structure allows us to obtain time-independent, energy-minimizer solutions to the full second-order equations of motion by solving appropriate first-order equations, also known as the “Bogomol’nyi equations” (see Jaffe-Taubes [11]). These particular solutions are called multivortices. For an overview of self-dual Chern-Simons theories see the monograph of Dunne [8]. In [9] and [10], Hong-Kim-Pac and Jackiw-Weinberg introduced a ChernSimons model, where the Maxwell term is neglected in order to achieve selfduality. We call this model the CS model. The CS multivortices have proved to be particularly interesting from the mathematical viewpoint, and have been extensively analyzed, see Caffarelli-Yang [2], Tarantello [17], NolascoTarantello [14]–[15], Ding-Jost-Li-Wang [6]–[7], Chae-Imanuvilov [3], and the references therein. Out of these studies unexpected connections to Riemannian geometry and sharp Sobolev embeddings have emerged. For example, connections between the asymptotics of CS multivortices and the geometric properties of the underlying surface have been found in [6]–[7] and [14]–[15]. Moreover, CS multivortices have been employed in [14] to construct minimizing sequences for sharp Sobolev-type embeddings. Of course, it is desirable to have a Chern-Simons multivortex theory which includes the Maxwell term as well. In [12], Lee-Lee-Min introduced a selfdual model including both the Chern-Simons term and the Maxwell term (the MCS model) where self-duality is achieved by the aid of a neutral scalar field. Lee-Lee-Min also showed by formal arguments that in the limit where the Chern-Simons action dominates the Maxwell action (the CS limit), the MCS model reduces to the CS model. At this point the natural question is, whether or not the CS limit may be rigorously justified. This problem has already been addressed by Chae-Kim [4] in a full space setting and by Chae-Kim [5] and by Ricciardi-Tarantello [16] on the flat torus (these results extend with no effort to general compact two-dimensional surfaces). In [4]–[5] and [16] it was shown that the MCS multivortex theory is by many ways equivalent to the CS multivortex theory. Because of these results, the CS limit appeared to be reasonable. However, concerning the CS limit only partial results were obtained. Indeed, only a particular class of “maximal” solutions, to which monotonicity arguments 2

apply, was considered in [5]. On the other hand, it was shown in [16] that the MCS model admits in general multiple solutions (see Theorem 1.1 below). The CS limit was shown to hold for an arbitrary sequence of solutions, but the convergence of the neutral scalar field was obtained only in the L2 -sense. The analytical difficulties are due to the fact that the Maxwell-term is of higher order with respect to the Chern-Simons term, and therefore dropping the Maxwell term results in a problem of the singular perturbation type. The aim of this note is to complete the results in [5] and [16], by showing that, in the case of compact surfaces, the CS limit holds for an arbitrary sequence of solutions in the C q -norm, for all q ≥ 0. Our arguments will also provide simplified proofs of some estimates in [16]. We now proceed to define the elliptic problem for MCS multivortices. Let Σ be a smooth compact Riemannian 2-manifold. In what follows, unless otherwise specified, all equations are defined on Σ. The elliptic system for MCS multivortices is given by: (1)

u

∆u =µ (e − N ) + 4π

n X

δpj

j=1

(2)

∆N = − µ2 (eu − N ) + µ eu (N − 1).

Here, eu and N are smooth real functions defined on Σ, n ≥ 1 is a positive integer, p1 , . . . , pn are given points in Σ, δpj , for j = 1, . . . , n, is the Dirac measure concentrated at pj and µ > 0 is a constant. All derivatives are taken covariantly with respect to the metric of Σ, in particular ∆ = div∇ is the Laplace-Beltrami operator. Physically, the field eu describes the density of superconducting particles on Σ, N is a neutral scalar field, p1 , . . . , pn are the (prescribed) vortex points and µ−1 is the coupling constant for the Maxwell action. Equation (1) implies that eu vanishes exactly at the vortex points p1 , . . . , pn , and hence that Σ is in the “normal state”, i.e. non-superconducting, exactly at p1 , . . . , pn . The CS limit corresponds to letting µ → +∞. This limit is meaningful by virtue of an existence result proved in [16], which “in words” states that system (1)–(2) admits (multiple) solutions for all large values of µ, provided the area of Σ, denoted |Σ|, is not too small compared to the number of vortex points. The precise statement is contained in the following theorem:

