Journal of Differential Equations

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1 J. Differential Equations ) Contents lists available at SciVerse ScienceDirect Journal of Differential Equations Sharp existence results for mean field equations with singular data D. Bartolucci a,,c.s.lin b a Department of Mathematics, University of Rome Tor Vergata, Via della Ricerca Scientifica n., 0033 Rome, Italy b Taida Institute for Mathematical Sciences and Department of Mathematics, National Taiwan University, Taipei, Taiwan article info abstract Article history: Received 25 June 200 Available online 6 January 202 MSC: primary 35B30, 35J65, 35J99, 35R05 secondary 35Q35, 49J99 Let Ω be a simply connected, open and bounded domain in R 2. We are concerned with the nonlinear elliptic problem m v = 8π Ω ev 4π α j δ p j in Ω, ev 0.) j= v = 0 on Ω, where α j > 0, δ p j denotes the Dirac mass with singular point p j and {p,...,p m } Ω. We provide necessary and sufficient conditions for the existence of solutions to 0.). Our result is the two dimensional version of the sharp existence/nonexistence result obtained in Druet 2002) [3] for elliptic equations with critical exponent in dimension 3. In particular, we prove that the set Ω+ m α) is open, where, for a given α = α,...,α m ) 0, + ) 0, + ), Ω+ m α) ={p,...,p m ) problem 0.) has a solution} Ω Ω. 202 Elsevier Inc. All rights reserved.. Introduction Let Ω R 2 be a simply connected, open and bounded domain and {p,...,p m } Ω be any finite subset. We are concerned with the existence of solutions for * Corresponding author. addresses: bartoluc@mat.uniroma2.it D. Bartolucci), cslin@math.ntu.edu.tw C.S. Lin) /$ see front matter 202 Elsevier Inc. All rights reserved. doi:0.06/j.jde

2 46 D. Bartolucci, C.S. Lin / J. Differential Equations ) v = λ v = 0 Ω ev ev 4π m α j δ p j in Ω, j= on Ω,.) in case λ = 8π and α j 0, + ), j {,...,m}. The analysis of.) has recently attracted a lot of attention due its many applications in mathematical physics. We refer the reader to [2,4,5,7 2,4,5,7,9 22], and the references quoted therein for further details. In particular we refer to [7,8,6] and the introduction of [4] for the application of.) to the analysis of vortex-type configurations in turbulent Euler flows. We will not discuss here issues related with non-smooth domains. Therefore, unless otherwise specified we will assume that Ω is of class C 2. We will denote by z = x + ix 2 and p j = p j, + ip j,2 the complex coordinates corresponding to x, x 2 ) Ω, p j,, p j,2 ) Ω, and by D ={z R 2 z < } the open unit disk. We will often need to use conformal mappings from D to Ω. To avoid any possible ambiguity, in case Ω itself is the unit disk, we will denote it by B.Inparticular,foranyfixedp Ω, wewill : Ω D to be its inverse. For any fixed α > 0, p Ω and any r > 0 small enough, let G Ω z, p) C 0 Ω \ B r p)) be the unique solution of denote by f p : D Ω, any Riemann map which satisfies f p 0) = p, and set g p = f p { GΩ z, p) = δ p in Ω, G Ω z, p) = 0 on Ω..2) The continuity assumption ensures that G Ω is uniquely defined by G Ω z, p) = 2π log g pz). Let GΩ denote the regular part of G Ω z, p), and set z = f p w). Then, we have G Ω z, p) = G Ω z, p) + log z p = 2π 2π log g pz) z p = 2π log f pw) f p 0), w and we define γ Ω z) = GΩ z, z) = 2π log g pz) 2 ) g = pz) 2π log w 2) f p w) to be the corresponding Robin function. We define v to be a solution for.) if u := v + 4π α j=,...,m jg Ω z, p j ) is an H 0 Ω), weak solution for u = λ Veu in Ω, Ω Veu.3) u = 0 on Ω, where V z) = exp 4π j=,...,m ) α j G Ω z, p j )..4) Two crucial results Theorems. and.4 below) will be used whose proofs will not be discussed here. We refer the reader to the remarks following the corresponding statements for further details.

3 D. Bartolucci, C.S. Lin / J. Differential Equations ) Theorem.. See [3].) Let log h be harmonic and continuous in Ω. For any λ 0, 8π) there exists one and only one solution u λ for u = λ hveu in Ω, Ω hveu u = 0 on Ω. In particular.5) admits at most one solution for λ = 8π.Moreover,foranyλ 0, 8π] the first eigenvalue of the linearized problem for.5) at u λ is strictly positive..5) Remark. Theorem. was proved in [3] in case h. It is straightforward to verify that the same proof works for any h satisfying the required assumptions as well. We omit the details of this proof here since it can be worked out by a step-by-step adaptation of the one already provided in [3]. Remark. We remark that uniqueness for.5) holds for example if Ω admits a finite number of conical or cuspidal non-exponential) points, see [9] and [3] for more details. Clearly u solves.3) if and only if it is a critical point for J λ u) = 2λ Ω u 2 + log Ω Ve u, u H 0 Ω),.6) where Ω = Ω Ω. As a consequence of the Moser Trudinger inequality [8] we see that J λ attains its maximum for any λ 0, 8π). Indeed, J λ is bounded from above, upper-semicontinuous and coercive for any λ 0, 8π). A subtle problem arises for λ = 8π,since J 8π is bounded from above but not coercive. Then the existence of solutions for.3), or equivalently of maximizers for J 8π is not granted. In the case where no singularities are contained in Ω, that is α j = 0 j {,...,m}, this problem has been solved in [9]. However the rest of our discussion is more delicate than that in [9] because of the Dirac masses in.). This issue affects the proofs of our main results in various ways. For example in [9] some results obtained in [8] were used which cannot be taken for grant in our situation. As a matter of fact, even the definitions of the main quantities involved in our analysis have to be carefully studied to take into account the role of the Dirac masses. Therefore we provide an ab initio and self-contained discussion of the problem which generalizes that of the regular case α j = 0 j {,...,m}. Let us define I λ Ω) = sup J λ u),.7) u H 0 Ω) and F m z; q,ω)= 8π GΩ z, q) + log [ V z) ],.8) F m z; Ω) = 4πγ Ω z) + log [ V z) ]..9) Our first result toward the understanding of the existence/nonexistence problem for λ = 8π is a generalization of the one obtained in [8] in the regular case. Indeed, we have the following

