Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials
aa r X i v : . [ m a t h . A P ] J u l COMPACTNESS, EXISTENCE AND MULTIPLICITY FOR THE SINGULAR MEANFIELD PROBLEM WITH SIGN-CHANGING POTENTIALS
FRANCESCA DE MARCHIS, RAFAEL LÓPEZ-SORIANO AND DAVID RUIZA
BSTRACT . In this paper we consider a mean field problem on a compact surface without bound-ary in presence of conical singularities. The corresponding equation, named after Liouville, ap-pears in the Gaussian curvature prescription problem in Geometry, and also in the ElectroweakTheory and in the abelian Chern-Simons-Higgs model in Physics. Our contribution focuses onthe case of sign-changing potentials, and gives results on compactness, existence and multiplicityof solutions.
1. I
NTRODUCTION
The classical problem of prescribing the Gaussian curvature on a compact surface Σ under aconformal change of the metric dates back to [12, 37]. Let us denote by g the original metric, ˜ g the new one and e v the conformal factor (that is, ˜ g = e v g ). The problem reduces to solving thePDE − ∆ g v + 2 K g ( x ) = 2 K ( x ) e v , where K g , K denote the curvature with respect to g and ˜ g , respectively. Observe that by theUniformization Theorem we can assume that K g is a constant. The solvability of this equationhas been studied for a long time, and it is not possible to give here a comprehensive list ofreferences.The above setting needs to be modified if one wants to prescribe also the appearance of con-ical singularities on the surface, a case that was first studied in [51]. We recall that a conformalmetric ˜ g has a conical singularity at p of order α ∈ ( − , + ∞ ) if there exist local coordinates z in C such that z ( p ) = 0 and(1.1) ˆ g ( z ) = e ψ | z | α | dz | , where ˆ g is the local expression of g and ψ is continuous in a neighborhood of in C and C outside the point p .In this case we are led with weak solutions of the problem(1.2) − ∆ g v + 2 K g = 2 K ( x ) e v − π m X j =1 α j δ p j , Mathematics Subject Classification.
Key words and phrases.
Prescribed Gaussian curvature problem, conical singularities, variational methods, Morsetheory. where δ p j denotes a Dirac delta at the point p j ∈ Σ (see Appendix of [3] for a rigorous deductionof (1.2)). Integrating the above equation and taking into account the Gauss-Bonnet formula, weobtain(1.3) πχ (Σ) = 2 ˆ Σ Ke v dV g − π m X j =1 α j . We now transform equation (1.2) into another one which admits a variational structure. Let G ( x, y ) be the Green function of the Laplace-Beltrami operator on Σ associated to g , i.e.(1.4) − ∆ g G ( x, y ) = δ y − | Σ | in Σ , ˆ Σ G ( x, y ) dV g ( x ) = 0 . We define(1.5) h m ( x ) = 4 π m X j =1 α j G ( x, p j ) = 2 m X j =1 α j log (cid:18) d ( x, p j ) (cid:19) + 2 πα j H ( x, p j ) , where H is the regular part of G . By the change of variable u = v + h m we can pass to the equation ( ∗ ) λ − ∆ g u = λ ˜ Ke u ´ Σ ˜ Ke u dV g − | Σ | ! in Σ , where(1.6) ˜ K = Ke − h m , and, according to (1.3), λ is given by(1.7) λ = 4 π ( χ (Σ) + m X j =1 α j ) . Notice that e v = e − h m e u which is consistent with (1.1). Observe also that ˜ K ( x ) ≃ d ( x, p j ) α j K ( x ) close to p j .This equation appears also in Physics, in the mathematical Glashow-Salam-Weinberg modelof the Electroweak Theory and in the abelian Chern-Simons-Higgs model, see [33, 39, 49, 52]. Inthis context, the points p j represent vortices of order α j ∈ N . There are by now many worksdealing with this problem, see [3–7, 9, 10, 17, 18, 24–26, 29, 38, 45, 46, 50]. In this framework, theequation receives the name of (singular) mean field equation. We highlight that under thisperspective the restriction (1.7) is not present.In this paper we are concerned with equation ( ∗ ) λ in the case in which K is a sign-changingfunction. We give existence and generic multiplicity results by means of variational methods.For that one also needs to show that solutions of ( ∗ ) λ are a priori bounded, which in this casemeans that they form a compact set. HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 3
Regarding compactness of solutions, one can pose the question as follows: given u n a se-quence of solutions of ( ∗ ) λ for λ = λ n → λ , is it uniformly bounded? This question has beenaddressed in [16,41] for the regular problem, and in [8,9] for the equation with vortices, alwaysfor positive potentials K ( x ) . In summary, if blow-up occurs then e u n concentrate around a fi-nite set of critical points, and a quantization argument shows that λ must belong to a certaindiscrete critical set. Here, the assumption on the positivity of K is not just a technical one, ascan be inferred from some recent examples of blowing-up solutions in [13, 30]. Those solutionsconcentrate around local maxima of K at level.The question of compactness has been elided in [42] by using energy estimates in a relatedproblem posed on a surface with boundary. This technique is however very much restricted tothe case considered there.For sign-changing functions K the first related compactness result is [20]. That paper isconcerned with the scalar curvature prescription problem, a higher dimension analogue of ourproblem which has also attracted much attention in the literature. The authors are able to showcompactness of solutions under the hypothesis(H1) K is a sign-changing C ,α function with ∇ K ( x ) = 0 for any x ∈ Σ with K ( x ) = 0 .An improvement of this technique has been given in [23]. The general idea is to first deriveuniform integral estimates, which allow one to obtain a priori estimates in the region { x ∈ Σ : K ( x ) < − δ } , for δ > small. Then the moving plane technique is used to compare thevalues of u on both sides of the nodal curve Γ = { x ∈ Σ : K ( x ) = 0 } . This, together withthe aforementioned integral estimate, implies boundedness in a neighborhood of Γ . Finally, werely on [8, 9, 41] for the region { x ∈ Σ : K ( x ) > δ } .The approach of [20] has been partially adapted to problem ( ∗ ) λ in [21, 29]. However, thoseresults use the stereographic projection to pass to a global problem in the plane and are hencerestricted to Σ = S . Moreover, the derivation of the integral estimate [21, Lemma 2.2], essentialin both papers, is not completely clear. One of the goals of this paper is to settle the question ofcompactness: we show compactness for ( ∗ ) λ in any compact surface under assumption (H1).Our approach follows the ideas of [23]. The main difficulty with respect to [23] comes fromthe fact that u n is neither positive nor uniformly bounded from below, a priori. This is a prob-lem for the integral estimate in [21, 23], and also for the use of the moving plane method nearthe nodal curve. In our proofs we first estimate the negative part of u n by using Kato inequal-ity. This is the key for the proof of the integral estimate and is also essential to perform thecomparison argument by the moving plane method.For what concerns existence and multiplicity of solutions, we shall restrict ourselves to thecase of positive orders α j . Our proofs make use of variational methods. Indeed, problem ( ∗ ) λ is the Euler-Lagrange equation of the energy functional(1.8) I λ ( u ) = 12 ˆ Σ |∇ u | dV g + λ | Σ | ˆ Σ u dV g − λ log ˆ Σ ˜ Ke u dV g , defined in the domain(1.9) X = (cid:26) u ∈ H (Σ) : ˆ Σ ˜ Ke u dV g > (cid:27) . If λ < π , then I λ is coercive and a minimizer exists, see [51]. Instead, I λ is not boundedfrom below if λ > π . This range is the main concern of this paper, and we shall use a Morse FRANCESCA DE MARCHIS, RAFAEL LÓPEZ-SORIANO AND DAVID RUIZ theoretical approach to find solutions of ( ∗ ) λ which are saddle-type critical points of I λ . Inorder to do that we study the topology of the energy sublevels of I λ . In a certain sense, afunction u at a low energy level concentrate around a certain number of points of Σ , and thetopology of the space of those configurations plays a crucial role in the min-max argument.This approach has been followed in different works. For instance, [31] considers an analogueproblem in dimension 4, whose ideas can be applied equally well to the regular mean fieldproblem (see [32]), whereas [3, 18, 45] deal with the singular problem. Let us point out thatall those papers consider positive functions K . For sign changing potentials, the location ofthe points is restricted to the set { x ∈ Σ : K ( x ) > } , and this fact changes dramatically thetopology of the space of configurations.Roughly speaking, to obtain existence of solutions one needs to show that the very lowenergy sublevels form a non-contractible set. If Σ = S this study has been carried out in [29],yielding existence of solutions. In this paper we consider the case of general Σ and we alsoprove multiplicity results. For multiplicity, we need to estimate the sum of the Betti numbersof the energy sublevels. This multiplicity result is valid under nondegeneracy assumptions,which are generic (in the couple ( K, g ) ) by a transversality argument.In general, min-max arguments yield existence of solutions provided that the well-knownPalais-Smale property is satisfied. In this type of problems the validity of that property is stillan open problem; however we can circumvent this difficulty by using the deformation lemmaestablished in [43], in the spirit of [48]. This technique has now become well known, and relieson compactness of solutions of approximating problems. At this point our aforementionedcompactness result comes into play.The rest of the paper is organized as follows. In Section 2 we set the notation, recall somepreliminary results and state the main theorems proved in the paper. Section 3 is devoted tothe proof of the compactness result, see Theorem 2.1. In Section 4 we give a description of thetopology of the energy sublevels. The estimation of the dimension of the homology groups ofthose sets is made in Section 5. Finally, the proofs of our main results are completed in Section 6.2. N OTATIONS , MAIN RESULTS AND PRELIMINARIES
In this section we fix the notation used in this paper, formulate the principal results obtainedand collect some preliminary known results.From now on (Σ , g ) will be a compact surface without boundary Σ equipped with a Rie-mannian metric g and d ( x, y ) will denote the distance between two points x, y ∈ Σ induced bythe ambient metric. B p ( r ) stands for the open ball of radius r > and center p ∈ Σ and Ω r = { x ∈ Σ : d ( x, Ω) < r } . Given f ∈ L (Σ) , we denote the mean value of f by − ´ Σ f = | Σ | ´ Σ f , where | Σ | is the area of Σ .Since the functional I λ is invariant under addition of constants, we can restrict its domain tofunctions with mean. In other words, we can consider I λ defined in ¯ X , where(2.1) ¯ X = { u ∈ X : ˆ Σ u dV g = 0 } . For a real number a , we introduce the following notation for the sublevels of the energy func-tional I λ (defined in (1.8)) restricted to ¯ XI aλ = { u ∈ ¯ X : I λ ( u ) ≤ a } . HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 5
The symbol ∐ will be employed to denote the disjoint union of sets.Throughout the paper, the sign ≃ refers to homotopy equivalences, while ∼ = refers to home-omorphisms between topological spaces or isomorphisms between groups.Given a metric space M and k ∈ N , we denote by Bar k ( M ) the set of formal barycenters oforder k on M , namely the following family of unit measures supported in at most k points(2.2) Bar k ( M ) = ( k X i =1 t i δ x i : t i ∈ [0 , , k X i =1 t i = 1 , x i ∈ M ) . We consider
Bar k ( M ) as a topological space with the weak ∗ topology of measures.Uninfluential positive constants are denoted by C , and the value of C is allowed to varyfrom formula to formula.2.1. Main results.
Let us define the sets Σ + = { x ∈ Σ : K ( x ) > } , Σ − = { x ∈ Σ : K ( x ) < } , Γ = { x ∈ Σ : K ( x ) = 0 } . Assumption (H1) implies that the nodal line Γ is regular and that(2.3) N + = { connected components of Σ + } < + ∞ . In what follows we will assume that:(H2) p j / ∈ Γ for all j ∈ { , . . . , m } .So we can suppose, up to reordering, that there exists ℓ ∈ { , . . . , m } such that(2.4) p j ∈ Σ + for j ∈ { . . . , ℓ } , p j ∈ Σ − for j ∈ { ℓ + 1 , . . . , m } .For K > it is known that the blow-up phenomena can only occur if the parameter λ takesthe form πr + P mj =1 π (1 + α j ) n j , with r ∈ N ∪ { } , n j ∈ { , } (see [9]). Therefore the setof solutions is compact if λ has a different form. In the next theorem we obtain an analogousconclusion without the sign restriction, where the set of critical values is(2.5) Λ = πr + ℓ X j =1 π (1 + α j ) n j : r ∈ N ∪ { } , n j ∈ { , } \ { } . Theorem 2.1.
Assume that α , . . . , α m > − and let K n be a sequence of functions with K n → K in C ,α sense, where K satisfies (H1), (H2). Let u n be a sequence of weak solutions of the problem (2.6) − ∆ g u n = ˜ K n e u n − f n in Σ , with f n → f in C ,α sense and ˜ K n = K n e − h m with h m given by (1.5) . Then, the following alternativeholds(1) either u n is uniformly bounded in L ∞ (Σ) ;(2) or, up to a subsequence, u n diverges to −∞ uniformly;(3) or, up to a subsequence, lim n → + ∞ ´ Σ ˜ K n e u n ∈ Λ , defined in (2.5) . FRANCESCA DE MARCHIS, RAFAEL LÓPEZ-SORIANO AND DAVID RUIZ
We point out that that the conical singularities located in Σ − do not play any role in thecompactness result. Observe also that equation ( ∗ ) λ can be written in the form (2.6) by addinga suitable constant to u n = u , if ˜ K n = ˜ K and f n = λ | Σ | .In order to state our existence result we introduce an additional assumption on K :(H3) N + > k or Σ + has a connected component which is not simply connected,where N + is defined in (2.3). Theorem 2.2.
Let α , . . . , α ℓ > , where ℓ is defined in (2.4) , and λ ∈ (8 kπ, k + 1) π ) \ Λ . If (H1),(H2), (H3) are satisfied then ( ∗ ) λ admits a solution. For
K > , and thus Σ + = Σ and N + = 1 , (H3) is satisfied if the surface Σ has positivegenus; this case has been covered in [3].If Σ + has trivial topology Theorem 2.2 is not applicable. We can give a result also in this case,following the ideas of [45]. For that, we define the set:(2.7) J λ = { p j ∈ Σ + : λ < π (1 + α j ) } and we introduce the hypothesis(H4) J λ = ∅ . Theorem 2.3.
Let α , . . . , α ℓ ∈ (0 , , where ℓ is defined in (2.4) , and λ ∈ (8 π, π ) \ Λ . If (H1), (H2),(H4) are satisfied then ( ∗ ) λ admits a solution. Remark 2.4.