3

Theorem 1.1 ([16]). The condition 4πn µ> |Σ| − 16πn

1+

s

16πn |Σ|

!

is necessary for the existence of solutions to system (1)–(2). For every given set of vortex points {p1 , . . . , pn } ⊂ Σ, there exists a constant γ0 > 0 (depending on the Green’s function for ∆ on Σ with poles in p1 , . . . , pn , in the “normalized” case |Σ| = 1) such that for all µ>

4πn γ0 |Σ|

system (1)–(2) admits at least two solutions. Henceforth we assume that Σ, p1 , . . . , pn and µ → +∞ are such that solutions for (1)–(2) exist. A formal argument to get an idea of what kind of asymptotics we should expect is as follows. If the right hand side in (1) remains bounded as µ → +∞, we must have N − eu → 0. On the other hand, system (1)–(2) implies that (u, N ) satisfies the equation: n

(3)

∆(u +

X 1 N ) = eu (N − 1) + 4π δpj . µ j=1

Inserting N = eu into (3) and dropping the term µ−1 N , we obtain: (4)

u

u

∆u = e (e − 1) + 4π

n X

δpj .

j=1

Equation (4) is precisely the equation for CS multivortices, as considered in [2], [17], [14]–[15], [6]–[7], [3]. Our main result is the following Theorem 1.2. Let (u, N ) be solutions for (1)–(2), with µ → +∞. There exists u˜ satisfying (4) such that for any q ≥ 0, up to subsequences, there holds: eu → eu˜

in C q (Σ)

N → eu˜

in C q (Σ). 4

The proof is based on the combination of an L∞ -estimate (a weaker version of which was already used in [16]), and on a new “energy identity” as in Lemma 2.2 below, which is the key ingredient to derive a priori bounds in H 1 (Σ) independently of µ → +∞. These estimates will be obtained in Section 2, where a direct proof of the strong H 1 -convergence will also be provided. In Section 3 we shall use the estimates of Section 2 as a basis for an induction argument which will yield bounds in the H k (Σ)∩L∞ (Σ) spaces, for all k ≥ 0. Then standard Sobolev embeddings will conclude the proof.

2

H 1-convergence

In order to work in Sobolev spaces, it is convenient to remove the singular term in (1)–(4) in the following standard way. Let u0 be the Green’s function uniquely defined on Σ by the properties (see [1]): (5)

∆u0 = 4π

(6)

Z

n X

δpj −

j=1

4πn |Σ|

u0 = 0.

Σ

Note that eu0 is smooth on Σ. Indeed, eu0 is clearly smooth away from the vortex points p1 , . . . , pn . Near a fixed pj , it blows up as log |x−pj |2 , therefore eu0 vanishes as |x − pj |2 in pj and in particular it is smooth near pj . Setting v = u − u0 , we have that system (1)–(2) reduces to the following system with smooth data: (7) (8)

4πn |Σ| 2 u0 +v ∆N = −µ (e − N ) + µ eu0 +v (N − 1)

∆v = µ (eu0 +v − N ) +

and (3) reduces to (9)

∆(v +

1 4πn N ) = eu0 +v (N − 1) + . µ |Σ|

Similarly, (4) is equivalent to the equation: (10)

∆v = eu0 +v (eu0 +v − 1) + 5

4πn . |Σ|

Henceforth, unless otherwise specified, all function spaces are defined on Σ, and all integrals are taken over Σ with respect to the Lebesgue measure. We denote by C > 0 a general constant independent of v, N, µ, whose value may vary from line to line. The following lemma provides L∞ -estimates independent of µ → +∞ for solutions to the MCS system (7)–(8). Recall that the solutions v, N are assumed to be smooth. Lemma 2.1. Suppose (v, N ) satisfies (7)–(8). Then eu0 +v < N < 1,

pointwise on Σ.

