Uniqueness of topological solutions of self-dual Chern-Simons equation with collapsing vortices
UUniqueness of topological solutions of self-dualChern-Simons equation with collapsing vortices
Genggeng Huang ∗ and Chang-Shou Lin † ∗ Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China † Taida Institute for Mathematical Sciences, Center for Advanced Study in Theoretical Sciences,National Taiwan University, Taipei, 10617, Taiwan
Abstract
We consider the following Chern-Simons equation,∆ u + 1 ε e u (1 − e u ) = 4 π N (cid:88) i =1 δ p εi , in Ω , (0.1)where Ω is a 2-dimensional flat torus, ε > δ p stands for the Diracmeasure concentrated at p . In this paper, we proved that the topological solutions of (0.1)are uniquely determined by the location of their vortices provided the coupling parameter ε is small and the collapsing velocity of vortices p εi is slow enough or fast enough comparingwith ε . This extends the uniqueness results of Choe [5] and Tarantello [22]. Meanwhile,for any topological solution ψ defined in R whose linearized operator is non-degenerate, weconstruct a sequence topological solutions u ε of (0.1) whose asymptotic limit is exactly ψ after rescaling around 0. A consequence is that non-uniqueness of topological solutions in R implies non-uniqueness of topological solutions on torus with collapsing vortices. This paper is devoted to study the following semi-linear elliptic equation with exponentialnonlinearity, ∆ u + 1 ε e u (1 − e u ) = 4 π N (cid:88) i =1 δ p εi , in Ω , (1.1)where Ω is a 2-dimensional flat torus, ε > δ p stands for the Diracmeasure concentrated at p . Through out the paper, we always normalize the volume of Ω as | Ω | = 1.(1.1) arises in the Abelian Chern-Simons model introduced by Jackiw-Weinberg [14] and Hong-Kim-Pac [13]. This model is given in the (2 + 1) − dimensional Minkowski space with metric g µν = diag (1 , − , − L = κ ε µνρ F µν A ρ + D µ φD µ φ − κ | φ | (1 − | φ | ) (1.2)where A µ ( µ = 0 , ,
2) is a real gauge field on R , φ is the complex-valued Higgs field, F µν = ∂ µ A ν − ∂ ν A µ is the curvature tensor, D µ = ∂ µ − √− A µ is the gauge covariant derivative, ε µνρ ∗ [email protected] † [email protected] a r X i v : . [ m a t h . A P ] D ec s totally skew symmetric tensor with ε = 1, and κ > φ, A ) is saturated, as in [14] and [13], one can get the followingBogomol’nyi type equation. (cid:40) ( D + iD ) φ = 0 ,F + κ | φ | ( | φ | −
1) = 0 . (1.3)As in Jaffe-Taubes [15], we let u = ln | φ | , and denote the zeros of φ by { p ε , · · · , p εN } , (1.3) canbe transformed to (1.1) with ε = κ , if we impose the periodic boundary condition(introduced by’t Hooft [20]). For the details of derivation of (1.1) and related models, we refer the readers toHong-Kim-Pac [13], Jackiw-Weinberg [14], Dunne [10], Tarantello [23] and Yang [24].A sequence of solutions u ε of (1.1) are called topological type if u ε ( x ) → , a.e. in Ω , as ε → , and called non-topological type if u ε ( x ) → −∞ , a.e. in Ω , as ε → . When the vortices don’t change with ε , we rewrite (1.1) as∆ u + 1 ε e u (1 − e u ) = 4 π N (cid:88) i =1 δ p i , in Ω . (1.4)The construction of topological vortex condensate u ( x ) of (1.4) was first done by Caffarelli-Yang [2] via both monotone scheme and variational method. Then Tarantello [21] further exploitedthe variational structure and got both topological (for general N ) and non-topological vortexcondensates(for N = 1). After that, many papers were devoted to find the non-topological vortexcondensate for N ≥
2. For these developments, we refer readers to [6–9, 16–19] and referencestherein. All these works reveal that the non-topological solution isn’t unique. As for the topologicalsolutions, Choe [5] and Tarantello [22] independently proved that the topological solution of (1.4)is unique when the coupling constant ε >
Theorem A.
There is a critical value of ε , say, ε ( p , · · · , p N ) > such that, for < ε <ε ( p , · · · , p N ) , (1.4) admits a unique topological solution. By Theorem A, we shall see that the critical value ε ( p , · · · , p N ) doesn’t only depend on thevortex number N but also on the location of the vortices on the flat torus Ω. For the physicalapplications, it is relevant to know to what extend the uniqueness property stated above dependson the smallness of the parameter ε , and hence on the location of the vortex points. In the end ofthe paper of Tarantello [22], the author considered a generalization of Theorem A, where the vortexpoints are allowed to vary with ε but no collapsing of vortices happens. So a natural question iswhether the critical value ε ( p , · · · , p N ) in Theorem A depends only on the vortex number N .We give a partial answer to this problem in this paper. First, we classify the points { p ε , · · · , p εN } according to their asymptotic behavior. Along a subsequence, we define A k,ε = { p εi | lim ε → | p εi − p εk | ε < + ∞} . (1.5)It follows directly from the definition of A k,ε that for any k, m , either A k,ε = A m,ε or A k,ε ∩ A m,ε = ∅ . Without loss of generality, we can take A i,ε , i = 1 , · · · , l such that A i,ε ∩ A j,ε = ∅ , i (cid:54) = j, ∪ li =1 A i,ε = { p ε , · · · , p εN } . (1.6)2 heorem 1.1. There exists C ( N ) > such that if for any set A i,ε , i = 1 , · · · , l defined in (1.5) , (1.6) , either ( a. ) lim ε → | p εk − p εm | ε ≥ C ( N ) , for all different points p εk , p εm ∈ A i,ε , or ( b. ) lim ε → | p εk − p εm | ε ≤ C ( N ) , for all different points p εk , p εm ∈ A i,ε (1.7) hold. Then equation (1.1) has a unique topological solution for all small ε > . The constant C ( N ) in Theorem 1.1 is determined by the following theorem which is due toChoe [5]. Theorem B. [5]
For any
N > , there exists C ( N ) > , such that (cid:80) lj =1 α j ≤ N and given { p · · · , p l } , either ( a. ) | p i − p j | ≥ C ( N ) , ∀ i (cid:54) = j, or ( b. ) | p i − p j | ≤ C ( N ) , ∀ i (cid:54) = j. (1.8) Then the equation ∆ u + e u (1 − e u ) = 4 π l (cid:88) i =1 α i δ p i , in R , (1.9) has a unique topological solution. Moreover, the linearized operator Lh = ∆ h + e u (1 − e u ) h (1.10) is an isomorphism from H ( R ) to L ( R ) satisfying (cid:107) Lh (cid:107) L ( R ) ≥ C (cid:107) h (cid:107) H ( R ) , ∀ h ∈ H ( R ) (1.11) for some constant C > . Theorem B tells us that the restriction (1.7) in Theorem 1.1 is natural as we don’t have theuniqueness of topological multivortex solutions of (1.9) in R . In the following, we will show, insome sense, the uniqueness of topological solutions of (1.1) is “equivalent” to the uniqueness oftopological solutions of (1.9). Theorem 1.2.