4 48 D. Bartolucci, C.S. Lin / J. Differential Equations ) Theorem.2. I 8π Ω) + max Ω and if the strict inequality holds in.0), theni 8π Ω) is attained. F m ; Ω)+ log B Ω,.0) Next, as a consequence of Theorem., we prove the following Theorem.3. Let {u λ } be the family of maximizers for.3) for λ 0, 8π). The following properties are equivalent: i) There exists C > 0 such that sup λ 0,8π) u λ C; ii) I 8π is attained; iii) Problem.3) admits a solution for λ = 8π. Here and in the rest of this paper we will denote by dz dz dw dw dτ z) =, dτ w) = 2 2 the volume element corresponding to the coordinates z Ω and w D respectively. Then we have Theorem.4. Let log h be harmonic and continuous in Ω. Assume that {u k } is a one-point blow-up sequence for u = λ hveu in Ω, Ω hveu u = 0 on Ω, that is, as k +, suppose that {u k } satisfies weakly in the sense of measures in Ω, for some q Ω. Put hve u k λ k 8π, λ k 8πδ q,.) Ω hveu k ε k := λ k Ω hve u k)..2) Then, as k +,wehaveε k 0 + and λ k 8π = hq)v q)ε k Ω where Hz, q) dτ z) z q 4 R 2 \Ω ) dτ z) + o),.3) z q 4 Hz, q) = hz)v z) hq)v q) exp 8π GΩ z, q) 8πγq) )..4)

5 D. Bartolucci, C.S. Lin / J. Differential Equations ) Remark 2. It is well known that q, the blow-up point, must be a critical point for F m ; q,ω)+ log[h] or equivalently, a critical point for Fm ; Ω) + log[h]. In particular, by using.) and the massquantization result in [5], we see that q / {p,...,p m }. Remark. We will not provide the proof of Theorem.4 here since it can be worked out by a step-bystep adaptation of the one already worked out in [9]. Indeed, one has just to use Remark 2 and the fact that log hz)v z)) is harmonic in Ω \{p,...,p m }. We refer the reader to [9] for more details concerning this point. Let q be any critical point for F m ; q,ω).foreach j =,...,m, we choose p D to satisfy j Putting z = f q w) and f j = f p j,wehave V f q w) ) = exp 4π j=,...,m f q p j ) = p j..5) )) ) α j G Ω fq w), f q p j = f fq w) ) 2α j. j j=,...,m It is well known that φ : D D is univalent if and only if it takes the form φ σ,θ w) = e iθ σ w σ w, for some σ D, and θ [0, 2π). Moreover, observe that f q : D Ω, f j : D Ω, f q p j ) = p j, f j 0) = p j. Then, f fq w) ) = φ j p j,θ w) = eiθ j j p j w p, j w for some θ j [0, 2π), and V f q w) ) = j=,...,m f fq w) ) 2α j = j p j w j 2α p, j=,...,m j w.6) where the values {p,...,p m } are determined implicitly by.5). We also define r = min p j.7) j {,...,m} to be the minimal radius attained by {p,...,p m }. We observe that, since f q0) = q, q / {p,...,p m } and f q is univalent, then p j 0, j {,...,m}, and then r > 0..8) Set Φw) = j=,...,m p j w p j w ) α j,.9)

6 420 D. Bartolucci, C.S. Lin / J. Differential Equations ) and B r = B r 0). Then, let + Φw) f q w) = c k w k, w B r, k=0 be the power series expansion for Φw) f q w) in B r.sinceq Ω is a critical point for F m ; q,ω), then w = 0 D is a critical point in the transplanted domain D for F m ; 0, D). Indeed, we observe that in the transplanted domain D, the weight function see also 3.) and 3.5) below) corresponding to problem.3) reads V f w)) f q w) 2. In particular, since G D w, 0) 0, we have Φw) f q w) 2 = exp F m w; 0, D) ), and then w = 0 must be a critical point for Φw) f q w). We conclude that c = 0, so that we can define: Sq) = + r 2 c 0 2 c k 2 + k r2k ) + π D\B r Φw) f q w) 2 w 4 dτ w)..20) Clearly, since f q is univalent, and in view of.8), we have c 0 = By using Theorem.4, we will prove j=,...,m ) p 2α j f q 0) ) j Theorem.5. Let {u k } be a one-point blow-up sequence for.3), which then satisfies, as k +, weakly in the sense of measures in Ω, for some q Ω. Put Ve u k λ k 8π, λ k 8πδ q,.22) Ω Veu k σ k := λ k Ω Ve u k). Then, as k +,wehaveσ k 0 + and with S defined by.20). λ k 8π = πσ k Sq) + o) ),.23) By using Theorem. and Theorems.4,.5, we will prove the main result of this paper. In particular our result yields the two dimensional version of the sharp existence/nonexistence result obtained in [3] for elliptic equations with critical exponent in dimension 3. Indeed, we have the following Theorem.6. Let q be a relative maximizer for Fm ; Ω),withSq) 0. Then q is the unique absolute maximizer of Fm ; Ω) and in particular the set of maximizers {u λ } for J λ with λ 0, 8π) satisfies.22) with concentration point q.