Proof. Let x¯, y¯ ∈ Σ satisfy eu0 +v (¯ x) = maxΣ eu0 +v and N (¯ y ) = maxΣ N . u0 +v Note that e satisfies the equation ∆eu0 +v = µ(eu0 +v − N )eu0 +v + |∇(u0 + v)|2 eu0 +v

(11)

and that the equation (8) for N may be written in the form: 1 1 (12) ∆N − µ2 (1 + eu0 +v )N = −µ2 (1 + )eu0 +v . µ µ Then
u0 +v u0 +v u0 +v 2 u0 +v 0 ≥ ∆e −N e =µ e +|∇(u0 + v)| e x ¯

x ¯

x ¯

and

1 u0 +v 1 u0 +v 2 2 0 ≥ ∆N = µ (1 + e )N − µ (1 + )e . µ µ y¯ y¯ y¯ Consequently, max eu0 +v = eu0 +v (¯ x) ≤ N (¯ x)

(13)

Σ

and max N = N (¯ y) ≤

(14)

(1 + µ1 )eu0 +v (¯ y)

Σ

1 + µ1 eu0 +v (¯ y)

.

Since the real function f defined by f (t) = (1 + µ1 )t/(1 + µ1 t) for all t > −µ is increasing, combining (13) and (14) we have that max N ≤ Σ

y) (1 + µ1 )eu0 +v (¯ y) 1 + µ1 eu0 +v (¯

≤ ≤

(1 + µ1 ) maxΣ eu0 +v 1+ (1 + 1+ 6

1 maxΣ eu0 +v µ 1 ) maxΣ N µ , 1 maxΣ N µ

≤

x) (1 + µ1 )N (¯ x) 1 + µ1 N (¯

which yields N ≤ 1. Since N ≡ 1 is not admissible, we conclude that N < 1 pointwise on Σ. Now we use the equation for eu0 +v − N : 1 (15) ∆(eu0 +v − N ) − µ2 (1 + eu0 +v )(eu0 +v − N ) =|∇(u0 + v)|2 eu0 +v µ + µ eu0 +v (1 − N ). Since 1 − N > 0, the right hand side in (15) is positive and therefore the maximum principle implies eu0 +v − N ≤ 0. Since eu0 +v − N ≡ 0 is not admissible, we conclude that eu0 +v < N , pointwise on Σ. We shall now derive an “energy identity”, which will provide bounds in H 1 independent of µ → +∞. Lemma 2.2. Let (v, N ) satisfy (7)–(8). Then the following identity holds: Z Z Z 2 2 u0 +v 2 (16) |∇N | + µ (e − N ) + |∇(u0 + v)|2 eu0 +v N Z =µ eu0 +v (N − eu0 +v ). Moreover, the following estimates hold, independently of µ → +∞: (i) kµ(eu0 +v − N )kL2 ≤ C (ii) kN kH 1 ≤ C (iii) kvkH 2 ≤ C. Proof. Multiplying (8) by eu0 +v − N and integrating, we have: Z Z Z u0 +v 2 u0 +v 2 (17) (e −N )∆N = −µ (e −N ) +µ eu0 +v (N −1)(eu0 +v −N ). Now, using (11) we have Z Z Z u0 +v u0 +v (e − N )∆N = e ∆N − N ∆N Z Z u0 +v = ∆e N + |∇N |2 Z Z
u0 +v 2 u0 +v = µ(e − N ) + |∇(u0 + v)| e N + |∇N |2 Z Z Z u0 +v u0 +v 2 u0 +v =µ (e − N )e N + |∇(u0 + v)| e N + |∇N |2 . 7

Inserting into (17) we obtain the desired identity (16). To obtain the a priori estimates, note that an integration of (7) yields Z (18) µ (N − eu0 +v ) = 4πn. Therefore, using the pointwise estimates of Lemma 2.1, we obtain from identity (16) that Z Z Z 2 2 u0 +v 2 (e − N ) + |∇(u0 + v)|2 eu0 +v N |∇N | + µ Z (19) ≤µ (N − eu0 +v ) = 4πn. Now, (19) immediately implies (i) and k∇N kL2 ≤ C. Since by Lemma 2.1 we also have 0 < N < 1, we obtain (ii). To see (iii), note that by (i) and (1) R 2 2 ≤ C, so we only have to estimate | v|. By Lemma 2.1 we we have kD vk L R R R have that v ≤ − u0 = 0. In order to bound v from below, we integrate (9) and use Lemma 2.1 and Sobolev embeddings to obtain: Z Z u0 +v 4πn = e (1 − N ) ≤ eu0 +v Z Z u0 −1 −1 ≤ max e exp{|Σ| v + kv − |Σ| vkL∞ } Σ Z ≤ max eu0 exp{|Σ|−1 v + CkD2 vkL2 }. Σ