Suppose ψ is a topological solution of (1.9) with its linearized operator L ψ satisfying (1.11) . Then there exists a topological solution u ε solves ∆ u ε + 1 ε e u ε (1 − e u ε ) = 4 π l (cid:88) i =1 α i δ εp i , in Ω , (1.12) and ˆ u ε ( x ) = u ε ( εx ) such that (cid:107) ˆ u ε − ψ (cid:107) L ∞ ( B d/ε (0)) → , for some constant d > small . (1.13)A direct consequence of Theorem 1.2 is the following corollary. Corollary 1.1.
Suppose, for some configuration { p , · · · , p l } , there exist two different topologicalsolutions u , u of (1.9) with their linearized operators L u , L u satisfying (1.11) . Then (1.12) possesses at least two topological solutions for small ε . ε N > N such that if 0 < ε < ε N , (1.1) admits a maximal solution. We will revisit the constructionof subsolutions in [2]. The uniqueness part of Theorem 1.1 is proved by contradiction. Supposethat there exist two sequences of distinct topological solutions u ,ε , u ,ε of (1.1). Then there exists x ε ∈ Ω such that | u ,ε ( x ε ) − u ,ε ( x ε ) | = | u ,ε − u ,ε | L ∞ (Ω) (cid:54) = 0 , and x ε → p as ε → A ε = u ,ε − u ,ε | u ,ε − u ,ε | L ∞ (Ω) . (1.14)Then A ε satisfies ∆ A ε + 1 ε e ˜ u ε (1 − e ˜ u ε ) A ε = 0 , in Ω (1.15)where ˜ u ε is between u ,ε and u ,ε . After a suitable scaling at x ε , (1.15) converges to a boundedsolution A of ∆ A + e U (1 − e U ) A = 0 , in R , (1.16)where U is a topological solution of∆ U + e U (1 − e U ) = 4 π l (cid:88) i =1 δ q i , in R . (1.17)Here l and vortices { q , · · · , q l } are determined by the rescaling region and the collapsing velocityof the vortices compared with the coupling constant ε . Then we can apply Theorem B to getcontradictions whenever { q , · · · , q l } satisfies the assumption (1.8).The proof of Theorem 1.2 depends on the perturbation method which follows from Choe [5].We consider topological solutions of (1.12) as a perturbation of ψ ε ( x ) = ψ ( xε ). Set u ε ( x ) = η ( x ) ψ ε ( x ) + ε v ε ( x ) and define an operator G ε ( v ) : H (Ω) → H (Ω). We will prove G ε is awell-defined contraction mapping in some suitable space B .This paper is organized as follows. In Section 2, we will collect some known results andestablish some preliminary estimates for the topological solutions which are important to showthe convergence of (1.15) to (1.16). In Section 3, we will prove Theorem 1.1 i.e. the existenceand uniqueness of the topological solution. Section 4 is devoted to the construction of topologicalsolutions which locally converge to a specified topological solution in R . Recall Green function G ( x, y ) on Ω, − ∆ x G ( x, y ) = δ y − , x, y ∈ Ω , ˆ Ω G ( x, y ) dx = 0 . (2.1)We list some properties of G ( x, y ) as follows:(a.) ∀ φ ∈ C (Ω), φ ( x ) = ´ Ω φ ( y ) dy − ´ Ω G ( x, y )∆ φ ( y ) dy .(b.) G ( x, y ) ∈ C ∞ (Ω × Ω \{ x = y } ), G ( x, y ) = − π ln | x − y | + γ ( x, y ) where γ ( x, y ) is the regularpart of G ( x, y ).(c.) | G ( x, y ) | ≤ C (1 + | ln | x − y || ), and G ( x, y ) ≥ − C , ∀ x, y ∈ Ω and some constants
C, C > u + e u (1 − e u ) = 0 , in R , ˆ R e u (1 − e u ) dx < ∞ . (2.2)Applying the method of moving plane as in Chen-Li [4], after a translation, all the solutions of(2.2) are radially symmetric. Then consider (cid:40) u (cid:48)(cid:48) ( r ; s ) + r u (cid:48) ( r ; s ) + e u ( r ; s ) (1 − e u ( r ; s ) ) = 0 , r > ,u (0; s ) = s, β ( s ) = ´ R e u ( r ; s ) (1 − e u ( r ; s ) ) dx. (2.3)By [3], β ( s ) is a well-defined continuously differentiable function on ( −∞ , β ( s ) is monotoneincreasing such that β ( s ) → + ∞ as s → β ( s ) → π as s → −∞ .Before the proof of the existence and uniqueness of topological solutions of (1.1), we needto show the uniform lower bound of the coupling parameter ε which guarantees the existence ofsolutions of (1.1). In [2], the authors proved the following theorem. Theorem C. [2]
There is a critical value of ε , say, ε ( p , · · · , p N ) > , satisfying, ε ( p , · · · , p N ) ≤ √ πN such that, for < ε ≤ ε ( p , · · · , p N ) , (1.4) has a solution, while for ε > ε ( p , · · · , p N ) , the equationhas no solution. The proof of Theorem C is based on the construction of sub- and supersolutions. Since theexistence of supersolutions always holds true, the delicate part is the construction of subsolu-tion(Lemma 3 in [2]). However, if we revisit the construction of subsulotions as in Lemma 3 in [2],we will find ε ( p , · · · , p N ) is independent of the location of vortex. Although, the construction issimilar, we present it here for the convenience of readers. Theorem 2.1.