There are many examples of applications of these results to the geometric problem com-mented in the Introduction. Just to exhibit an example, let us consider the problem of prescribing aconformal metric in
Σ = T with gaussian curvature K and one conical point p of order α . Assumethat assumptions (H1), (H2) are satisfied. Then Theorem 2.2 implies that the problem is solvable if oneof the following assumptions are satisfied:(1) α ∈ (2 k, k + 2) with k ∈ N and Σ + has more than k connected components.(2) α ∈ (2 k, k + 2) with k ∈ N and Σ + has a component which is not simply connected.Let us now consider the same problem but with m conical points, all of them of order α . Then Theorem2.3 implies that the geometric problem is solvable if α ∈ (0 , and < m α < α and at least oneconical point is placed in Σ + .Many other examples can be constructed. In order to now state our multiplicity results we introduce some more notation. Let us denoteby A i the non-contractible connected components of Σ + and C h be the contractible ones, i =1 , . . . , N , h = 1 , . . . , M and N, M ∈ N ∪ { } , N + M = N + . Obviously, Σ + = N a i =1 A i ∐ M a h =1 C h . Recall that a bouquet of g circles is a set B g = S gj =1 S ′ j where S ′ j are simple closed curvesverifying that S ′ i ∩ S ′ j = { q } . If A i has genus g i and b i boundary components, it is well knownthat A i can be retracted to an inner bouquet B g i , where g i = 2 g i + b i − . Instead, C h ishomotopically equivalent to any point y h ∈ C h . Therefore HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 7 (2.8) Σ + ≃ N a i =1 B g i ∐ { y , . . . , y M } , with g i = 2 g i + b i − for i = 1 , . . . , N .Define M as the space of all Riemannian metrics on Σ equipped with the C ,α norm and(2.9) K ℓ = (cid:26) K : Σ → R : K satisfies (H1), (H2) p , . . . , p ℓ ∈ Σ + , p ℓ +1 , . . . , p m ∈ Σ − (cid:27) , also equipped with the C ,α norm. Theorem 2.5.
Let ℓ ∈ { , . . . , m } and let us assume that α , . . . , α ℓ > . If λ ∈ (8 kπ, k + 1) π ) \ Λ , k ∈ N , then for a generic choice of function K and metric g (namely for ( K, g ) in an open and densesubset of K ℓ × M ), then { solutions of ( ∗ ) λ } ≥ X q ≥ d q , where if k + 1 − M ≤ N , then d q = (cid:16) N + M − N + M − p (cid:17) X a + . . . + a N = k − p + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − p (1 ≤ p ≤ k + 1) , otherwise; while if k + 1 − M ≥ N , then d q = (cid:16) N + M − N + M − p (cid:17) X a + . . . + a N = k − p + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − p (1 ≤ p ≤ N ) , (cid:16) N + M − sM − s (cid:17) X a + . . . + a N = k − N − s + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − N − s (1 ≤ s ≤ M ) , otherwise; with s a,g = (cid:0) a + g − g − (cid:1) and g i defined in (2.8) .Moreover we adopt the following convention: if N = 0 X a + . . . + a N = ha i ≥ s a ,g . . . s a N ,g N = (cid:26) if h = 0 , if h = 0 .Notice that if k + 1 − M = N the two formulas coincide. We point out that { solutions of ( ∗ ) λ } → + ∞ as N + = N + M → + ∞ .The above result gives no information if Σ + has trivial topology; however, our second mul-tiplicity result can be applied also in this case. FRANCESCA DE MARCHIS, RAFAEL LÓPEZ-SORIANO AND DAVID RUIZ
Theorem 2.6.
Let ℓ ∈ { , . . . , m } and let us assume that α , . . . , α ℓ ∈ (0 , . If λ ∈ (8 π, π ) \ Λ , thenfor a generic choice of function K and metric g (namely for ( K, g ) in an open and dense set of K ℓ × M ),then { solutions of ( ∗ ) λ } ≥ N + − N X i =1 g i + | J λ | , where the set J λ is defined in (2.7) and g i in (2.8) . Remark 2.7.
We point out that by standard elliptic estimates any solution u of ( ∗ ) λ is classical if all the α j ’s are positive, while if α j ∈ ( − , for j ∈ J ⊂ { , . . . , m } , then u ∈ C (Σ \ {∪ j ∈ J p j } ) ∩ C (Σ) . Topological and Morse-theoretical preliminaries.
In this subsection we recall a classical theorem on Morse inequalities. Furthermore we givea short review of basic notions of algebraic topology needed to get the multiplicity estimates.Finally, we state a recent result concerning the topology of barycenter sets with disconnectedbase space.Given a pair of spaces ( X, A ) we will denote by H q ( X, A ; Z ) the relative q-th homologygroup with coefficient in Z and by ˜ H q ( X ; Z ) = H q ( X, x ; Z ) the reduced homology withcoefficient in Z , where x ∈ X . We adopt the convention that ˜ H q ( X ; Z ) = 0 for any q < .Finally, if X , Y , are two topological spaces and f : X → Y is a continuous function, we willdenote by f ∗ : H q ( X ; Z ) → H q ( Y ; Z ) the pushforward morphism induced by f .Let us first recall a classical result in Morse theory. Theorem 2.8. (see e.g. [19], Theorem 4.3) Suppose that H is a Hilbert manifold, I ∈ C ( H ; R ) satisfiesthe ( P S ) -condition at any level c ∈ [ a, b ] , where a , b are regular values for I . If all the critical points of I in { a ≤ I ≤ b } are nondegenerate, then { critical points of I in { a ≤ I ≤ b } with index q } ≥ dim( H q ( { I ≤ b } , { I ≤ a } ; Z )) for any q ≥ ,where we call (Morse) index of u ∈ H the number of negative eigenvalues (counted with multiplicity)of the selfadjoint operator d I ( u ) . In what follows we collect some well-known definitions and results in algebraic topologyand we refer to [36] for further details.
Wedge sum.
Given spaces C and D with chosen points c ∈ C and d ∈ D , then the wedgesum C ∨ D is the quotient of the disjoint union C ∐ D obtained by identifying c and d to asingle point. If { c } (resp. { d } ) is a closed subspace of C (resp. D ) and is a deformation retractof some neighborhood in C (resp. D ), then(2.10) ˜ H q ( C ∨ D ; Z ) ∼ = ˜ H q ( C ; Z ) M ˜ H q ( D ; Z ) , see [36, Corollary 2.25]. Smash Product.
Inside a product space X × Y there are copies of X and Y , namely X × { y } and { x } × Y for points x ∈ X and y ∈ Y . These two copies of X and Y in X × Y intersect HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 9 only at the point ( x , y ) , so their union can be identified with the wedge sum X ∨ Y . The smashproduct X ∧ Y is then defined to be the quotient X × Y /X ∨ Y . For the reduced homology ofthe smash product the following formula holds, [36, page 276],(2.11) ˜ H q ( C ∧ D ; Z ) ∼ = M i + j = q − ( ˜ H i ( C ; Z ) ⊗ ˜ H j ( D ; Z )) . Unreduced suspension.
The unreduced suspension (often, as in [36], denoted by SC ) is de-fined to be(2.12) Σ C = ( C × [0 , / { ( c , ≃ ( c , and ( c , ≃ ( c , for all c , c ∈ C } . For the reduced homology of the unreduced suspension the following formula holds, [36, page132, ex. 20],(2.13) ˜ H q +1 (Σ C ; Z ) ∼ = ˜ H q ( C ; Z ) . Join.