So, −1

exp{|Σ|

Z

v} ≥

maxΣ

4πn , 2 eCkD vkL2

eu0

and (iii) follows. Using Lemma 2.1 and Lemma 2.2, we now prove the strong H 1 -convergence result: Proposition 2.1. Let (v, N ) be solutions for (7)–(8) with µ → +∞. There exists a solution v˜ for (10) such that, up to subsequences: v → v˜, strong H 1 N → eu0 +˜v , strong H 1 8

µ(eu0 +v − N ) → eu0 +˜v (eu0 +˜v − 1), strong L2 . ˜ ∈ H1 Proof. By the a priori estimates of Lemma 2.2, there exist v˜ ∈ H 2 , N ˜ and W ∈ L2 such that v + v˜ weak H 2 , strong H 1 and a.e. on Σ, N + N 1 p u0 +v weak H , strong L for all p ≥ 1, and a.e. on Σ, and µ(e − N) + W , 2 u0 +v u0 +˜ v 1 weak L . It is readily checked that e →e , strong H and a.e. on Σ. ˜ = eu0 +˜v a.e. on Σ. Claim (a). N Indeed, by (7) we have: Z Z Z 4πn 1 u0 +v ∇v · ∇ϕ + (e − N )ϕ + ϕ=0 ∀ϕ ∈ C ∞ . µ µ|Σ| ˜ strong L2 , letting Since k∇vk2 ≤ C, eu0 +v → eu0 +˜v strong L2 and N → N µ → +∞ we find Z ˜ )ϕ = 0 (eu0 +˜v − N ∀ϕ ∈ C ∞ , and thus Claim (a) follows. Claim (b). v˜ satisfies (10) a.e. on Σ. From (9) we have: Z Z Z Z 1 4πn u0 +v ∇v · ∇ϕ + ∇N · ∇ϕ + e (N − 1)ϕ+ ϕ=0 µ |Σ| ∀ϕ ∈ C ∞ . Since v → v˜ strong H 1 , k∇N k2 ≤ C, moreover eu0 +v → eu0 +˜v strong L2 and, by Claim (a), N → eu0 +˜v strong L2 , letting µ → +∞ we obtain Claim (b). Claim (c). W = eu0 +˜v (eu0 +˜v − 1) a.e. on Σ. Indeed, we have from (8) that: Z Z Z 1 u0 +v ∇N · ∇ϕ − µ(e − N )ϕ + eu0 +v (N − 1)ϕ = 0 ∀ϕ ∈ C ∞ . µ Since k∇N k2 ≤ C, µ(eu0 +v − N ) + W weak L2 , eu0 +v → eu0 +˜v strong L2 and N → E u0 +˜v strong L2 , letting µ → +∞, we find: Z Z − W ϕ + eu0 +˜v (eu0 +˜v − 1)ϕ = 0 ∀ϕ ∈ C ∞ . 9