7 D. Bartolucci, C.S. Lin / J. Differential Equations ) Remark. By using the notion of stability of critical points for Fm ; Ω) introduced in [4], we can prove that if q is any critical point for Fm ; Ω),withSq) 0, then q istheuniqueabsolutemaximizer and the conclusion of Theorem.6 holds true, in this case as well. The proof can be worked out by the same argument used to prove Theorem.6. We skip it here to avoid technicalities. We remark that this result has been firstly obtained in [9] in case α j = 0 j {,...,m}. As an immediate consequence of Theorem.6, we have Corollary.7. Let {u k } be a sequence of solutions which satisfies.). If λ k 8π =o)ε k as k +, where ε k has been defined in.2), thenλ k < 8π. Remark. Corollary.7 is false if Ω is a torus, see [2]. For a given α = α,...,α m ) 0, + ) 0, + ), letusdefine Ω m + α) = { p,...,p m ) problem 0.) has a solution } Ω Ω. Then, another immediate consequence of Theorem.6 is the following, Corollary.8. The set Ω+ m α) is open. A subtle and interesting problem to solve is that of the topology of Ω+ m α). Wewillinvestigate this problem in another paper. Meanwhile, we have the following Conjecture. If Ω is convex then Ω+ m α) is simply connected. Conjecture 2. If Ω = B,p = 0, α 2 = =α m and {p 2,...,p m } is symmetric, then 0.) has a solution. As a matter of fact, the analysis of the sign of Sq) as a function of {α j } and {p j }, j {,...,m}, is a delicate problem on its own. Even in the case where Ω take up very simple geometries, a complete characterization of the existence/nonexistence problem for 0.) is nontrivial. We illustrate this fact in Sections 5, 6, 7 by discussing some explicit examples, where we will use various consequences of Theorem.6. Indeed, we have Theorem.9. I 8π Ω) is attained if and only if Sq)>0 for an absolute maximizer q of Fm ; Ω). As a consequence of Theorems.9 and., we also have Theorem.0. I 8π Ω) is attained if and only if I 8π > + max Ω Fm ; Ω)+ log B Ω. This result is the two dimensional version of the sharp existence/nonexistence result obtained in [3] for elliptic equations with critical exponent in dimension 3. Moreover, as an immediate consequence of Theorems.6 and.9, we have Corollary.. If Fm ; Ω) admits more than one absolute maximizer, then I 8π Ω) is attained. This paper is organized as follows. In Section 2 we will prove Theorem.2 and some results which will be used in the proofs of Theorems.6,.9,.0. In Section 3 we will prove Theorems.5 and.6. In Section 4 we will prove Theorems.9,.0 and Corollary.8. In Sections 5, 6 and 7 we will analyze some examples by using the results obtained so far.

8 422 D. Bartolucci, C.S. Lin / J. Differential Equations ) Preliminary results Proof of Theorem.2. We first prove.0). Actually, we will prove two more refined results, which will be used in the proof of Theorems.3,.6,.9,.0. Lemma 2.. Let q be any critical point for Fm ; Ω) in Ω and g q : Ω D any inverse Riemann map which satisfies g q q) = 0. Define ) + ε v ε w) = 2log, ε + w 2 w D, 2.) and u ε z) = v ε gq z) ). Then, as ε 0 +,wehave J 8π u ε ) = + Fm q; Ω)+ log B Ω + ε c 0 Sq) + O ε 2). 2 Proof. Clearly, letting f q = gq,wehave J 8π u ε ) = 6π D v ε 2 + log Ω + log D V fq w) ) f q w) ) 2 e v ε. It is easy to verify that 6π D ε v ε 2 = log + ε ) + + ε. On the other side, we have D V fq w) ) f q w) 2 e v ε = I ε, r ) + I 2 ε, r ), where we define I ε, r ) = V fq w) ) f q w) 2 e vε dτ w), B r and I 2 ε, r ) = D\B r V fq w) ) f q w) 2 e vε dτ w).

9 D. Bartolucci, C.S. Lin / J. Differential Equations ) Clearly e v ε w 4 uniformly in D \ B r,asε 0 +, and it is not difficult to verify that I 2 ε, r ) = D\B r Φw) f q w) 2 w 4 dτ w) + Oε), as ε 0 +. Moreover, since c = 0, we have I ε, r ) = D\B r = 2π c c k w k=0 r 0 e vε dτ w) e v ε ρ dρ + 2π = 2π c ε) 2 2ε = 2π c 0 2 2ε 2π c r c k 2 ) + ε)2 + 2ε + r 2 ) + 2π 2r 2 0 ρ 2k 3 dρ + Oε) c k 2 2k ) r2k ) + Oε) + c + 2π k 2 2k ) r2k ) + Oε), as ε 0 +. Thus, we may use the above expansions to conclude that [ J 8π u ε ) = + log ε + log log ε + log π c ε 2π c π Ω 2r 2 Φw) f ) q + w) 2 dτ w) + w 4 O ε 2)] D\B r = + log π c ε Ω π c 0 2 2π c c + 2π k 2 2r 2 2k ) r2k ) Φw) f ) q + w) 2 dτ w) + w 4 O ε 2) D\B r = + log c log π Ω + c 0 2 Sq)ε + O ε 2) = + Fm q; Ω)+ log B Ω + c 0 2 Sq)ε + O ε 2), where we used the fact that c 0 2 = Φ0) f q 0) 2 = exp F m 0; 0, D) ) exp Fm 0; D) ) = exp Fm q; Ω) ). + c k 2 2k ) r2k ) See also 3.5) below for more details concerning the last identity.

10 424 D. Bartolucci, C.S. Lin / J. Differential Equations ) Since the choice of the critical point q in Lemma 2. is arbitrary, it follows immediately that.0) holds. To conclude the proof of Theorem.2, we will also need the following Lemma 2.2. I λ Ω) is continuous and increasing for λ 0, 8π]. Proof. Clearly J λ u) < J λ2 u) for any u H 0 Ω) and λ <λ 2, and then I λ I λ2. For anyλ 0 0, 8π), and for any sequence λ n λ 0,let{u n } be a sequence of maximizers, which of course exist because I λ is attained in 0, 8π). Then, I λn = J λn u n ) J λn u), u H 0 Ω). 2.2) Passing to the limit we easily conclude that lim inf n + I λn J λ0 u), u H 0 Ω), i.e. lim inf n + I λn I λ0. On the other side, since {u n } is bounded in H 0 Ω), we also conclude that I λ0 J λ0 u n ) = J λ0 u n ) J λn u n ) + J λn u n ) = o) + I λn, as n +, and we obtain lim sup n + I λn I λ0.ifλ n 8π, by using 2.2) we conclude once more that lim inf I λ n + n I 8π, and this time we may use the monotonicity of I λ to obtain lim sup I λn I 8π. n + Observe that, for any ρ PΩ), where PΩ) = { ρ L Ω) ρ 0, ρ log ρ L Ω), ρ L Ω) = } the functional, f λ ρ, V,Ω)= λ 2 Ω ρg Ω ρ Ω ρ log ρ + Ω ρ log V Ω, 2.3) is well defined, where G Ω denotes the Green function defined in.2) and G Ω ρ the standard convolution. By arguing as [7], we also have the following Lemma 2.3. Let {u k } be a sequence of maximizers for J λk as λ k 8π. Assume that {u k } satisfies λ k Ve u k 8πδ q, 2.4) Ω Veu k for some q Ω. Then, the concentration point q is an absolute maximizer for Fm ; Ω) and lim sup k + J λk u k ) + Fm q; Ω)+ log B Ω. 2.5)