The join of two spaces C and D is the space of all segments joining points in C to pointsin D . It is denoted by C ∗ D and is the identification space C ∗ D = C × [0 , × D/ ( c, , d ) ∼ ( c ′ , , d ) , ( c, , d ) ∼ ( c, , d ′ ) ∀ c, c ′ ∈ C, ∀ d, d ′ ∈ D. Being C ∗ D ≃ Σ( C ∨ D ) , [36, page 20, ex. 24], we have that(2.14) ˜ H q ( C ∗ D ; Z ) ∼ = ˜ H q (Σ( C ∨ D ); Z ) . At last, we present a recent result obtained in [1, Theorem 5.19] concerning the space offormal barycenters on a disjoint union of spaces.
Proposition 2.9.
For C , D two disjoint connected spaces and k ≥ , Bar k ( C ∐ D ) has the homologyof Bar k ( C ) ∨ Σ Bar k − ( C ) ∨ Bar k ( D ) ∨ Σ Bar k − ( D ) ∨∨ k − _ ℓ =1 ( Bar k − ℓ ( C ) ∗ Bar ℓ ( D )) ∨ k − _ ℓ =2 (Σ Bar k − ℓ ( C )) ∗ Bar ℓ − ( D ) .
3. C
OMPACTNESS OF SOLUTIONS
In this section we present the proof of Theorem 2.1, a compactness result for solutions u n ofthe general problem (2.6).As commented previously, most of the results in this direction consider only the case ofpositive K , like for instance [16, 41] for the regular case and [8, 9] for the singular one. In orderto prove Theorem 2.1, and following [20], we will first derive an a priori estimate in the regionin which K is negative, and later we will give such estimates in the nodal region of K . Proposition 3.1.
Given δ > , there exists C > such that u n ( x ) ≤ C for all x ∈ Σ − \ Γ δ , n ∈ N . Proposition 3.2.
There exist ε, C > , such that u n ( x ) ≤ C for all n ∈ N and x ∈ Γ ε . The proof of Theorem 2.1 will be finally accomplished by studying the possible blow-up ofthe sequence u n in Σ + \ Γ ε .One of the difficulties in our study is that we do not know a priori whether the term(3.1) ˆ Σ | ˜ K n | e u n dV g is bounded or not. By standard regularity results, this would give a priori W ,p estimates( p ∈ (1 , ) on u n . Instead, if we integrate (2.6) we only obtain that ´ Σ ˜ K n e u n is bounded.Our first lemma shows that such kind of estimate is indeed possible for u − n = min { u n , } .This fact will be useful to prove both Propositions 3.1 and 3.2. Lemma 3.3.
Under the conditions of Theorem 2.1, define v n = u − n − − ´ Σ u − n . Then there exists C > such thata) k v n k L p ≤ C for any p ∈ [1 , + ∞ ) ;b) v n ( x ) ≥ − C for any x ∈ Σ .Proof. We apply the well-known Kato inequality to the operator ∆ g (see for instance [2, Theo-rem 5.1])(3.2) − ∆ g u − n ≥ ( − ∆ g u n ) χ { u n ≤ } = (cid:16) ˜ K n e u n − f n (cid:17) χ { u n ≤ } ≥ − Ch ( x ) , where(3.3) h ( x ) = 1 + X j ≥ ℓ +1 α j < d ( x, p j ) α j . Observe that h ∈ L q (Σ) for q ∈ [1 , δ ) if δ > is sufficiently small.Since the Radon measures µ n = − ∆ g u − n ≥ − Ch ( x ) are given as a divergence (in the sense ofdistributions), then ´ Σ d µ n = 0 . From that we conclude that ´ Σ d | µ n | is bounded. We use theGreen’s representation for v n and Hölder inequality to obtain: | v n ( x ) | p ≤ (cid:18) ˆ Σ d | µ n | (cid:19) p − ˆ Σ | G ( x, y ) | p d | µ n ( y ) | . We now integrate in x and make use of Fubini Theorem, taking into account that the Greenfunction of ∆ g in Σ belongs to L p : k v n k pL p ≤ (cid:18) ˆ Σ d | µ n | (cid:19) p − ˆ Σ ˆ Σ | G ( x, y ) | p d | µ n ( y ) | dx = (cid:18) ˆ Σ d | µ n | (cid:19) p − ˆ Σ ˆ Σ | G ( x, y ) | p dx d | µ n ( y ) | ≤ C (cid:18) ˆ Σ d | µ n | (cid:19) p . This concludes the proof of a).For the proof of b), we write the Green function of ∆ g in Σ as G ( x, y ) = − π log( rd ( x, y )) +˜ H ( x, y ) , where ˜ H : Σ × Σ → R is a bounded function. Here we have chosen r ∈ (0 , diam (Σ) − ) .Then, HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 11 v n ( x ) = ˆ Σ G ( x, y ) d µ n ( y ) = − π ˆ Σ log( rd ( x, y )) d µ + n ( y ) − π ˆ Σ log( rd ( x, y )) d µ − n ( y ) + ˆ Σ ˜ H ( x, y ) d µ n ( y ) . By the choice of r > , − π ˆ Σ log( rd ( x, y )) d µ + n ( y ) ≥ . Moreover, by (3.2), − π ˆ Σ log( rd ( x, y )) d µ − n ( y ) ≥ − C π ˆ Σ log( rd ( x, y )) h ( y )) dy ≥ − C, and finally (cid:12)(cid:12)(cid:12)(cid:12) ˆ Σ ˜ H ( x, y ) d µ n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ˜ H k L ∞ ˆ Σ d | µ n | ≤ C. (cid:3) As a first consequence of Lemma 3.3, we present an integral estimate in domains entirelycontained in the positive or negative region. This result is an extension of the Chen-Li integralestimate for positive solutions, see [23]. In our case u n may change sign, but we can performthe estimate thanks to Lemma 3.3. Lemma 3.4.
Under the conditions of Theorem 2.1, for every open subdomain Σ completely containedin Σ + or Σ − , there exists C > so that (cid:12)(cid:12)(cid:12)(cid:12) ˆ Σ ˜ K n e u n dV g (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. Proof.
Take Σ a smooth domain such that Σ ⊂ Σ ⊂ Σ ⊂ Σ ± . Let ϕ be the first eigenfunctionof the Laplace operator in Σ , that is, ( − ∆ g ϕ = λ ϕ in Σ ,ϕ = 0 on ∂ Σ . Next, we multiply (2.6) by ϕ , and integrate by parts over Σ to obtain(3.4) ˆ Σ ˜ K n ϕ e u n = − ˆ Σ u n ∆ g ( ϕ ) + O (1) . Let us denote f = ∆ g ( ϕ ) = 2( |∇ ϕ | − λ ϕ ) . Observe that ´ Σ f = 0 . Then ˆ Σ u − n f = ˆ Σ (cid:18) u − n − − ˆ Σ u − n (cid:19) f, so that, by Lemma 3.3, a),(3.5) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Σ u − n f (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13) u − n − − ˆ Σ u − n (cid:13)(cid:13)(cid:13)(cid:13) L (Σ ) k f k L ∞ (Σ ) ≤ C. On the other hand, for any γ > , ˆ Σ u + n | f | ≤ C ˆ Σ u + n ≤ C ˆ Σ u + n | ϕ ˜ K n | γ | ϕ ˜ K n | γ . By Young inequality we obtain(3.6) ˆ Σ u + n | f | ≤ ε ˆ Σ | u + n | γ ϕ | ˜ K n | + C ε ˆ Σ | ϕ ˜ K n | γ − γ . We can take γ > sufficiently small so that the second integral term in the right hand side isfinite (recall that, by Hopf lemma, ϕ ∼ d ( x, ∂ Σ ) near the boundary). Then, by (3.4), (3.5) and(3.6) ˆ Σ | ˜ K n | ϕ e u n ≤ C + ε ˆ Σ | u + n | γ ϕ | ˜ K n | . We now use the inequality ( t + ) γ ≤ C + e t to conclude that ˆ Σ | ˜ K n | ϕ e u n ≤ C, finishing the proof. (cid:3) In order to prove Proposition 3.1, we will need the following result, which is based on amean value inequality for subharmonic functions.