Thus Claim (c) follows. We shall now obtain the strong convergences as a consequence of the following Claim (d). As µ → +∞, ˜ k2 2 + kW k2 2 . k∇N k2L2 + kµ(eu0 +v − N )k2L2 → k∇N L L To see this, we rewrite identity (16) in the form: Z 2 u0 +v 2 (20) k∇N kL2 + kµ(e − N )kL2 =µ eu0 +v (N − eu0 +v ) Z − |∇(u0 + v)|2 eu0 +v N. Now since eu0 +v → eu0 +˜v strong L2 and since, by Claim (c), µ(eu0 +v − N ) + eu0 +˜v (eu0 +˜v − 1) weak L2 , we have Z Z u0 +v u0 +v µ e (N − e ) → e2(u0 +˜v) (1 − eu0 +˜v ). Furthermore, we can expand: Z Z 2 u0 +v |∇(u0 + v)| e N = (∆eu0 +v − ∆(u0 + v)eu0 +v )N Z Z u0 +v = − ∇e · ∇N − µ(eu0 +v − N )eu0 +v N. Therefore, since ∇eu0 +v → ∇eu0 +˜v strong L2 , ∇N + ∇eu0 +˜v weak L2 , µ(eu0 +v − N ) + eu0 +˜v (eu0 +˜v − 1) weak L2 , eu0 +v → eu0 +˜v strong Lp for all p ≥ 1, N → eu0 +˜v strong Lp for all p ≥ 1, we find that, as µ → +∞, Z Z Z 2 u0 +v u0 +˜ v 2 |∇(u0 + v)| e N → − |∇e | + eu0 +˜v (1 − eu0 +˜v )e2(u0 +˜v) . Taking limits in (20), we have that, as µ → +∞: Z Z 2 u0 +v 2 2(u0 +˜ v) u0 +˜ v k∇N kL2 + kµ(e − N )kL2 → e (1 − e ) + |∇eu0 +˜v |2 Z − eu0 +˜v (1 − eu0 +˜v )e2(u0 +˜v) Z Z u0 +˜ v 2 = |∇e | + e2(u0 +˜v) (1 − eu0 +˜v )2 ˜ k2 2 + kW k2 2 . =k∇N L L 10

Therefore, Claim (d) is also proved. Now by standard functional analytic results, the H 1 -convergence of N and the L2 -convergence of µ(eu0 +v − N ) are strong.

3

H k -convergence

We shall now bound v and N independently of µ → +∞ in the Sobolev spaces H k , for all k ≥ 2. The bounds will follow from an iteration in the spaces X k = H k ∩ L∞ , endowed with the norms: kukX k = kukH k + kukL∞ . By the Gagliardo-Nirenberg estimate (see e.g. [13]), we have: kDj uk 2k ≤ CkukX k , j

∀ u ∈ X k,

where Dj u denotes all derivatives of u of order j, 0 ≤ j ≤ k, and C > 0 does not depend on u. From the above estimate, using the H¨older inequality, one easily sees that if u1 , . . . , us ∈ X k and β1 , . . . , βs are multi-indices such that |β1 | + · · · + |βs | = k, then the product Dβ1 u1 · · · Dβs us ∈ L2 and kDβ1 u1 · · · Dβs us kL2 ≤ C k ku1 kX k · · · kus kX k . In particular, X k is an algebra: if u1 , u2 ∈ X k , then the product u1 u2 ∈ X k and (21)

ku1 u2 kX k ≤ Cku1 kX k ku2 kX k ,

∀u1 , u2 ∈ X k ,

for some C > 0 independent of u1 , u2 ∈ X k . It is also readily seen that (22)

kev kX k ≤ CekvkL∞ (1 + kvkkX k )

∀v ∈ X k .

Now we observe that the equation satisfied by µ(eu0 +v − N ) and the equation for N as in (12) have a similar structure. Indeed, µ(eu0 +v − N ) satisfies the equation (23)

1 u0 +v e )µ(eu0 +v − N ) µ 1 = µ2 (eu0 +v (1 − N ) + |∇(u0 + v)|2 eu0 +v ). µ

∆µ(eu0 +v − N ) − µ2 (1 +

11

Equations (12) and (23) are of the form: (24)

∆u − µ2 (1 +

1 c)u = µ2 f µ

with c = eu0 +v , for some smooth f uniformly bounded for µ → +∞. The following two lemmas provide estimates independent of µ for equation (24): Lemma 3.1. Suppose u is a smooth solution of (24), where c ≥ 0 and f are also smooth. Then (25)

kukLp ≤ kf kLp ,

∀ 1 ≤ p ≤ ∞.