There exists ε ( N ) ≥ N − cN for some constant c > , such that, for any configura-tion { p , · · · , p N } , (1.1) has a maximal solution provided ε < ε ( N ) .Proof. The proof of Theorem 2.1 is also based on the monotone scheme. The only thing we needto take care of is the construction of subsolutions. Set u ( x ) = − π N (cid:88) i =1 G ( x, p i ) . Then by property ( c ) of G ( x, y ), u ( x ) ≤ πC N . We want to construct a subsolution v ,∆ v + 1 ε e v + u (1 − e v + u ) ≥ πN. (2.4)Define a smooth function f δ ( x ) as follows: f δ ( x ) = (cid:40) ∀ x ∈ ∪ Ni =1 B δ ( p i );0 , ∀ x ∈ Ω \ ∪ Ni =1 B δ ( p i )and 0 ≤ f δ ( x ) ≤ δ is a parameter which will be determined later. By a direct computation,we get C ( δ ) = ˆ Ω πN f δ ≤ π N δ . Denote g δ ( x ) = 8 πN f δ ( x ) − C ( δ ). Consider the following equation:∆ w = g δ , in Ω , ˆ Ω wdx = 0 . (2.5)5ince (cid:107) g δ (cid:107) L ∞ (Ω) ≤ πN (cid:107) f δ (cid:107) L ∞ (Ω) + C ( δ ) ≤ πN + 32 π N δ , by standard W ,p estimates andSobolev embedding, we get (cid:107) w (cid:107) L ∞ (Ω) ≤ C (cid:107) w (cid:107) W , (Ω) ≤ C (cid:107) g δ (cid:107) L (Ω) ≤ C ( N + N δ ) . (2.6)Hence we can choose 0 < C ≤ π ( C + C )( N + N δ ) such that w = w − C , u + w ≤ − π ( C + C )( N + N δ ) < ln 12 . (2.7)Let δ = √ πN . Then ∀ x ∈ B δ ( p i ), we have∆ w = g δ ≥ πN (2 − πN δ ) ≥ πN ≥ ε e u + w ( e u + w −
1) + 4 πN. (2.8)Set µ = inf { e u + w | x ∈ Ω \ ∪ Ni =1 B δ ( p i ) } ,µ = sup { e u + w | x ∈ Ω \ ∪ Ni =1 B δ ( p i ) } .µ < by (2.7). It remains to estimate µ . By the choice of δ , we now have w ≥ − C N . Byproperty ( c ) of G ( x, y ), we get u ( x ) = − π N (cid:88) i =1 G ( x, p i ) ≥ C N (ln δ −
1) = − C N (ln N + 1) , ∀ x ∈ Ω \ ∪ Ni =1 B δ ( p i ) . The lower bound estimates of w and u imply that µ ≥ N − C N . So in order to show∆ w = g δ ≥ ε e u + w ( e u + w −
1) + 4 πN, x ∈ Ω \ ∪ Ni =1 B δ ( p i ) , (2.9)we only need to guarantee ε µ ≥ πN . This yields ε ≤ N − C N . This also implies w is asubsolution of (2.4) for any configuration { p , · · · , p N } , ∀ < ε ≤ N − C N . Then the arguments in[2] imply the existence of maximal solutions.
Lemma 2.1.
Let u ε be a sequence of solutions of (1.1) . Then up to a subsequence, one of thefollowing holds true:(i) u ε → −∞ , a.e. as ε → .(ii) u ε → , a.e. as ε → . Moreover, u ε → in L p (Ω) , ∀ p ≥ .Proof. Set u ,ε as ∆ u ,ε = 4 π N (cid:88) i =1 δ p εi − πN, ˆ Ω u ,ε dx = 0 . Let d ε = ´ Ω u ε dx and u ε = u ,ε + w ε + d ε . Then w ε satisfies∆ w ε + 1 ε e u ε (1 − e u ε ) = 4 πN. Claim: ∃ C q > (cid:107)∇ w ε (cid:107) L q (Ω) ≤ C q , ∀ q ∈ (1 , q (cid:48) = qq − >
2. Then (cid:107)∇ w ε (cid:107) L q (Ω) ≤ sup { ˆ Ω ∇ w ε ∇ ϕ | ϕ ∈ W ,q (cid:48) (Ω) , ˆ Ω ϕdx = 0 , (cid:107)∇ ϕ (cid:107) L q (cid:48) (Ω) ≤ } .
6y Poincar´ e inequality and Sobolev embedding theorem, we see that (cid:107) ϕ (cid:107) L ∞ (Ω) ≤ C ( (cid:107) ϕ (cid:107) L q (cid:48) (Ω) + (cid:107)∇ ϕ (cid:107) L q (cid:48) (Ω) ) ≤ C (cid:107)∇ ϕ (cid:107) L q (cid:48) (Ω) . In view of the equation w ε satisfying, one gets (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω ∇ w ε · ∇ ϕdx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω ∆ w ε ϕdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) ϕ (cid:107) L ∞ (Ω) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω ε | e u ε (1 − e u ε ) | dx + 4 πN (cid:12)(cid:12)(cid:12)(cid:12) ≤ C which implies (cid:107)∇ w ε (cid:107) L q (Ω) ≤ C q . By Poincar´ e inequality, we also have (cid:107) w ε (cid:107) L q (Ω) ≤ C (cid:107)∇ w ε (cid:107) L q (Ω) . Therefore, up to a subsequence, there exists w ∈ W ,q (Ω) and ∀ p ≥ w ε (cid:42) w in W ,q (Ω) , w ε → w in L p (Ω) , w ε → w a.e..It also follows from the definition of u ,ε that u ,ε → u in L p (Ω) for all p ≥
1. We consider thefollowing two possible cases.1. lim sup ε → e dε ε ≤ C for some constant C .2. lim sup ε → e dε ε = + ∞ .If Case 1 happens, then e u ε = e w ε + u ,ε + d ε ≤ Cε e w ε + u ,ε → u ε → −∞ a.e..Now we assume Case 2 happens. Since u ε < ´ Ω u ,ε dx = 0, we see that 0 ≤ e d ε ≤
1. Thismeans we can find A ≥ ε → sup e d ε = A. By Fatou’s lemma, we get4 πN ε e − d ε = ˆ Ω e w ε + u ,ε (1 − e u ε ) ≥ ˆ Ω e w + u (1 − Ae w + u ) dx. Since ´ Ω w + u dx = 0, we get A ≡ . This ends the proof of Lemma 2.1.Since our arguments always are along a subsequence, without loss of generality, in the argumentsbelow, we always assume P n = { p ε n , · · · , p ε n N } = { p ,n , · · · , p N,n } → { p , · · · , p N } = P,u n → u n + 1 ε n e u n (1 − e u n ) = 4 π N (cid:88) i =1 δ p i,n , in Ω . (2.10)Set d = inf p i (cid:54) = p j | p i − p j | . Then we have the following lemma. Lemma 2.2.