Lemma 3.5.
Let w be a function defined in Σ ⊂ Σ , x ∈ Σ , and assume that − ∆ g w ( x ) ≤ A for all x ∈ Σ , for some positive value A > . Take R > such that R < min (cid:26) d ( x , ∂ Σ ) , diam (Σ ) (cid:27) . Then there exists
C > depending only on Σ and A such that sup x ∈ B x ( R/ w ( x ) ≤ C − ˆ B x ( R ) w ! . Proof.
Define v as the solution of the problem (cid:26) − ∆ g v = − A, in Σ , v = 0 , on ∂ Σ .Clearly v is smooth and w + v is a subharmonic function. We now apply the mean valueinequality for subharmonic functions (see [40, Theorem 2.1] for its version on manifolds) toconclude. (cid:3) HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 13
Proof of Proposition 3.1.
Take Σ ⊂ Σ ⊂ Σ − , x ∈ Σ and fix r > sufficiently small. We applyLemma 3.5 to w = u + and we obtain sup B x ( r ) u + n ( x ) ≤ C + C ˆ B x (4 r ) u + n = C + C ˆ B x (4 r ) u + n p ( − ˜ K n ) /p ( x )( − ˜ K n ) /p ( x ) ≤ C + C ˆ B x (4 r ) e unp ( − ˜ K n ) /p ( x )( − ˜ K n ) /p ( x ) ≤ C + C ˆ B x (4 r ) − ˜ K n ( x ) e u n ! /p ˆ B x (4 r ) − ˜ K n ) p − ( x ) ! p − p . It suffices to choose a large enough p and use Lemma 3.4 to conclude that sup B x ( R ) u + n ( x ) < C . (cid:3) We now turn our attention to Proposition 3.2. The proof follows the argument of [23], withthe main difference that our solutions u n are not positive. This difficulty can be bypassedthanks to the following lemma, whose proof is based on Lemma 3.3. Lemma 3.6.
Under the hypotheses of Theorem 2.1, and given δ > , there exists C > such that (3.7) u n ( x ) − u n ( x ) ≤ C for every n ∈ N , x ∈ Σ − \ Γ δ , x ∈ Σ . Moreover, for any r > , there exists C > such that (3.8) |∇ u n ( x ) | ≤ C ∀ x ∈ Σ − \ (Γ δ ∪ m [ i = ℓ +1 B p i ( r )) . Proof.
By Lemma 3.3, b), we have that(3.9) u n ( x ) − − ˆ Σ u − n ≥ u − n ( x ) − − ˆ Σ u − n ≥ C. Taking into account Lemma 3.5, we have that for any fixed r ∈ (0 , δ ) , u n ( x ) − − ˆ Σ u − n ≤ C − ˆ B x ( r ) (cid:18) u n ( x ) − − ˆ Σ u − n (cid:19)! Moreover, by Proposition 3.1, u n ( x ) ≤ u − n ( x ) + C for all x ∈ B x ( r ) . Making use of Lemma3.3, a), we conclude(3.10) u n ( x ) − − ˆ Σ u − n ≤ C − ˆ B x ( r ) (cid:12)(cid:12)(cid:12)(cid:12) u − n ( x ) − − ˆ Σ u − n (cid:12)(cid:12)(cid:12)(cid:12)! ≤ C This estimate, together with (3.9), shows that (3.7) holds true.We now turn our attention to the proof of (3.8). Given r > , take any p > and fix x suchthat B x ( r ) ⊂ Σ − \ (Γ δ ∪ S mi = ℓ +1 B p i ( r )) . Recall the inequality (see [35, Theorem 9.11]) (cid:13)(cid:13)(cid:13)(cid:13) u n − − ˆ Σ u − n (cid:13)(cid:13)(cid:13)(cid:13) W ,p ( B x ( r )) ≤ C || ˜ K n e u n − f n || L p ( B x ( r )) + (cid:13)(cid:13)(cid:13)(cid:13) u n − − ˆ Σ u − n (cid:13)(cid:13)(cid:13)(cid:13) L p ( B x ( r )) ! . Combining (3.10) and Lemma 3.3, b), u n − − ´ Σ u − n is uniformly bounded in L ∞ ( B x ( r )) , whereasProposition 3.1 implies that ˜ K n e u n − f n is uniformly bounded on B x ( r ) . Therefore, (cid:13)(cid:13)(cid:13)(cid:13) u n − − ˆ Σ u − n (cid:13)(cid:13)(cid:13)(cid:13) W ,p ( B x ( r )) ≤ C. In particular (3.8) holds. (cid:3)
Proof of Proposition 3.2.