11 D. Bartolucci, C.S. Lin / J. Differential Equations ) Proof. We may assume without loss of generality that B q) Ω. Putting ρ k = Veu k Ω Veu k and substituting in 2.3), we see that, for any k large enough, we have J λk u k ) = fλ k ρ k, V,Ω) = λ k ρ k G B q) ρ k 2 + B q) Ω sup ρ PB ) ρ k log V Ω λ k 2 B q) B q) ρ k log ρ k + o) + λ k 2 ρ k G B q) ρ k B q) ρ k GΩ ρ k f λ k ρ,, B ) + 4πγ Ω q) + log V q) + log B Ω + o) = sup fλ k ρ,, B ) + Fm q; Ω)+ log B ρ PB ) Ω + o), where we have used 2.4) and the fact that B q) ρ k G B q) ρ k G B q)q, q) = G B 0, 0) = 0, as k +. It has been proved in [8] that, for any λ 0, 8π), { sup fλ ρ,, B ) = sup v 2 + log ρ PB ) 2λ v H 0 B ) B B } v e. Since the maximizers for this variational problem are well known and take up the simple radial expression 2.), the same explicit evaluation of Lemma 2., shows that sup fλ k ρ k,, B ) + 4πγ B 0) =, ρ P as k +. Thus, 2.5) follows by using this result and the estimate above. In particular we can use Lemma 2.2 to conclude that I 8π = lim I λ k lim sup k + k + J λk u k ) + Fm q; Ω)+ log B Ω. 2.6) At this point, by using.0), we immediately conclude that q is an absolute maximizer for F ; Ω). We can finally conclude the proof of Theorem.2. By using Lemma 2.2, we see that I 8π = lim k + I λk, whenever λ k 8π.Let{u k } be the corresponding sequence of maximizers. We argue by contradiction, and suppose that the strict inequality holds in.0), but I 8π is not attained. If {u k } happens to be uniformly bounded, then a bootstrap argument shows that we may find a subsequence which converges in H 0 Ω) to a solution u = u 8π. As a consequence of Lemma 2.2, we see

12 426 D. Bartolucci, C.S. Lin / J. Differential Equations ) that u 8π must be a maximizer for J 8π, so that I 8π is attained. Thus we may assume without loss of generality that {u k } is unbounded. The Brezis Merle theory [6] for Liouville type equations with singular data [5] then shows that there exists a subsequence which satisfies 2.4). Thus, we may apply Lemma 2.3 and conclude once more that 2.6) holds for some absolute maximizer q. It follows by.0) that indeed the equality sign holds in.0), which is the desired contradiction. Proof of Theorem.3. i) ii) If the maximizers are uniformly bounded, and since solutions to.3) are unique for λ 0, 8π), it is readily seen that they must converge, as λ 8π to a solution u 8π for.3) with λ = 8π. Lemma 2.2 implies that u 8π is a maximizer for J 8π.ThusI 8π is attained. ii) iii) If I 8π is attained, clearly.3) admits a solution. iii) i) Assume by contradiction that the maximizers {u λ } where not uniformly bounded. Theorem. asserts that the first eigenvalue of the linearized problem for.3) at u = u 8π is strictly positive. It then follows that the implicit function theorem can be applied and we may find a branch of uniformly bounded solutions {v λ } for any λ 8π < 0 small enough. Thus, for any λ 8π < 0 small enough, we conclude that u λ > v λ, which is a contradiction to the uniqueness of solutions for λ 0, 8π). 3. Proof of Theorems.5.6 Proof of Theorem.5. Let f q : D Ω be a Riemann map such that f q 0) = q and set ũ k w) = u k f q w)). Then{ũ k } satisfies ũ = λ V f qw)) f q w) 2 eũ D V f qw)) f qw) 2 eũ in D, ũ = 0 on D. 3.) Thus,.3) and.4) read λ k 8π = V f q 0) ) f q 0) 2 σ k D Hw) dτ w) w 4 R 2 \D ) dτ w) + o), w 4 Hw) = V f qw)) f q w) 2 V f q 0)) f q0) 2, where V f q )) satisfies.6). Clearly σ k = σ k.thus, λ k 8π = σ k [ D R 2 \D = σ k [ D V f q w)) f q w) 2 V f q 0)) f q 0) 2 dτ w) w 4 V f q 0)) f q 0) 2 w 4 ] dτ w) + o) Φw) f q w) 2 Φ0) f q 0) 2 w 4 dτ w) Φ0) f q 0) 2 = σ k [I r ) + I 2 r ) + π ] ) c r π c 0 2, R 2 \D ] dτ w) w 4

13 D. Bartolucci, C.S. Lin / J. Differential Equations ) where we define I r ) = B r Φw) f q w) 2 Φ0) f q 0) 2 dτ w), w 4 and I 2 r ) = D\B r Φw) f q w) 2 dτ w). w 4 It is not difficult to verify that r I r ) = 2π 0 + c k 2 r 2k 3 dr = 2π + c k 2. 2k ) r2k ) The conclusion follows collecting together the above expansions. Proof of Theorem.6. Let q be a relative maximizer for Fm ; Ω),withSq) 0. We divide the proof in three steps. Step. If Sq)<0 and if q is a strict maximum for Fm ; Ω),thenq is an absolute maximizer and the maximizers {u λ } for J λ with λ 0, 8π) satisfy.22) with concentration point q as λ 8π. Since the maximum is strict, then it is non-degenerate according to the definitions in [4] and [5]. Then, there exists a sequence of -point blow-up solutions for.3), satisfying.22) with concentration point q. If Sq) <0, Theorem.5 implies that λ k < 8π for any k large enough. By using Theorem., we conclude that {u k } coincides with a subset of the maximizer s set {u λ } for J λ with λ 0, 8π). Thus, the implications i) iii) of Theorem.3 and the Brezis Merle theory readily imply that {u λ } must satisfy.22) at point q as λ 8π.Moreoverq is an absolute maximizer by Lemma 2.3. Step 2. If Sq) 0, then q is an absolute maximizer of Fm ; Ω) and the maximizers {u λ } for J λ with λ 0, 8π) satisfy.22) with concentration point q as λ 8π. We have the following Lemma 3.. Let q be a relative maximum point for Fm ; Ω),withSq) 0. For any η 0, ) define f η) q ) = f q η ), Ω η = f η) q D),andΦ η) ) = Φη ). For any η > 0 small enough, we have q Ω η, {p,...,p m } Ω η, { p } η,..., p m D, 3.2) η and there exists a sequence of solutions {ũ η) n } for ũ = λ Φη) w) f η) q w)) 2 eũ Ω Φη) w) f η) in D, q w)) 2 eũ ũ = 0 on D, 3.3)