Suppose u n → a.e. solves (2.10) . Then for any R n → + ∞ , R n ε n ≤ d , we have (cid:107) u n (cid:107) L ∞ (Ω \∪ Ni =1 B Rnεn ( p i,n )) → . roof. First, we need to show for any compact set K ⊂ Ω \ P , (cid:107) u n (cid:107) L ∞ ( K ) → . Set δ = d ( K, P ). Since u n < u n is subharmonic in B δ ( x ) for any x ∈ K .By using the mean value theorem and Lemma 2.1, we get0 ≤ − u n ( x ) ≤ | B δ | (cid:107) u n (cid:107) L (Ω) → , as n → ∞ , ∀ x ∈ K. Next, we classify the points { p ,n , · · · , p N,n } according to their asymptotic behavior. Define A k,n as A k,n = { p i,n | lim n →∞ | p i,n − p k,n | ε n < + ∞} . It’s obvious that for any i, j , either A i,n = A j,n or A i,n ∩ A j,n = ∅ along a suitable subsequence.Without loss of generality, we can take A i,n , i = 1 , · · · , l such that A i,n ∩ A j,n = ∅ , i (cid:54) = j, ∪ li =1 A i,n = { p ,n , · · · , p N,n } . Set r n = 14 min(inf i (cid:54) = j d ( A i,n , A j,n ) ε n , R n ) → + ∞ . So it’s enough to prove that (cid:107) u n (cid:107) L ∞ (Ω \∪ li =1 B rnεn ( q i,n )) → , q i,n ∈ A i,n , i = 1 , · · · , l. It follows from our choice that ∪ li =1 B r n ε n ( q i,n ) covers the set { p ,n , · · · , p N,n } and B r n ε n ( q i,n ) ∩ B r n ε n ( q j,n ) = ∅ for i (cid:54) = j . Hence, the set B d (0) \ ∪ li =1 B r n ε n ( q i,n ) is pathwise connected. Nowsuppose u n ( x n ) ≤ − c < x n ∈ Ω \ ∪ li =1 B r n ε n ( q i,n ). Then by our first step in the proof, withoutloss of generality, we may assume x n → ∈ P . Choosing smooth curves γ n ⊂ B d (0) \ ∪ li =1 B r n ε n ( q i,n ) joining x n and y ∈ ∂B d (0), by intermediate value theorem, we get z n ∈ γ n , u n ( z n ) = s < β ( s ) > πN with β ( s ) defined in (2.3) since it’s already known u n ( y ) →
0. Setˆ u n ( x ) = u n ( ε n x + z n ). Then ˆ u n ( x ) solves (cid:40) ∆ˆ u n + e ˆ u n (1 − e ˆ u n ) = 0 , in B r n / (0) , ˆ u n ( x ) < , ˆ u n (0) = s < , ´ B rn/ (0) e ˆ u n (1 − e ˆ u n ) ≤ πN. As (cid:12)(cid:12)(cid:12) e ˆ un (1 − e ˆ un )ˆ u n (cid:12)(cid:12)(cid:12) ≤ C uniformly, by Harnack inequality(Theorem 8.20 in [11]), we have ˆ u n is locallyuniformly bounded. Applying standard W ,p and Schauder’s estimates, one getsˆ u n → ˆ u, in C loc ( R )and ˆ u solves (2.2) with ˆ u (0) = s . Then4 πN ≥ ˆ R e ˆ u (1 − e ˆ u ) dx ≥ β ( s ) > πN. This yields a contradiction and proves present lemma.
Lemma 2.3.
Suppose u n is a topological solution of (1.1) . Then up to a subsequence, for any R n → + ∞ , R n ε n ≤ d , we have ε kn | D k u n | L ∞ (Ω \∪ Ni =1 B Rnεn ( p i,n )) → , for any k ∈ N and faster that any other power of R n . roof. Set Ω n = Ω \ ∪ Ni =1 B R n ε n ( p i,n ), Ω (cid:48) n = Ω \ ∪ Ni =1 B R n ε n ( p i,n ). By Lemma 2.2, we have ˆ Ω n | u n | dx ≤ (1 + | u n | L ∞ (Ω n ) ) ˆ Ω n | u n | | u n |≤ (1 + | u n | L ∞ (Ω n ) ) ˆ Ω n (1 − e u n ) ≤ (1 + | u n | L ∞ (Ω n ) ) e | u n | L ∞ (Ω n ) ˆ Ω n e u n (1 − e u n ) ≤ Cε n . (2.11)In getting (2.11), we have used | − e t | ≥ | t | | t | , ∀ t ∈ R . Then for any x ∈ Ω (cid:48) n , we note that u n issubharmonic in B R n ε n ( x ). By mean value theorem and (2.11), we get | u n ( x ) | ≤ CR n ε n ˆ B Rnεn ( x ) | u n ( y ) | dy ≤ CR n , ∀ x ∈ Ω (cid:48) n . Define 0 ≤ ϕ n ∈ C ∞ (Ω) such that ϕ n ≡ , in ∪ Ni =1 B R n ε n ( p i,n ) , ϕ ≡ , in Ω (cid:48) n , | D k ϕ n | ≤ C k ( R n ε n ) − k , k = 1 , , · · · . Then ˆ Ω (cid:48) n ε n e u n (1 − e u n ) ≤ ε n ˆ Ω n e u n (1 − e u n ) ϕ n = − ˆ Ω u n ∆ ϕ n dx ≤ CR n ε n ˆ Ω n | u n | ≤ CR n . (2.12)Repeating the procedure of (2.11) and (2.12), we have ˆ Ω n | u n ( x ) | dx ≤ C m ε n R mn , ˆ Ω n ε n e u n (1 − e u n ) ≤ C m R mn . This implies (cid:107) u n (cid:107) L ∞ (Ω n ) ≤ C m R mn , (cid:107) ε n e u n (1 − e u n ) (cid:107) L ∞ (Ω n ) ≤ C m R mn ε n . Set ˆ u n ( x ) = u n ( ε n x ). Then ˆ u n solves∆ˆ u n + e ˆ u n (1 − e ˆ u n ) = 0 , in Ω n ε n . Since | ˆ u n | ≤ C m /R mn in Ω n /ε n , by standard W ,p − estimates and Schauder estimates, we get | D k ˆ u n | L ∞ (Ω n /ε n ) ≤ C k,m R mn , ∀ k ≥ . Scaling back to u n ( x ), one get | D k u n | L ∞ (Ω \∪ Ni =1 B Rnεn ( p i,n )) ≤ C m R mn ε kn . If no confuse occurs, in the remaining part of this paper, A k,n , q k,n , k = 1 , · · · , l , r n alwaysmean the terminologies defined in Lemma 2.2. Suppose p ,n , · · · , p l , ,n ∈ A ,n , l , is the number ofelements in A ,n and p ,n = 0 after a shift of coordinates. Then set v n = u n ( x ) − (cid:80) l , i =1 ln | x − p i,n | .We have the following important a priori estimates.9 emma 2.4. Suppose u n is a topological solution of (2.10) . Then ε n e u n (1 − e u n ) → π (cid:88) i ∈ I l i δ p i , weakly in the sense of measure in Ω , where I ⊂ { , · · · , N } is a set of indices identifying all distinctvortices in { p , · · · , p N } , l i ∈ N is the multiplicity of p i , i ∈ I . And for any ˜ r n ≤ r n , ˜ r n → + ∞ ,we have ˆ B ˜ rnεn (0) ε n (1 − e u n ) = 4 πl , +4 π l , (cid:88) i =1 p i,n ·∇ v n ( p i,n )+ o (1) , ˆ Ω \∪ li =1 B ˜ rnεn ( q i,n ) ε n (1 − e u n ) = o (1) . Proof.
The first part is easy. Since for any ϕ ∈ C ∞ (Ω), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε n ˆ Ω e u n (1 − e u n ) ϕ − π N (cid:88) i =1 ϕ ( p i,n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − ˆ Ω ∆ u n ϕ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − ˆ Ω ∆ ϕu n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) ∆ ϕ (cid:107) L ∞ (Ω) (cid:107) u n (cid:107) L (Ω) → , as ε n → P n → P .Denote f n = 2 l , (cid:88) i =1 ln | x − p i,n | . And v n solves∆ v n + 1 ε n e u n (1 − e u n ) = 0 , in B ˜ r n ε n (0) . (2.13)By Lemma 2.3 and the property of p i,n , i = 1 , · · · , l , , we have |∇ v n + 2 l , ν ˜ r n ε n | = O ( 1˜ r n ε n ) , on ∂B ˜ r n ε n (0) (2.14)where ν is the outward normal of ∂B ˜ r n ε n (0). In view of ˆ B ˜ rnεn (0) ( x · ∇ v n )∆ v n = ˆ ∂B ˜ rnεn (0) ( x · ∇ v n )( ∇ v n · ν ) − ˆ ∂B ˜ rnεn (0) ( x · ν ) |∇ v n | = 12 ˆ ∂B ˜ rnεn (0) (cid:18) O (cid:18) r n (cid:19) − l , (cid:19) (cid:18) O (cid:18) r n ε n (cid:19) − l , ˜ r n ε n (cid:19) = 4 πl , + O (cid:18) r n (cid:19) . (2.15)And ˆ B ˜ rnεn (0) ( x · ∇ v n ) 1 ε n e u n (1 − e u n )= ˆ B ˜ rnεn (0) ( x · ∇ u n ) 1 ε n e u n (1 − e u n ) − ( x · ∇ f n )( − ∆ v n )= ˆ B ˜ rnεn (0) ε n (1 − e u n ) − ε n ˆ ∂B ˜ rnεn (0) ( x · ν )(1 − e u n ) + ˆ B ˜ rnεn (0) ( x · ∇ f n )∆ v n . (2.16)And ˆ B ˜ rnεn (0) ( x · ∇ f n )∆ v n − πl , + 2 lim δ → l , (cid:88) i =1 ˆ B ˜ rnεn (0) \ B δ ( p i,n ) ∇ · [ p i,n · (ln | x − p i,n | − ln(˜ r n ε n ))]∆ v n + o (1)= − πl , − π l , (cid:88) i =1 p i,n · ∇ v n ( p i,n ) + 2 π l , (cid:88) i =1 ˆ ∂B ˜ rnεn (0) ( p i,n · ν )(ln | x − p i,n | − ln(˜ r n ε n ))∆ v n − π l , (cid:88) i =1 ˆ ∂B ˜ rnεn (0) ∇ ( p i,n · ∇ v n ) · ν (ln | x − p i,n | − ln(˜ r n ε n )) + o (1) (2.17)= − πl , − π l , (cid:88) i =1 p i,n · ∇ v n ( p i,n ) + o (1) . In order to get (2.15)-(2.17), we have used the estimates in Lemma 2.3 and ln | x − p in |− ln(˜ r n ε n ) → ∂B ˜ r n ε n (0). Combining (2.15)-(2.17), we get1 ε n ˆ B ˜ rnεn (0) (1 − e u n ) = 4 πl , + 4 π l , (cid:88) i =1 p i,n · ∇ v n ( p i,n ) + o (1) . (2.18)Now we want to show p i,n · ∇ v n ( p i,n ) is uniformly bounded for i = 1 , · · · , l , . By Green’s repre-sentation formula, one gets u n − u ,n = ˆ Ω u n ( y ) dy + ˆ Ω G ( x, y ) 1 ε n e u n (1 − e u n ) dy. (2.19)Since G ( x, y ) = − π ln | x − y | + γ ( x, y ) where γ ( x, y ) is the regular part, one can expect that for x ∈ B Rε n (0) for some fixed R large enough, |∇ u n − l , (cid:88) i =1 x − p i,n | x − p i,n | |≤ C + ˆ Ω | x − y | ε n e u n (1 − e u n ) dy + 2 l (cid:88) i = l , +1 x − p i,n | x − p i,n | (2.20) ≤ C + ˆ B εn ( x ) | x − y | ε n e u n (1 − e u n ) dy + ˆ Ω \ B εn ( x ) | x − y | ε n e u n (1 − e u n ) dy + o (cid:18) ε n (cid:19) ≤ C + Cε n . By (2.20) and the uniform bound of | p i,n | ε n , i = 1 , · · · , l , , we can get p i,n · ∇ v n ( p i,n ) is uniformlybounded for i = 1 , · · · , l , .From | ε n e u n (1 − e u n ) | L (Ω) = 4 πN and u n <