Proof. For p = ∞ the statement is an immediate consequence of the maximum principle. Indeed, let x ∈ Σ be such that u(x) = maxΣ u. Then 0 ≥ ∆u |x = µ2 (1 +

1 c)u|x + µ2 f |x , µ

and therefore, recalling that c ≥ 0, we find: u(x) ≤ −

f |f | ≤ max |f |. 1 |x ≤ max Σ 1 + 1c Σ 1 + µc µ

Similarly, we see that if x ∈ Σ satisfies u(x) = minΣ u, then u(x) ≥ − max |f |, Σ

and thus we conclude that kukL∞ ≤ kf kL∞ , as desired. If 1 ≤ p < ∞, we multiply (24) by |u|p−2 u and integrate to obtain: Z Z Z 1 p−2 2 2 p 2 (p − 1) |u| |∇u| + µ (1 + c)|u| = −µ f |u|p−2 u. µ Consequently, Z Z Z 1 2 p p−2 2 2 µ (1 + c)|u| = − (p − 1) |u| |∇u| − µ f |u|p−2 u µ 2 ≤ µ kf kLp kukp−1 Lp . Since c ≥ 0, it follows that kukpLp ≤ kf kLp kukp−1 Lp and thus we obtain (25). 12

Lemma 3.2. Suppose u is a smooth solution of (24). Then for all k there exists C > 0 independent of µ → +∞ such that 1 kukX k ≤ C kckX k kukX k + kf kX k . µ Consequently, if there holds kckX k ≤ C 0 for some constant C 0 > 0 independent of µ → +∞, then there exist constants µk > 0, Ck > 0 such that kukX k ≤ Ck kf kX k ,

∀µ ≥ µk .

Proof. We fix a multi-index α such that 0 ≤ |α| ≤ k. Differentiating (24) by Dα , multiplying by Dα u and integrating we find: Z Z α α 2 2 α 2 α 2 Dα uDα f. (26) k∇D ukL2 + µ kD ukL2 = −µ D (cu)D u − µ Now, using the H¨older inequality and (21) we have Z | Dα (cu)Dα u| ≤kDα (cu)k2 kDα uk2 ≤ kcukH k kukH k ≤kcukX k kukH k ≤ CkckX k kuk2X k and |

Z

Dα uDα f | ≤ kDα uk2 kDα f k2 ≤ kukH k kf kH k ≤ kf kX k kukX k .

Hence (26) implies that 1 kDα uk2L2 ≤ C kckX k kuk2X k + kukX k kf kX k . µ Since α is an arbitrary multi-index satisfying |α| ≤ k, we conclude that 1 kuk2H k ≤ C kckX k kuk2X k + kukX k kf kX k , µ for some C > 0 independent of µ → +∞. On the other hand, by Lemma 3.1 we have kukL∞ ≤ kf kL∞ . Now the Lemma follows easily. 13

The next estimate will be used to bound the right hand side in (23): Lemma 3.3. If ∇v ∈ X k , then |∇(u0 + v)|2 eu0 +v ∈ X k and k|∇(u0 + v)|2 eu0 +v kX k ≤ CekvkL∞ (1 + kvkkX k )(1 + k∇vk2X k ). Proof. Since |∇u0 |2 eu0 = ∆eu0 +

4πn u0 e , |Σ|

we have that |∇u0 |2 eu0 is smooth and furthermore: (27)

|∇(u0 + v)|2 eu0 +v =(|∇u0 |2 + 2∇u0 · ∇v + |∇v|2 )eu0 +v 4πn u0 =ev (∆eu0 + e + 2∇eu0 · ∇v + eu0 |∇v|2 ). |Σ|

Now, using (21) and (22) we have: kev ∇vkX k ≤Ckev kX k k∇vkX k ≤ekvkL∞ (1 + kvkkX k )k∇vkX k and kev |∇v|2 kX k ≤Ckev kX k k∇vk2X k ≤ekvkL∞ (1 + kvkkX k )k∇vk2X k . Using (27) the desired estimate easily follows. We can finally prove the H k -estimates: Proposition 3.1. Let (v, N ) satisfy system (7)–(8), with µ → +∞. Then for every k ≥ 1 there exist constants µ ¯ > 0 and C > 0 independent of µ ≥ µ ¯ such that (28)

kvkH k + kN kH k ≤ C

∀µ ≥ µ ¯.

Proof. We first show that (29)

∀k ≥ 0 ∃µk > 0, Ck > 0 : kµ(eu0 +v − N )kXk ≤ Ck , ∀µ ≥ µk .