14 428 D. Bartolucci, C.S. Lin / J. Differential Equations ) which satisfies λ n 8π Φ η) w) f η) q w)) 2, λ eũn n Ω Φη) w) f η) q w)) 2eũn 8πδ ) Proof. Since Ω η = f η) q D), Ω η Ω and f η) q f q,asη, it is clear that 3.2) holds for any η small enough. Clearly q is a relative maximum point for Fm ; Ω) if and only if w = 0 is a relative maximum point for F m fq w); Ω ) = log [ w 2) 2 V fq w) ) f q w) 2 ] Fm w; D) = 4πγ D w) + log Ṽ w); Ṽ w) = V f q w) ) f q w) 2 = Φw) f q w) ) Of course, this is equivalent to the fact that w = 0 is a relative maximum point for w 2 ) 2 Φw) f q w) 2. η) We are going to prove that w = 0 is a strict maximum point for F m w; D) = 4πγ D w) + log Φ η) w) f η) q ) w) 2. Let {a k } and {b k } be the coefficients of the series expansions for f q and Φ in D and B r respectively. We can assume without loss of generality that c 0 = b 0 a = Φ0) f q 0)>0. Then, using the fact that c = 0, it is easy to verify that w 2 ) 2 Φw) f q w) 2 = c c 0 2Re [ c 2 w 2] 2 c 0 2 w 2 + O w 3) = c c 0 w, Aw +O w 3), where, setting w = w + iw 2, we denote by w the vector w, w 2 ), by, the standard scalar product and, setting c 2 = ξ + iξ 2,wedefineA = A{c k }) to be the 2 2 matrix whose entries are a = ξ c 0, a 2 = ξ 2 = a 2, a 22 = ξ c 0. As far as we are concerned with the analysis of the definiteness of A, we can assume without loss of generality that ξ 0. Since w = 0 is a maximum point, then, only one of the following situations may occur. Either A) det A > 0 and a < 0, a 22 < 0, or A2) det A = 0, and a < 0, a 22 < 0, or A3) det A = 0, and at least one of a and a 22 is zero. Indeed, it is readily seen that any other case would imply that w = 0 cannot be a local maximum point. Letting {c k η)} be the coefficients of the power series relative to Φ η) w) f η) q w)) 2, and since c k = k n + )a n+ b k n, n=0

15 D. Bartolucci, C.S. Lin / J. Differential Equations ) then we conclude that c k η) = η k+ c k. Thus, in particular c η) = 0, because c = 0. We are going to prove that for any η 0, ), then η) A η = A{c k η)}) is definite negative, i.e. that w = 0 is a strict maximum point for F m w; D). Itis enough to set up the worst case, that is A3). In this case, observe that det A η = η 2 c 2 0 η6 ξ 2 + ξ 2 2 ) > η 2 [ c 2 0 ξ 2 + ξ 2 2 )] = 0. Moreover, since c 0 > 0 and ξ 0, we can exclude the case where c 0 = ξ.thenc 0 = ξ, and we conclude that ξ > 0 and η 2+ ξ ηc 0 < η 2+ ξ ηc 0 = η η 2 ξ c 0 ) < ηξ c 0 ) = 0. Then, det A η > 0 and a η)<0, i.e. w = 0 is a strict maximum point for any η 0, ). η) Thus, since w = 0 is a strict maximum point for F m w; D), we can apply the results in [4] and [5] to conclude that for any such η, there exists a sequence of solutions for 3.3) which blows up as in 3.4). We are left to prove that λ n 8π. First of all, we need to apply Theorem.4 to problem 3.3) on Ω = B. To this end, we just need to verify that Φ η) w) f η) q w)) 2 takes the form hv for some log h harmonic and continuous in B and V taking the form.4) for a suitable set of m point singularities in B. Indeed, by using.5),.9) and 3.2), we have Φ η) w) f η) q w) ) 2 = = j=,...,m j=,...,m [ = exp 4π p j ηw ηp j w p j η w p j η w j=,...,m ) α j 2 f η) q w) ) 2 3.6) ) α j 2 j=,...,m p η j η w) ) α j 2 η) f ηp j w q w) ) 2 α j G B w, p ))] j hw) 3.7) η where loghw)) = log f η) q w)) j=,...,m η p j η w) ηp )α j 2 is easily seen to satisfy the required assumptions. Thus, we may apply Theorem.4 to problem 3.3) on Ω = B. In particular, since the base j w domain is B D and the blow-up point is w = 0, by arguing as in the proof of Theorem.5, we verify that.23) holds true for λ n 8π, that is, λ n 8π has the same sign of S = S η 0), where S η 0) = η2 r 2 + c [ = η 2 + r 2 c η 2k+) c k 2 k η 2k c k 2 r 2k ) + k r2k ) + D\Bηr D\Bηr Φηw) f ηw) 2 dτ w) w 4 ] Φηw) f ηw) 2 dτ ηw) ηw 4