0, we see that for Ω n = Ω \ ∪ li =1 B ˜ r n ε n ( q i,n )1 ε n ˆ Ω n (1 − e u n ) ≤ sup Ω n (1 − e u n ) inf Ω n e u n ˆ Ω n ε n e u n (1 − e u n ) → . Denote ˆ u n ( x ) = u n ( ε n x ) , p i,n ε n → ˆ p i , i = 1 , · · · , l , . emma 2.5. Suppose ˆ u ( x ) is the unique topological solution of ∆ˆ u + e ˆ u (1 − e ˆ u ) = 4 π l , (cid:88) i =1 δ ˆ p i , in R , ˆ u < , sup R \∪ l , i =1 B (ˆ p i ) |∇ ˆ u | < + ∞ , ˆ R (1 − e ˆ u ) dx < + ∞ where ˆ p i satisfies the assumptions in Theorem B. Set ˆ v n ( x ) = ˆ u n ( x ) − l , (cid:88) i =1 ln | x − p i,n ε n | | x − p i,n ε n | = ˆ u n − h n , ˆ v ( x ) = ˆ u ( x ) − l , (cid:88) i =1 ln | x − ˆ p i | | x − ˆ p i | = ˆ u − h. Then lim ε → sup B rn (0) | ˆ v n − ˆ v | = 0 .Proof. Claim: sup B rn (0) \∪ l , i =1 B (ˆ p i ) | ˆ u n | ≤ C. Suppose the claim isn’t true. We can pick up z n ∈ B r n (0) \ ∪ l , i =1 B (ˆ p i ) such that ˆ u n ( z n ) → −∞ .By Lemma 2.3, z n must be uniformly bounded. Otherwise, one can define ˜ r n = | z n | → + ∞ andΩ n = Ω \ ∪ li =1 B ˜ r n ε n ( q i,n ). Then Lemma 2.3 tells us that (cid:107) u n (cid:107) L ∞ (Ω n ) → u n ( z n ε n ) = ˆ u n ( z n ) → −∞ as z n ε n ∈ Ω n . Because ˆ u n solves∆ˆ u n + e ˆ u n (1 − e ˆ u n ) = 0 , in B r n (0) \ ∪ l , i =1 B (ˆ p i ) , ˆ u n < (cid:12)(cid:12)(cid:12) e ˆ un (1 − e ˆ un )ˆ u n (cid:12)(cid:12)(cid:12) ∈ L ∞ , by Harnack inequality, we haveˆ u n → −∞ , locally in any compact set K ⊂ B r n (0) \ ∪ l , i =1 B (ˆ p i ) . This implies that ˆ B R (0) (1 − e ˆ u n ) ≥ πR → + ∞ , as R → + ∞ which contradicts to ˆ B R (0) (1 − e ˆ u n ) ≤ ε n ˆ B rnεn (0) (1 − e u n ) ≤ C. This proves our claim. Since ˆ v n solves∆ˆ v n + e ˆ u n (1 − e ˆ u n ) = l , (cid:88) i =1 | x − p i,n ε n | ) = g n , in B r n (0) . By the claim, we have (cid:107) ˆ v n (cid:107) L ∞ ( ∂B R ) ≤ C R , for R ≥ max ≤ i ≤ l , | ˆ p i | + 1 . Applying maximum principle to ∆ˆ v n ± (4 l , + 1) ≷
0, one gets (cid:107) ˆ v n (cid:107) L ∞ ( B R ) ≤ C R . From thestandard W ,p estimates, we have ˆ v n → ˆ v in W ,ploc ( R ) for p >
1. By Sobolev embedding theorem,we have ∇ ˆ v n ( p i,n ε n ) → ∇ ˆ v (ˆ p i ). This implies ˆ B rn (0) (1 − e ˆ u n ) = 4 πl , + 4 π l , (cid:88) i =1 ˆ p i · ∇ ˆ v (ˆ p i ) + o (1) . ˆ B rn (0) ( e ˆ u n − e ˆ u ) = ˆ B rn (0) ( e ˆ u n − + ( e ˆ u − − ˆ B rn (0) (1 − e ˆ u n )(1 − e ˆ u ) ≤ o (1) . This again implies o (1) = ˆ B rn (0) \ B R (0) ( e ˆ u n − e ˆ u ) ≥ min B rn (0) \ B R (0) ( e u n , e u ) ˆ B rn (0) \ B R (0) (ˆ u n − ˆ u ) , or ´ B rn (0) \ B R (0) (ˆ u n − ˆ u ) = o (1). Direct computation yields that | h n − h | L ( B rn (0) \ B R (0)) + | g n − g | L ( B rn (0)) = o (1) , as n → ∞ . By the local convergence of ˆ v n → ˆ v and standard W ,p estimates, we havelim ε n → sup B rn (0) | ˆ v n − ˆ v | → . Proof of Theorem 1.1 . By Theorem 2.1, for any configuration { p , · · · , p N } , ∃ ε N > ∀ < ε ≤ ε N , (1.4) admits a maximal solution. Suppose u n is a sequence of maximal solutions of(1.1) with configuration { p ,n , · · · , p N,n } and coupling parameter ε n →
0. By Lemma 2.1, we haveeither u n → u n → −∞ a.e.. Set v n = u n − u ,n . Since w constructed in Theorem 2.1 isa subsolution of ∆ v n + 1 ε n e v n + u ,n (1 − e v n + u ,n ) = 4 πN, by the monotone decreasing property of maximal solutions with respect to ε , we have v n ≥ w which is uniformly bounded from below. This implies u n = v n + u ,n → u ,n , u ,n . Set φ n ( x ) = ( u ,n − u ,n )( x ) | u ,n − u ,n | L ∞ (Ω) , | φ n ( x n ) | = (cid:107) φ n (cid:107) L ∞ (Ω) = 1 . We consider the following two cases.(I) ˜ r n = min( | x n − p ,n | ε n , · · · , | x n − p N,n | ε n ) → + ∞ . By Lemma 2.3, one has (cid:107) u ,n (cid:107) L ∞ ( B ˜ rnεn ( x n )) , (cid:107) u ,n (cid:107) L ∞ ( B ˜ rnεn ( x n )) → . Set ˆ φ n ( x ) = φ n ( ε n x + x n ), ˆ u i,n ( x ) = u i,n ( ε n x + x n ), i = 1 ,