Since the proof is of local nature, we first pass to a problem in a planardomain. Given a point p ∈ Γ , we take a small neighborhood U of p and an isothermal coordi-nate system y = ( y , y ) centered at p such that the metric g takes the form g = e ϕ ( dy + dy ) in Ω ′ ⊂ R , where ϕ is smooth and ϕ (0) = 0 . Consequently, for a subdomain Ω ⊂ Ω ′ , u n satisfies − ∆ u n = e ϕ ( y ) ˜ K n ( y ) e u n − e ϕ ( y ) f n ( y ) , in Ω where ∆ is the usual laplacian.Let us define u ,n as the unique solution for the problem(3.11) (cid:26) ∆ u ,n = e ϕ f n , in Ω , u ,n = 0 , on ∂ Ω .If we now write the equation in the new variable u n − u ,n , which will be denoted again by u n , we obtain:(3.12) − ∆ u n = W n ( y ) e u n in Ω ,where W n ( y ) = e ϕ ( y ) e u ,n ( y ) ˜ K n ( y ) .Moreover, W n → W in C ,α (Ω) . Assumption (H1) is translated to W in the form W is a C ,α (Ω) function, changes sign and ∇ W ( x ) = 0 in Γ = { x ∈ Ω : W ( x ) = 0 } . Our proof is based on the method of moving planes, which allows us to compare the valuesof u n close to Γ . For the sake of clarity, we drop the subindex n in the notation of the rest of thisproof.By the assumptions on W for small δ > , there exists β > s.t.(3.13) |∇ W ( y ) | ≥ β for any y with | W ( y ) | ≤ δ. For a given point x ∈ Γ , take a ball B ⊂ { y ∈ Ω : W ( y ) > } tangent to Γ at x . Applyingthe Kelvin transform which leaves ∂B invariant and taking a neighborhood of x , we obtain aset (renamed as Γ ) which is strictly convex (recall that Γ is a C ,α curve).Next, through a translation and a rotation we define the new system as x = ( x , x ) and x = γ ( x ) which corresponds to the curve Γ . Let Ω ε = { x < γ ( x ) + ε } ∩ { x > − ε } and ∂ l Ω ε = { x : x − γ ( x ) = ε } . In summary, it is possible to choose a conformal map such that,for some ε > small, the following hold (see figure):(i) x becomes the origin;(ii) Ω ε is located to the left of the line x = ε and it is tangent to it;(iii) ∂ l Ω ε is uniformly convex; HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 15 (iv) ∂W∂x ≤ − β , for every x ∈ Ω ε ;(v) Ω ε ∩ { p , . . . , p m } = ∅ . Γ ∂ l Ω ε x = − ε x = ε x x x Ω ε is the whole shaded region Ω ε ∩ { W < } Ω ε ∩ { W > } Let m = min x ∈ ∂ l Ω ε u ( x ) and M = max x ∈ ∂ l Ω ε u ( x ) . We define ˜ u ∈ C (Ω ε ) such that:a) ˜ u = u in ∂ l Ω ε ;b) m ≤ ˜ u ≤ M ;c) |∇ ˜ u | ≤ C in Ω ε .Observe that c) is possible by (3.8).Let w be the harmonic function(3.14) (cid:26) ∆ w = 0 , in Ω ε , w = ˜ u, on ∂ Ω ε .Due to (3.7), the oscillation of u on ∂ l Ω ε is bounded, i.e.,(3.15) M − m = max ∂ l Ω ε u − min ∂ l Ω ε u ≤ C. Consequently, the oscillation of w is also bounded in Ω ε . We also define a new auxiliary func-tion v as(3.16) v ( x ) = u ( x ) − w ( x ) + C ( ε + γ ( x ) − x ) , for some C > to be determined. It is clear that the function v verifies(3.17) ∆ v + f ( x, v ( x )) − C γ ′′ ( x ) = 0 , in Ω ε ,with f ( x, v ( x )) = W ( x ) e v ( x )+ w ( x ) − C ( ε + γ ( x ) − x ) . We claim that for a suitable C (3.18) v ( x ) ≥ in Ω ε and v ( x ) = 0 on ∂ l Ω ε . The boundary condition is direct. In order to prove the first part, we distinguish two cases: • Case 1 : ε < x − γ ( x ) ≤ ε Taking into account (v), by (3.8) we have that (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C and (cid:12)(cid:12)(cid:12)(cid:12) ∂w∂x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. Consequently,(3.19) ∂v∂x = ∂u∂x − ∂w∂x − C ≤ C − C . It suffices to choose C sufficiently large to obtain that ∂v∂x is negative. Since v = 0 on ∂ l Ω ε , it is clear that (3.18) holds. • Case 2 : x − γ ( x ) ≤ ε and x ≥ − ε By (3.7), we have that v ( x ) = u ( x ) − w ( x ) + C ( ε + γ ( x ) − x ) ≥ min Ω ε u − max ∂ l Ω ε u + C ε ≥ − C + C ε . So, choosing C sufficiently large, (3.18) holds.Now we are ready to apply the method of moving planes to v in the x direction. Thus, westart from x = ε and move the line perpendicular to x − axis towards the left. Namely, let T λ = (cid:8) x ∈ R : x ≥ λ (cid:9) the half-plane, M λ = (cid:8) x ∈ R : x = λ (cid:9) its boundary and x λ = (2 λ − x , x ) the reflection point of x with respect to the line M λ . Our goal is to prove that(3.20) v ( x λ ) ≥ v ( x ) , for every x ∈ T λ ∩ Ω ε for λ ∈ (cid:2) ε − ε , ε (cid:3) , with some ε ∈ (0 , ε ) to be determined. By (3.18) and(3.19), (3.20) holds for λ ∈ (cid:0) ε, ε (cid:3) .By a standard argument (see [34]), we see that the moving planes argument can be carriedon provided that(3.21) f ( x, v ) ≤ f ( x λ , v ) for every x ∈ Ω ε with ε > λ > ε − ε . It is easy to check that (3.21) holds if(3.22) ∂f ( x, v ( x )) ∂x = e u (cid:18) ∂W∂x + W (cid:18) ∂w∂x + C (cid:19)(cid:19) ≤ , if x ∈ Ω ε , x > − ε . In other words, if f is monotone decreasing along the direction (1 , , near x .If W ( x ) ≤ , it is enough to choose C > − ∂w∂x to verify ∂W∂x + W (cid:18) ∂w∂x + C (cid:19) ≤ W (cid:18) ∂w∂x + C (cid:19) ≤ . HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 17
In the case that W ( x ) > by the assumptions on W , for every ε > there exists a neighbor-hood V ε of Γ such that W ( x ) ≤ ε .Since ∂w∂x + C is bounded from above, then ∂W∂x + W (cid:18) ∂w∂x + C (cid:19) ≤ − β ε (cid:18) ∂w∂x + C (cid:19) ≤ − β ε C, therefore we can take ε small enough to obtain the desired conclusion. We choose ε < ε .In this way, the method of moving planes works up to λ = ε − ε . Therefore, (3.20) impliesthat v ( x ) is monotone decreasing in the (1 , − direction. In fact, we can repeat the previousargument rotating the x − axis by a small angle.Thus, there exist ε > and a fixed cone ∆ such that for any x ∈ B x ( ε ) we have v ( y ) ≥ v ( x ) , ∀ y ∈ ∆ x , and(3.23) ∅ 6 = ∆ x ∩ Σ + ⊂ Ω ε where ∆ x denotes a translation of the cone ∆ with x at its vertex. By (3.16), we can transformthe previous inequality into(3.24) u ( y ) + C ( ε ) ≥ u ( x ) ∀ y ∈ ∆ x . Moreover, there exists η > , such that for any x ∈ B x ( ε ) the intersection of the cone ∆ x withthe set Σ + \ Γ η has a positive measure, and the lower bound of the measure depends only on η and the C norm of K . Namely, setting Σ x = ∆ x ∩ (Σ + \ Γ η ) we have that for any x ∈ B x ( ε ) (3.25) | Σ x | ≥ η > . Thanks to this, the proof can be concluded combining (3.24) and the integral bound of Lemma3.4. Indeed by virtue of (3.23) and property (v)(3.26) min Σ x ˜ K ≥ η > , hence e u ( x ) = − ˆ Σ x e u ( x ) dy (3.24) ≤ C − ˆ Σ x e u ( y ) dy (3.26) ≤ C − ˆ Σ x ˜ Ke u ( y ) dy Lemma 3.4 ≤ C | Σ x | (3.25) ≤ C. (cid:3) Proof of Theorem 2.1.