We use induction on k: 14

Claim (a). There exist µ1 and C1 > 0 independent of µ ≥ µ1 , such that kµ(eu0 +v − N )kX 1 ≤ C1

∀µ ≥ µ1 .

Indeed, recall that by Lemma 2.2-(iii) we have kvkH 2 ≤ C. Therefore, by Sobolev embeddings, k∇vkLp ≤ Cp , for all p ≥ 1. Consequently, by the expansion (27), we have thatk|∇(u0 + v)|2 eu0 +v kLp ≤ Cp , for all p ≥ 1, and therefore by Lemma 3.1 applied to (23) we obtain that kµ(eu0 +v − N )kp ≤ C. Now the W 2,p -estimates applied to (7) yield kvkW 2,p ≤ Cp for all p ≥ 1. In particular, we get k∇vkL∞ ≤ C and hence k∇vkX 1 ≤ C. Applying again Lemma 3.1 with p = ∞ to (23) we find that kµ(eu0 +v − N )kL∞ ≤ C. Since k∇vkX 1 ≤ C, we have by Lemma 3.3 that k|∇(u0 + v)|2 eu0 +v kX 1 ≤ C. So by Lemma 3.2 applied to (23), and recalling that keu0 +v (1 − N )kX 1 ≤ C by the estimates (21)–(22), we conclude that there are constants µ1 > 0 and C1 > 0 independent of µ ≥ µ1 , such that kµ(eu0 +v − N )kX 1 ≤ C1 for all µ ≥ µ1 , as desired. Claim (b). If there exist constants µk > 0 and Ck > 0 such that kµ(eu0 +v − N )kX k ≤ Ck for µ ≥ µk , then there exist µk+1 ≥ µk and Ck+1 > 0 such that kµ(eu0 +v − N )kX k+1 ≤ Ck+1 for all µ ≥ µk+1 . Indeed, suppose kµ(eu0 +v − N )kX k ≤ Ck for all µ ≥ µk and let µ ≥ µk . Then, by standard elliptic estimates, kvkX k+2 ≤ C. By (22) we get keu0 +v kX k+2 ≤ C and by Lemma 3.2 applied to (12), we conclude kN kX k+2 ≤ C. Since k∇vkX k+1 ≤ C we also have by Lemma 3.3 that k|∇(u0 +v)|2 eu0 +v kX k+1 ≤ C. So from (23), using Lemma 3.2, we conclude that there are µk+1 > 0 and Ck+1 > 0 such that kµ(eu0 +v − N )kX k+1 ≤ Ck+1 for all µ ≥ µk+1 , as desired. Now (29) follows by Claim (a), Claim (b) and induction on k. To prove (28), for any given k we find µk−2 > 0 and Ck−2 > 0 such that (29) holds for k − 2 and we set µ ¯ := µk−2 . Then equation (7) and standard elliptic estimates yield kvkH k ≤ C for all µ ≥ µ ¯. Now equation (22) and Lemma 3.2 applied to (12) yield kN kX k ≤ C for all µ ≥ µ ¯, in particular kN kH k ≤ C for all µ ≥ µ ¯. Hence (28) follows, and the proof of Proposition 3.1 is complete. It is now finally possible to prove our main result. Proof of Theorem 1.2. For given solutions (u, N ) to system (1)–(2) with µ → +∞, recall that v is defined by v = u − u0 , with u0 the unique solution to (5)–(6). By Proposition 2.1 there exists a solution v˜ for (10) such that as µ → +∞, passing to a subsequence, v → v˜ strong H 1 and N → eu0 +˜v strong 15

H 1 . For any given q ≥ 0 we find µ ¯ > 0 and C > 0 such that (28) holds with k = q + 2. Then by the embedding theorems of Sobolev and Morrey, we have that v → v˜ strong C q and N → eu0 +˜v strong C q . Setting u˜ = u0 + v˜, we have that eu → eu˜ in C q and N → eu˜ in C q . Since u˜ is a solution for (4), Theorem 1.2 is completely proved. Acknowledgments. I would like to thank Professor Yanyan Li and Professor Gabriella Tarantello for fruitful and stimulating discussions. I am grateful to the Rutgers University Mathematics Department, where this work was completed, for hospitality.

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