2. Then ˆ φ n satisfies∆ ˆ φ n + e ˆ u n (1 − e ˆ u n ) ˆ φ n = 0 , in B ˜ r n (0) , (3.1)where ˆ u n is between ˆ u ,n , ˆ u ,n . From standard W ,p estimates and Schauder’s estimates, weobtain a subsequence ˆ φ n → ˆ φ in C loc ( R ) with ˆ φ satisfying∆ ˆ φ − ˆ φ = 0 , in R , | ˆ φ (0) | = (cid:107) ˆ φ (cid:107) L ∞ = 1 . r n ≤ C < + ∞ . Without loss of generality, we may assume p ,n = 0 and | x n | ε n ≤ C . Setˆ φ n ( x ) = φ n ( ε n x + x n ) . Then as in Case (I), by Lemma 2.5 and Theorem B, we obtain asubsequence ˆ φ n → ˆ φ in C loc ( R ) with ˆ φ satisfying∆ ˆ φ + e ˆ u (1 − e ˆ u ) ˆ φ = 0 , in R , | ˆ φ (0) | = (cid:107) ˆ φ (cid:107) L ∞ = 1 . Here ˆ u is the unique topological solution of (cid:40) ∆ˆ u + e ˆ u (1 − e ˆ u ) = 4 π (cid:80) l , i =1 δ ˆ p i , in R , ´ R (1 − e ˆ u ) < ∞ , ˆ u < . In order to obtain contradiction, we need to show ˆ φ ∈ H ( R ) both in Case (I) and (II). Let η ( x ) ∈ C ∞ c ( R ), η ≡ B (0), η ≡ B c (0), η R ( x ) = η ( xR ). Taking η R ˆ φ as a test function, weget ˆ R η R |∇ ˆ φ | + e u η R ˆ φ ≤ ˆ R η R |∇ ˆ φ | + CR ˆ B R \ B R ˆ φ + ˆ R e ˆ u (1 − e ˆ u ) η R ˆ φ ≤ C. This implies ˆ φ ∈ H ( R ). Taking η R ∆ ˆ φ as a test function, one gets, ˆ R η R | D ij ˆ φ | = ˆ R η R D i η R D i ˆ φD jj ˆ φ − ˆ R η R D j η R D i ˆ φD ij ˆ φ + ˆ R e ˆ u (1 − e ˆ u ) η R ˆ φ ∆ ˆ φ ≤ ˆ R η R | D ij ˆ φ | + C ≤ C. Now we have ˆ φ ∈ H ( R ), by Theorem B, ˆ φ ≡ | ˆ φ (0) | = 1. Suppose ψ ( x ) is a topological solution of∆ ψ + e ψ (1 − e ψ ) = 4 π l (cid:88) i =1 α i δ p i , in R . (4.1)The corresponding linearized operator L is ∆ + e ψ (1 − e ψ ) and we assume that (cid:107) Lh (cid:107) L ( R ) ≥ C (cid:107) h (cid:107) H ( R ) , ∀ h ∈ H ( R ) for some constant C > . (4.2)Also by Han [12], for R > max | p i | + 1, we have | ψ ( x ) | + |∇ ψ ( x ) | ≤ c e − c | x | , ∀| x | ≥ R, for some constants c , c > . (4.3)Consider a cut-off function η ( x ) with η ( x ) ≡ B δ (0), η ( x ) ≡ B c δ (0) for some δ > F ε ( v ) as follows F ε ( v ) = ∆ v + 1 ε e ηψ ε + ε v (1 − e ηψ ε + ε v ) − ηε e ψ ε (1 − e ψ ε ) + ε − (2 ∇ η · ∇ ψ ε + ψ ε ∆ η ) , (4.4)where ψ ε ( x ) = ψ ( xε ). By a direct computation, it can be checked that u ε = ηψ ε + ε v ε is a solutionof ∆ u ε + 1 ε e u ε (1 − e u ε ) = 4 π l (cid:88) i =1 α i δ εp i , in Ω (4.5)provided F ε ( v ε ) = 0 and ε is small enough. 14 emma 4.1. There is a constant ε > such that if < ε < ε , we have(I) (cid:107) F ε (0) (cid:107) L (Ω) ≤ c e − c /ε for some constants c , c > .(II) DF ε (0) is an isomorphism from H (Ω) onto L (Ω) , moreover, we have (cid:107) DF ε (0) h (cid:107) L (Ω) ≥ C (cid:107) h (cid:107) H (Ω) , ∀ h ∈ H (Ω) for some constant C > . (III) (cid:107) DF ε ( v ) h − DF ε (0) h (cid:107) L (Ω) ≤ Cε (cid:107) h (cid:107) H (Ω) , for (cid:107) v (cid:107) H (Ω) ≤ , ∀ h ∈ H (Ω) .Proof. By definition of F ε ( v ), we have F ε (0) = 1 ε ( e ηψ ε (1 − e ηψ ε ) − ηe ψ ε (1 − e ψ ε )) + 1 ε (2 ∇ η · ∇ ψ ε + ψ ε ∆ η ) . Since the support of ∇ η , ∆ η is contained in B δ (0) \ B δ (0) and ψ ε ( x ) , ∇ ψ ε decay to 0 exponentiallyfast as e − c/ε for | x | ≥ δ by (4.3), we have (cid:12)(cid:12)(cid:12)(cid:12) ε (2 ∇ η · ∇ ψ ε + ψ ε ∆ η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − c /ε . Since η ≡ B δ (0), the first term of F ε (0) vanishes in B δ (0) and again by the exponential decayproperty of ψ ε in | x | ≥ δ , one get (cid:107) F ε (0) (cid:107) L ∞ (Ω) ≤ c e − c /ε which implies (I) is true.We prove (II) by contradiction. Suppose there exists h n ∈ H (Ω) such that (cid:107) h n (cid:107) H (Ω) = 1 , (cid:107) DF ε n (0) h n (cid:107) L (Ω) = o (1) . Set η ( x ) = η (4 x ) and ˜ h n = (1 − η ) h n . Then ˜ h n solves∆˜ h n − ε n e ηψ ε (2 e ηψ ε − h n = (1 − η ) DF ε n (0) h n − ∇ η · ∇ h n − ∆ η h n . (4.6)Multiplying (4.6) with ˜ h n and integrating by parts, we get (cid:107)∇ ˜ h n (cid:107) L (Ω) + 1 ε n (cid:107) ˜ h n (cid:107) L (Ω) ≤ C (cid:107) ˜ h n (cid:107) L (Ω) ( (cid:107) h n (cid:107) H ( B cδ/ (0)) + (cid:107) DF ε n (0) h n (cid:107) L ( B cδ/ (0)) ) (4.7)for some constant C >