16 430 D. Bartolucci, C.S. Lin / J. Differential Equations ) [ < η 2 + r 2 c 0 2 c k 2 + k r2k ) + Bη\B r ] Φw) f w) 2 dτ w) < η 2 Sq) 0. w 4 Thus, λ n < 8π for any n large enough. By using Lemma 3. we can conclude the proof of Step 2. We argue by contradiction and suppose that there is no blowing up sequences of solutions for.3) as λ 8π blowing up at q. The uniqueness result for.3), the Brezis Merle theory and the implications i) ii) of Theorem.3 then show that the set of maximizers {u λ } for λ 0, 8π), that is the unique solutions for.3), must satisfy either one of the following alternatives: B-) u λ C for any λ 0, 8π), B-2) {u λ } blows up at q 0 q as λ 8π. We first discuss case B-). If B-) holds true, then, in view of the implications i) iii) of Theorem.3, we may assume as well that.3) admits a solution u 8π for λ = 8π.SetC 8π = u 8π. Observe that, for any η small enough, and if {ũ η) n } denotes the sequence given by Lemma 3., then, using 3.6) and 3.7), we see that u n z) = ũ η) n g η) q z)) satisfies h η V η e u n u n = λ n Ω h η V η e u in Ω n η, u n = 0 on Ω η, 3.8) where V η z) = exp 4π j=,...,m ) α j G η z, p j ), 3.9) h η z), uniformly as η, 3.0) and G η, p) denotes the Green function on Ω η. Clearly log h η is harmonic in Ω η and continuous in Ω η uniformly with respect to η, asη.inparticular,{u n } satisfies.22) on Ω η with concentration point q as λ n 8π and S η q) <0 see the final part of the proof of Lemma 3.). Thus, setting η k as k +, and by arguing as in Step, we see that for any fixed k N, theset of maximizers {u k,λ)} for λ 0, 8π), blows up as λ 8π with concentration point q, and that q is an absolute maximizer for Fm ; Ω ηk ). Observe that u k = u k z,λ) depends smoothly on λ. Thisis due to the fact that the first eigenvalue of the linearized operator for 3.8) is strictly positive, and the solution is unique, see Theorem.. For any large constant C > 2C 8π, and for any k N, thereexists λ k 0, 8π) such that and sup u k z,λ k ) = C, 3.) Ω sup u k z,λ k ) = C > 2 sup u k z,λ k ), 3.2) B 2δ q) Ω\B δ q) where δ is any small positive number such that 8δ <distq, p), for any other critical point p of F m ; Ω). Thus, we may extract a subsequence of u k,λ), which, in view of 3.9) and 3.0), converges uniformly to a solution uz,λc)) of.3), where λc) = lim k + λ k 8π. Since.3) admits

17 D. Bartolucci, C.S. Lin / J. Differential Equations ) a unique solution for λ = 8π, and since C = u,λc)) > 2 u 8π = 2C 8π,thenλC)<8π necessarily. Thus, putting C = j and letting j +, as a consequence of 3.2), we have a sequence of blow-up solutions for.3) with concentration point q, asλ j 8π.Thisisthedesiredcontradiction in case B-) holds true. In case B-2) holds true, we repeat this argument to obtain a sequence of solutions u k,λ) for 3.8) which satisfies 3.). Thus, we may extract a subsequence of u k,λ), which converges uniformly to a solution uz,λc)) of.3), where λc) = lim k + λ k 8π. This time we observe that, since B-2) holds true, then the implication i) iii) of Theorem.3 implies that.3) does not admit any solution for λ = 8π.ThusλC) <8π and we may argue as above to conclude that a sequence of blow-up solutions for.3) with concentration point q, asλ j 8π exists, in this case as well. This is the desired contradiction. Thus, the conclusion of Step 2 follows by using Lemma 2.3 once more. Step 3. By using Step 2, we conclude that if Sq) 0, then q is an absolute maximizer of Fm ; Ω) and the maximizers {u λ } for J λ with λ 0, 8π) satisfy.22) with concentration point q as λ 8π. We argue by contradiction and assume that there exists another absolute maximizer for Fm ; Ω),say q 0 q. IfSq 0 ) 0, then we may apply Step 2 and come up with another sequence of solutions, say {u 0) k } blowing up at q 0 as λ k 8π.Since{u λk } and {u 0) } satisfy.22) with concentration points k q and q 0 respectively, then we would have at least two distinct solutions for.3) for some λ k < 8π, which is a contradiction to Theorem.. On the other side, if Sq 0 )>0, then Lemma 2. shows that I 8π > +max Ω Fm q; Ω)+log B Ω, and then Theorem.2 implies that I 8π is attained. Thus, the implications i) ii) of Theorem.3 imply that the maximizers {u λ } for J λ with λ 0, 8π) must be uniformly bounded, which is of course a contradiction. 4. Proofs of Theorems.9,.0 and of Corollary.8 Proof of Corollary.8. We argue by contradiction and assume that for some given m and α = α,...,α m ) the set Ω+ m α) is not open. Then, there exist p,...,p m ) such that I 8π is attained and a sequence p n),...,pn) m ) such that p n),...,pn) m ) p,...,p m ) as n + and I 8π n) is not attained for any n N. Then, by using Theorem.6, we see that for any n N there exists only one n) absolute maximizer for F m ; Ω), which we will denote by q n. We may fix compact subsets K K {Ω \{p,...,p m }} with the property that {q n } K and {p n),...,pn) m } {Ω \ K } for any n N. n) Clearly F m z; Ω) depends smoothly on p n),...,pn) m ) for z K.Inparticularq n is continuous as a function of p n),...,pn) m ). Since, by using Theorem.6, we have Sq n ) 0, and since S is continuous as a function of q, we conclude that Sq) 0, for some absolute maximizer satisfying q n q. Thus Theorem.6 implies that q is unique and I 8π is not attained, which is of course a contradiction. Proof of Theorem.9. It follows by Lemma 2. that if Sq) >0 for some absolute maximizer of F m ; Ω),thenI 8π > + max Ω Fm ; Ω)+ log B Ω. It then follows by Theorem.2 that I 8π is attained. To prove the opposite implication, we argue by contradiction and suppose that I 8π is attained and Sq) 0 for all absolute maximums of Fm ; Ω). Then, by using Theorem.6, we see that there exists only one absolute maximizer, say q 0, and the maximizers {u λ }, λ 0, 8π), blow up at q 0. It follows by the implications i) ii) of Theorem.3 that I 8π cannot be attained. Proof of Theorem.0. If I 8π Ω) > + max Ω Fm ; Ω)+ log B Ω,thenI 8π is attained by Theorem.2. On the other side, Theorem.9 asserts that ii) of Theorem.3 that is, I 8π is attained) is equivalent to Sq)>0 for an absolute maximizer q of F m. It follows immediately by Lemma 2. that I 8π Ω) > + max Ω Fm ; Ω)+ log B Ω.