Take ε > δ > and the open set Σ = Σ + \ Γ δ , where ε is given byProposition 3.2. By Propositions 3.1 and 3.2, u n is uniformly bounded from above in Σ \ Σ .Moreover, by Lemma 3.4, ´ Σ ˜ K n e u n is bounded. By [9] there are two possibilities: Case 1 : u n is bounded from above in Σ . Therefore, ˜ K n e u n − f n ∈ L p (Σ \ Σ ) for some p > .Elliptic regularity estimates imply that u n − − ´ Σ u n ∈ W ,p (Σ) , so u n − − ´ Σ u n ∈ L ∞ (Σ) . If − ´ Σ u n is bounded, we obtain (1); if, on the contrary, a subsequence of − ´ Σ u n diverges negatively, weobtain (2). Case 2 : The sequence u n is not bounded from above. Applying the results of [9] concern-ing the blow–up analysis for (2.6) in Σ , we can assume that there exists a finite blow-up set S = { q , . . . , q r } ⊂ Σ . Moreover, by enlarging δ if necessary, we can assume that u n → −∞ uniformly in ∂ Σ , and ˜ K n e u n ⇀ r X i =1 β ( q i ) δ q i in the sense of weak convergence of measures in Σ , with β ( q i ) ≥ π .Now, let us define v the solution of the problem ( − ∆ g v = C h in Σ \ Σ ,v = 0 on ∂ Σ , where h is defined in (3.3) and C is a positive constant such that C h is an uniform upperbound of the term ˜ K n e u n − f n in Σ \ Σ . Since h ∈ L p (Σ \ Σ ) for some p > , then v ∈ L ∞ (Σ \ Σ ) by standard regularity results. By the maximum principle, for any C > there exists n ∈ N such that u n ≤ v − C in Σ \ Σ for n ≥ n . This implies that u n → −∞ uniformly in Σ \ Σ ; inparticular,(3.27) ˜ K n e u n ⇀ r X i =1 β ( q i ) δ q i in the sense of weak convergence of measures in Σ . It is worth to point out that, at this point of the proof, we can not apply yet the quantizationpart of the concentration-compactness Theorem of [9] unless we check the bounded oscillationcondition on ∂ Σ .By (3.27), employing the Green’s representation formula for u n , we have that u n − u n → r X i =1 β ( q j ) G ( x, q j ) + h m , uniformly on compact sets of Σ \ ( S ∪ { p , . . . , p ℓ } ) , where h m is defined in (1.5). Therefore,the sequence u n − u n admits uniformly bounded oscillation on any compact set of Σ \ ( S ∪{ p , . . . , p ℓ } ) . Indeed, there exists a constant C > such that max ∂ Σ u n − min ∂ Σ u n < C. By virtue of this condition, we can apply the quantization result of [9] to conclude that, upto subsequence, lim k → + ∞ ˆ Σ ˜ K n e u n ∈ Λ . (cid:3)
4. T
OPOLOGICAL DESCRIPTION OF THE ENERGY SUBLEVELS AND M ORSE INEQUALITIES
In this section we study the topology of the energy sublevels of I λ . We shall observe a changein the topology of the sublevels between high and low ones. This fact will be decisive to provethe existence and multiplicity theorems. HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 19
First, we define a continuous projection Ψ from low sublevels of I λ onto a compact topolog-ical space, whose character is inherited from the geometry of the function K . By using ideasfrom [45] we give a more accurate description for k = 1 depending on the order of the conicalpoints.Next, we define the reverse map ϕ from the corresponding space onto I − Lλ = { u ∈ X : I λ ( u ) ≤ − L } with L > large enough such that the composition Ψ ◦ ϕ is homotopically equiv-alent to the identity map.Finally, we adapt the well-known Morse inequalities for I λ which will be crucial to prove themultiplicity theorems.Since this scheme has been used several times, see [27, 29, 31, 45], we will be sketchy andfocus on the main differences concerning the sign-changing case.4.1. Topological description of the low sublevels of I λ . The first result allows us to project continuously functions u with a low energy level onto theset of formal barycenters on a union of bouquets and a simplex contained in Σ + . Proposition 4.1.
Let λ ∈ (8 kπ, π ( k + 1)) , k ∈ N , and assume (H1), (H2). Then for L > sufficientlylarge there exists a continuous projection Ψ : I − Lλ −→ Bar k ( Z N,M ) , where (4.1) Z N,M = N a i =1 B g i ∐ Y M ⊂ Σ + \ { p , · · · , p ℓ } and Y M = { y , . . . , y M } , where B g i ⊂ A i is a bouquet of g i circles, with g i defined in (2.8) , and y h ∈ C h .Moreover, if ˜ K + e un ´ Σ ˜ K + e un dV g ⇀ σ , for some σ ∈ Bar k ( Z N,M ) , then Ψ( u n ) → σ . Remark 4.2.
Under assumption (H3), the topological set
Bar k ( Z N,M ) is not contractible. In case N = 0 , Bar k ( Z N,M ) is the ( k − -skeleton of ( M − -symplex, which is not contractible if k < M (see Exercise 16 in Section 2.2 of [36]).Proof of Proposition 4.1. This lemma is proved in the spirit of [31], but following closely the ap-proach of [11].
Claim: If I λ ( u n ) → −∞ , up to a subsequence, σ n := ˜ K + e u n ´ Σ ˜ K + e u n dV g ⇀ σ ∈ Bar k (Σ + ) . Suppose by contradiction that there exist k + 1 points x , . . . , x k +1 ⊂ supp ( σ ) . Take r > such that B x i (2 r ) ∩ B x j (2 r ) = ∅ for i = j . Therefore, there exists ε > such that σ ( B x i (2 r )) > ε .As a consequence, σ n ( B x i ( r )) ≥ ε ; this implies that for any ˜ ε > :(4.2) log ˆ Σ ˜ K + e u n − | Σ | ˆ Σ u n ≤ k + 1) π − ˜ ε ) ˆ Σ |∇ u n | dV g + C, where C is a positive constant which depends on ε, r, ˜ ε . This kind of improvements of theMoser-Trudinger inequality were first obtained by Chen and Li in [22]. We also refer to [45, Proposition 2.2 and Proposition 2.3], where a formulation of this inequality for nonnegativefunctions was given in the case k = 2 (the case k > is analogous).Taking ˜ ε sufficiently small, (4.2) violates the hypothesis that I λ ( u n ) diverges negatively,yielding a contradiction.By the claim, given a neighborhood V of Bar k (Σ + ) in the weak topology of measures, thereexists L > O large enough such that if L > L , then(4.3) ˜ K + e u ´ Σ ˜ K + e u dV g ∈ V, ∀ u ∈ I − Lλ . In the appendix of [11], it is proved that
Bar k (Σ + ) is a Euclidean Neighborhood Retract. Ob-serve that the weak topology of measures is metrizable on bounded sets, see [15, Theorem 3.28].By [14, Lemma E.1], there exists V a neighborhood of Bar k (Σ + ) in the weak topology of mea-sures, and a continuous retraction X : V → Bar k (Σ + ) . Next, by (4.3), we define ˜Ψ as ˜Ψ : I − Lλ −→ V X −→ Bar k (Σ + ) u ˜ K + e u ´ Σ ˜ K + e u dV g P ki =1 t i δ x i . Observe that we can retract continuously A i onto B g i and C h onto a single point y h ∈ C h .Consequently, we can define the retraction(4.4) r : Σ + −→ Z N,M . We are now in conditions to define the map Ψ as the composition of ˜Ψ with the function r ∗ : Bar k (Σ + ) −→ Bar k ( Z N,M ) , the pushforward induced by the function r , then Ψ : I − Lλ −→ Bar k ( Z N,M ) u P i s i δ x i , where the values s i are given by ˜Ψ . Since r is a retraction, if ˜ K + e un ´ Σ ˜ K + e un dV g ⇀ σ , for some σ ∈ Bar k ( Z N,M ) , then Ψ( u n ) → σ . (cid:3) On the other hand, for λ ∈ (8 kπ, π ( k +1)) , k ∈ N , and Z a compact subset of Σ + \{ p , . . . , p ℓ } we consider test functions concentrated in at most k points of Z with arbitrary low energy. For γ > small enough, we consider a smooth nondecreasing cut-off function χ γ : R + → R + suchthat(4.5) χ γ ( t ) = (cid:26) t for t ∈ [0 , γ ]2 γ for t ≥ γ .For µ > and σ = P ki =1 t i δ x i ∈ Bar k ( Z ) , we define φ µ,σ : Σ → R φ µ,σ ( x ) = log X t i (cid:18) µ µχ γ ( d ( x, x i ))) (cid:19) ,ϕ µ,σ ( x ) = φ µ,σ ( x ) − ˆ Σ φ µ,σ dV g . HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 21
By the results in [29, Lemma 4.11, Lemma 4.12], which hold independently of the genus ofthe surface, we have that
Lemma 4.3.