0. From (4.7) and (cid:107) h n (cid:107) H (Ω) ≤
1, one has (cid:107) ˜ h n (cid:107) L (Ω) ≤ Cε n . Since η ≡ B cδ/ (0), (4.7) implies (cid:107)∇ h n (cid:107) L ( B cδ/ (0)) + 1 ε n (cid:107) h n (cid:107) L ( B cδ/ (0)) ≤ (cid:107)∇ ˜ h n (cid:107) L (Ω) + 1 ε n (cid:107) ˜ h n (cid:107) L (Ω) ≤ Cε n . (4.8)Let η ( x ) = η (2 x ) and ¯ h n = (1 − η ) h n . Again, by the same arguments as ˜ h n , we obtain (cid:107)∇ ¯ h n (cid:107) L (Ω) + 1 ε n (cid:107) ¯ h n (cid:107) L (Ω) ≤ Cε n ( (cid:107) h n (cid:107) H ( B cδ/ (0)) + (cid:107) DF ε n (0) h n (cid:107) L ( B cδ/ (0)) ) = o ( ε n ) . (4.9)By (4.9), it follows that (cid:107) ¯ h n (cid:107) L (Ω) = o ( ε n ) which implies (cid:107) ∆¯ h n (cid:107) L (Ω) = o (1). Then by W , estimates and 1 − η ≡ B cδ (0), we have (cid:107) h n (cid:107) H ( B cδ (0)) = o (1) which implies (cid:107) h n (cid:107) H ( B δ (0)) =1 + o (1).Set ˆ h n ( x ) = η ( ε n x ) h n ( ε n x ) and η n ( x ) = η ( ε n x ). By a direct computation, ˆ h n satisfies∆ˆ h n + e ψ (1 − e ψ )ˆ h n =2 ∇ ˆ h n · ∇ η n + ˆ h n ∆ η n + ε n η n [ DF ε n (0) h n ]( ε n x )+ [ e η n ψ (2 e η n ψ − − e ψ (2 e ψ − h n . (4.10)15et ˆΣ = { δε n ≤ | x | ≤ δε n } , Σ = { δ ≤ | x | ≤ δ } . Since the last term in (4.10) vanishes in ˆΣ c , thenby (4.9) and our assumption, (cid:107) ˆ h n (cid:107) H ( R ) ≤ C ( ε n (cid:107)∇ ˆ h n (cid:107) L (ˆΣ ) + (cid:107) ˆ h n (cid:107) L (ˆΣ ) + ε n (cid:107) DF ε n (0) h n (cid:107) L ( B δ (0)) ) ≤ C ( ε n (cid:107)∇ h n (cid:107) L (Σ ) + ε − n (cid:107) h n (cid:107) L (Σ ) + ε n (cid:107) DF ε n (0) h n (cid:107) L ( B δ (0)) ) = o ( ε n ) . This contradicts to (cid:107) h n (cid:107) H ( B δ (0)) = 1 + o (1). The second part of (II) follows immediately. Since DF ε (0) is a self-adjoint operator from H (Ω) → L (Ω), we obtain that DF ε is an isomorphismfrom H (Ω) to L (Ω).The estimate for (cid:107) DF ε ( v ) h − DF ε (0) h (cid:107) L (Ω) follows from DF ε ( v ) h − DF ε (0) h = 1 ε e ηψ ε ( e ε v − h − ε e ηψ ε ( e ε v − h and the embedding of H (Ω) (cid:44) → L ∞ (Ω). Proof of Theorem 1.2.
We define a functional G ε : H (Ω) → L (Ω) by G ε ( v ) = v − [ DF ε (0)] − F ε ( v ) . (4.11)For any fixed point v ε of G ε in H (Ω), u ε = ηψ ε + ε v ε is a solution of (1.12). It suffices to prove G ε admits a fixed point in H (Ω) for ε > B = { v ∈ H (Ω) || v | H (Ω) ≤ } .Claim: G ε : B → B is a well-defined contraction mapping provided ε > ε > (cid:107) DG ε ( v ) h (cid:107) L (Ω) ≤ (cid:107) [ DF ε (0)] − (cid:107)(cid:107) ( DF ε ( v ) − DF ε (0)) h (cid:107) L (Ω) ≤ Cε | h | H (Ω) for v ∈ B . Then by Lemma 4.1, we have (cid:107) G ε (0) (cid:107) H (Ω) ≤ C (cid:107) F ε (0) (cid:107) L (Ω) ≤ Ce − c/ε . For any v , v ∈ B , for ε > (cid:107) G ε ( v ) (cid:107) H (Ω) ≤ (cid:107) G ε (0) (cid:107) H (Ω) + (cid:107) G ε ( v ) − G ε (0) (cid:107) H (Ω) ≤ (cid:107) G ε (0) (cid:107) H (Ω) + (sup v ∈B (cid:107) DG ε ( v ) (cid:107) ) (cid:107) v (cid:107) H (Ω) ≤ C ( e − c/ε + ε ) , and (cid:107) G ε ( v ) − G ε ( v ) (cid:107) H (Ω) ≤ (sup v ∈B (cid:107) DG ε ( v ) (cid:107) ) (cid:107) v − v (cid:107) H (Ω) ≤ Cε (cid:107) v − v (cid:107) H (Ω) . Hence, if ε > G ε : B → B is a well-defined contraction mapping, which implies G ε has a unique fixed point v ε ∈ B . By a direct computation, one can see that u ε = ηψ ε + ε v ε satisfies (1.12). Also from Sobolev embedding H (Ω) (cid:44) → L ∞ (Ω) and ψ ε decaying exponentially to0, ∀| x | ≥ r , ∀ r >
0, we see u ε is a topological solution. Acknowledgement
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