18 432 D. Bartolucci, C.S. Lin / J. Differential Equations ) The case Ω = B and m = In this section we discuss the case where Ω = B and m = and set α = α and p = p. The case p = 0 has been already discussed in [4], where it has been proved by other methods that solutions for.3) blowing up as λ 8π,mustobeytoλ>8π for any λ 8π small enough. This is of course what we expect, since.3) admits in this case an explicit branch of radial solutions for any λ 0, 8π + α)) for any α > 0, and then Theorem.3 implies indeed that λ>8π for one-point blow-up solutions. In particular we have a third proof of this fact in this particular case. It is provided by Corollary., since it is trivial to verify that F admits a full inner circle as set of absolute maximizers. Thus, let us restrict our attention to the case where p 0. We have the following Theorem 5.. For any p B \{0}, thereexist0 < α α + < + such that if α α then I 8π is not attained while if α > α + then I 8π is attained. Proof. By using the rotation invariance of the problem, let us assume that p = ξ, ξ 0, ). A straightforward evaluation shows that, for any α > 0, the critical points q = q x + iq y of F, which in this case reads see.9)) F z; B ) = 2log z 2) z + 2α log ξ zξ, are solutions of q y = 0, Pq x ) = 0, where Pq x ) = 2ξq 3 x 2 + ξ 2 ) + α ξ 2 ))q 2 x + 2ξq x + α ξ 2 ). Since P) = 2 ξ) 2,thenP clearly admits at least one zero in, + ), and since F x; B ) has at least one absolute maximum in each interval,ξ) and ξ, ), we conclude that F z; B ) admits only two critical points in B, that is q = q ξ, α),ξ), q + = q + ξ, α) ξ, ). We observe however that P ) = 2 + ξ) 2, P0) = α ξ 2 ), Pξ) = α ξ 2 ) 2 < P0) and P) = 2 ξ) 2. Since P is of degree 3 and admits already one root in ξ, ) and one root in, + ), we conclude that At this point we may check that for any α > 0, q, 0). F q )> F q + ), ξ 0, ). Using φ ξ z) = z ξ zξ, z B, we see that φξ maps the family of disks of radius {B ρ } ρ 0,) conformally onto a family of balls {B R z)}, where R = Rξ, ρ) 0, ) is strictly increasing from 0 + to and z = zξ, ρ), zξ, ρ)

19 D. Bartolucci, C.S. Lin / J. Differential Equations ) ,ξ), is strictly decreasing from ξ + to 0 + as ρ increases from 0 + to.inparticular,b R z ) B R2 z 2 ) whenever R i = Rξ, ρ i ) and z i = zξ, ρ i ) with ρ 2 > ρ.letr ± and z ± satisfy q ± B R± z ± ). Clearly φ ξ q ) > φ ξ q + ), if and only if B R+ z + ) B R z ). We observe at this point that for any small α, F is close to F0 z; B ) := 4πγz) in C 2 norm on any compact subset of B \{ξ}. Inparticular, since P0) 0 and Pξ) 0, as α 0 +, we conclude that q 0 and q + ξ + as α 0 +. Indeed, by using the implicit function theorem, it is easy to prove that q is decreasing and q + is increasing with α. For later use, we remark that we may also conclude easily that q ) +,as α +. Thus, we may choose α small enough to guarantee that B R+ z + ) B R z ) and q < q +, so that, in particular, F q ) = 2log q 2) + 2α log φ ξ q ) > F q + ). Then q is the unique absolute maximizer for F and α small enough. Let us apply Theorem.9 and work directly in the base domain by using.23), Sq) = + r 2 c 0 2 c k 2 + k r2k ) + π D\B r Φw) f w) 2 dτ w), w 4 where f w) = q w q w, f w) = q 2 p ) α q w), Φw) = w 2 p, w Observe that, as α 0 +,wehave p = q ξ q ξ, r = p. q 0, p ξ), r ξ, and f w), Φw), uniformly in D \ B r, for any r >ξ. Then, we have π D\B r Φw) f w) 2 w 4 dτ w) ξ 2, as α 0+. Moreover, since the {c k } are nothing but the coefficients of the Cauchy product series for Φ f in B r, it is readily seen that and then, r 2 + c c k 2 k r2k ) ξ, 2 as α 0+, S q ξ, α) ), as α 0 +.

20 434 D. Bartolucci, C.S. Lin / J. Differential Equations ) We conclude that, for any fixed ξ 0, ), thereexistsα = α ξ) such that for any α α, I 8π is not attained. Next, we will prove that there exists α + = α + ξ) such that for any α > α +, I 8π is attained. To obtain this result, we will use Theorem.0. Let us define [ ] Ĩα) = I 8π B ) + max F ; B ). B Clearly we have [ ] Ĩα)> J 8π 0) + max F ; B ) B log φ ξ z) ) 2α dτ z) 2log q 2) 2α logφ ξ q ). B Observe that, since q B R z ), z 0,ξ) and q, 0), then, for any α, the sector { Cq ) = z q, ) 3, arg z 4 π, 5 )} π 4 satisfies Cq ) B \ B R z ). On the other side, by the definition of B R z ), we conclude that φ ξ z) > φ ξ q ) for any z B \ B R z ).Thus, log B φ ξ z) ) 2α dτ z) 2α logφ ξ q ) π > log B +log q 2)), 4 and in particular, Ĩα)> log 4 log q 2) > 0, whenever q > 4e. Since, as already remarked above, q ) + monotonically as α increases from 0 + to +, we conclude that there exists α + ξ) such that for any α > α + ξ), Ĩα)>0. Then, the definition of Ĩα) and Theorem.0 together imply that.3) admits a solution for λ = 8π for any such α. 6. The case where Ω is symmetric and m = We consider the case where Ω is symmetric with respect to the x and x 2 axes with m = one singularity p = p Ω, α = α. Of course, since Ω is simply connected, we have 0 Ω. Thenwe have Theorem 6.. If p = 0,thenI 8π is attained for any α > 0. Remark. In particular it is not true that for α small I 8π is attained if and only if it is attained for the problem with α = 0. Indeed, in case Ω = B with no singularities, it is well known that I 8π is not attained. We also point out that, in view of Remark, this result holds true for example on any regular N-polygon with N N even.

21 D. Bartolucci, C.S. Lin / J. Differential Equations ) Proof of Theorem 6.. By using the symmetry with respect to the x axis, see for example [, p. 232], we see that if a Riemann map is normalized in such a way that gq) = 0 and arg[g q)] > 0, for some real q Ω, wehavegw) = gw). Thus, according to our notations, let us set g = g q.inviewofthe symmetry with respect to the x 2 axis, by taking q = 0 and by using this result after a rotation, we conclude that g q is symmetric with respect to the x and x 2 axes. In particular, it follows that γ Ω z) and G Ω z, 0) are symmetric with respect to the x and x 2 axes. At this point, let us set p = 0, so that F z; Ω) = 4πγ Ω z) 4παG Ω z, 0). Clearly F is symmetric with respect to the x and x 2 axes. Since F z; Ω) as z 0 and z Ω, it admits at least one interior absolute maximizer, which we denote by q α Ω \{0}. Clearly, if either q α {x = 0} or q α {x 2 = 0} then F admits at least two absolute maximizers. If q α / {x = 0} {x 2 = 0} then F admits at least four absolute maximizers. In any case, as a consequence of Corollary., I 8π is attained for any α > The case where Ω = B and m = 2 We consider the case Ω = B with m = 2 symmetric singularities p = p = p 2 and α = α = α 2. In this situation we obtain Theorem 7.. For any p B \{0}, there exists α > 0 such that I 8π is attained for any α > α. Proof. Without loss of generality let p = 0,ξ) for some ξ 0, ). Then F 2 z; B ) = 2log z 2) z + 2α log ξ zξ + 2α log z + ξ + zξ. Clearly F2 is symmetric with respect to the x and x 2 axes. As a consequence, by arguing as in Section 6, we may use Corollary. and conclude that I 8π is attained whenever z = 0 does not happen to be the unique absolute maximizer for F2. Thus, our next aim will be to prove that there exists α = αξ) > 0 such that F 2 0; B )< F2 iξ 2 ; B ), 7.) for any α > α. Indeed, observe that 7.) is equivalent to that is 2log ξ 2α < 2log [ ξ 2 ) 2ξ 2 + ξ 4 ) α ], + ξ 4 ) α < ξ 2). 7.2) 2 Elementary considerations show that there exists a unique ξ = ξα) 0, ) such that + ξ 4 ) α = ξ 2), 2 and that 7.2) is satisfied for any ξ<ξ. In particular we can verify immediately that ξα) is increasing. Since +ξ 4 2 ) α 0 pointwise for ξ [0, ) as α +,thenξ as α +.Thus,forany

22 436 D. Bartolucci, C.S. Lin / J. Differential Equations ) ξ 0, ) there exists α = αξ) > 0 such that 7.) is satisfied for any α > α and we conclude that z = 0 cannot be a maximizer for F2.Inparticular F2 admits at least two maximizers and then I 8π is attained for any α > α. We conclude this series of examples with a result concerning the case where Ω is any bounded and simply connected domain with a finite number of conical type points see [9] for more details), with m singularities{p,...,p m } and α = α,...,α m ).LetΓ denote the set of absolute maximizers for γ Ω and assume that {p,...,p m } Γ =.Foranyq Γ,letS 0 q) be the quantity.20) corresponding to the regular case α = 0. We refer to [9] for more details concerning this point. Theorem 7.2. Suppose that S 0 q) 0 for at least one q Γ.Then,forany α small enough, I 8π is attained if and only if it is attained for α = 0. Proof. We first claim that there exists a compact subset K {Ω \{p,...,p m }} with the property that any absolute maximizer for Fm z; Ω), which we denote by q α,satisfies{q α } K as α 0. We argue by contradiction and suppose that there exists a sequence α n 0, such that there exists n) a sequence of maximizers {q αn } for F m z; Ω) such that q αn converges to one of the p j s as α n 0. We may assume without loss of generality that q αn p as α n 0. Let V n = V α n, z) be the weight defined by.4). Since V n foranyz Ω, and since q αn is a maximizer, for any n N we have 4πγ Ω z) + log [ V n z) ] = F n) m z; Ω) F n) m q αn ; Ω) 4πγ Ω q αn ), z Ω. In particular we conclude that for any n N and for any fixed q Γ,wehave As n + we conclude that 4πγ Ω q) + log [ V n q) ] 4πγ Ω q αn ). 4πγ Ω q) 4πγ Ω p ), which is the desired contradiction. At this point we observe that the same argument shows that there exists at least one maximizer q α such that q α q Γ as α 0. Then we see that Sq α ) S 0 q) because S is continuous as afunctionofq α as far as {q α } K. We claim that S 0 q) 0. Indeed, if S 0 q) = 0, then a result in [9] asserts that q is the unique maximizer for γ Ω. This is a contradiction to our assumption that S 0 q) 0 for at least one absolute maximizer of γ Ω.ThusS 0 q) 0, and we conclude that for any small α, I 8π is attained if and only if it is attained for α = 0. ThisisbecauseS 0 plays the same role see [9]) for the regular problem that S plays in the singular one. References [] L.V. Ahlfors, Complex Analysis, third ed., McGraw Hill, 979. [2] D. Bartolucci, C.C. Chen, C.S. Lin, G. Tarantello, Profile of blow up solutions to mean field equations with singular data, Comm. Partial Differential Equations ) 2004) [3] D. Bartolucci, C.S. Lin, Uniqueness results for mean field equations with singular data, Comm. Partial Differential Equations ) [4] D. Bartolucci, E. Montefusco, On the shape of blow up solutions to a mean field equation, Nonlinearity ) [5] D. Bartolucci, G. Tarantello, Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Comm. Math. Phys ) [6] H. Brezis, F. Merle, Uniform estimates and blow-up behaviour for solutions of u = V x)e u in two dimensions, Comm. Partial Differential Equations 6 8 9) 99) [7] E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys ) [8] E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two dimensional Euler equations: A statistical mechanics description. II, Comm. Math. Phys )

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