Let λ ∈ (8 kπ, k + 1) π ) , k ∈ N , and let Z be a compact subset of Σ + \ { p , . . . , p ℓ } .Then we can choose γ > such that: (i) given L > there exists a large µ ( L ) > satisfying that for µ ≥ µ ( L ) , ϕ µ,σ ∈ ¯ X , where ¯ X isdefined in (2.1) , and I λ ( ϕ µ,σ ) < − L for any σ ∈ Bar k ( Z ) ; (ii) for any σ ∈ Bar k ( Z ) ˜ K + e ϕ µ,σ ´ Σ ˜ K + e ϕ µ,σ dV g ⇀ σ as µ → + ∞ . The following two results allow us to deal with the case in which Σ + has only simply con-nected components and N + ≤ k ; however, it is restricted to λ ∈ (8 π, π ) . Proposition 4.4.
Let λ ∈ (8 π, π ) and assume (H1), (H2). Let (4.6) J λ,A i = J λ ∩ A i for i = 1 , . . . , N , J λ,C h = J λ ∩ C h for h = 1 , . . . , M , where J λ is defined in (2.7) , and let us assume that, up to reordering, M λ ∈ { , . . . , M } is such that J λ,C h = ∅ if h ∈ { , . . . , M λ } and J λ,C h = ∅ if h ∈ { M λ + 1 , . . . , M } .Then for L > sufficiently large there exists a continuous projection Ψ : I − Lλ → Bar ( W N,M,J λ ) , where (4.7) W N,M,J λ = N a i =1 B g i + | J λ,Ai | ∐ M λ a h =1 B | J λ,Ch | ∐ ˆ Y M λ ,B g i + | J λ,Ai | ⊂ A i , B | J λ,Ch | ⊂ C h are bouquets of g i + | J λ,A i | and | J λ,C h | circles respectively, with g i defined in (2.8) , for i = 1 , . . . , N and h = 1 , . . . , M λ , and ˆ Y M λ = ` Mh = M λ +1 { y h } with y h ∈ Y h for h = M λ + 1 , . . . , M .Moreover, if ˜ K + e u n ´ Σ ˜ K + e u n dV g ⇀ σ, for some σ ∈ Bar ( W N,M,J λ ) ,then Ψ( u n ) → σ . Remark 4.5.
Observe that W N,M,J λ is not contractible if and only if either N ≥ or M λ > , i.e. if(H3) holds, or if J λ = ∅ , namely (H4) holds.Proof. For
L > sufficiently large and r > , we apply some results of [29]; more specifically,by Propositions 4.4., 4.7., 4.8., and Remark 4.10, of [29], we construct the continuous projection β : I − Lλ → Σ + \ [ p i ∈ J λ B p i ( r ) , with the property that if ˜ K + e un ´ Σ ˜ K + e un dV g ⇀ δ x for some x ∈ Σ + \ [ p i ∈ J λ B p i ( r ) then β ( u n ) → x .Notice that I λ is bounded from below on the functions belonging to ˜Ψ − ( J λ ) .We can rewrite Σ + \ [ p i ∈ J λ B p i ( r ) as N a i =1 A ′ i ∐ M λ a h =1 C ′ h ∐ M a h = M λ +1 C h . where A ′ i = A i \ S p i ∈ J λ,Ai B p i ( r ) and C ′ h = C h \ S p i ∈ J λ,Ch B p i ( r ) .The sets A ′ i can be retracted to an inner bouquet B g i + | J λ,Ai | ⊂ A ′ i and C ′ h to B | J λ,Ch | ⊂ C ′ h , ina similar way to the proof of Proposition 4.1, we can define a retraction r : Σ + \ [ p i ∈ J λ B p i ( r ) −→ W N,M,J λ . Finally, we can define Ψ as the composition of β with the pushforward r ∗ : Bar (Σ + \ S p i ∈ J λ B p i ( r )) −→ Bar ( W N,M,J λ ) , then Ψ : I − Lλ −→ Bar ( W N,M,J λ ) u δ x . Since r is a retraction, the second part of the proposition is proved. (cid:3) Next, for λ ∈ (8 π, π ) , we introduce appropriate test functions that will allow to map acompact subset W of Σ + \ J λ into low sublevels of I λ .Let ˜ α = max n ≤ ℓ | p n / ∈ J λ α n or ˜ α = 0 if J λ = { p , . . . , p ℓ } or ℓ = 0 . For any α ∈ ( ˜ α, λ π − , µ > and p ∈ W , we define φ µ,p,α : Σ → R , φ µ,p,α ( x ) = 2 log (cid:18) µ α µχ γ ( d ( x, p ))) α ) (cid:19) ,ϕ µ,p,α ( x ) = φ µ,p,α ( x ) − ˆ Σ φ µ,p,α dV g . where χ γ is defined in (4.5).Since W is a compact subset of Σ + \ J λ the results in [29, Lemma 4.13, Lemma 4.14] holds,namely Lemma 4.6.
Let λ ∈ (8 π, π ) and let W be a compact subset of Σ + \ J λ . Then we can choose γ > such that: (i) given any L > , there exists a large µ ( L ) > satisfying that, for any µ ≥ µ ( L ) , ϕ µ,p,α ∈ ¯ X and I λ ( ϕ µ,p,α ) < − L for any p ∈ W ; (ii) for any p ∈ W , ˜ K + e ϕ µ,p,α ´ Σ ˜ K + e ϕ µ,p,α dV g ⇀ δ p as µ → + ∞ . HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 23
Topological characterization of the high sublevels of I λ . The compactness result Theorem 2.1, combined with the deformation Lemma [43, Proposi-tion 2.3], allows us to prove the next alternative bypassing the Palais-Smale condition, whichis not known for the functional I λ . Lemma 4.7.
Let λ / ∈ Λ and assume (H1), (H2). If I λ has no critical levels inside some interval [ a, b ] ,then I aλ is a deformation retract of I bλ . Remark 4.8.
Actually the deformation lemma in [43] is originally proved for the regular case and for K positive, but it adapts in a straightforward way to the singular one, even for K sign-changing.Indeed, in the proof of Proposition 2.3 of [43] a certain deformation is used following a flow in the domainof the functional, and I λ decreases along that flow. In our case I λ is not defined in the whole Sobolevspace but on X , but I λ ( u ) → + ∞ as u approaches the boundary of ¯ X , so that ¯ X is positively invariantunder this flow. Hence Proposition 2.3 of [43] is applicable and gives Lemma 4.7. In turn, since Theorem 2.1 implies that the functional I λ stays uniformly bounded on thesolutions of ( ∗ ) λ , the above Lemma can be used to show that it is possible to retract the wholespace ¯ X onto a high sublevel I bλ (see [44, Corollary 2.8], also for this issue minor changes arerequired). Lemma 4.9.
Let λ / ∈ Λ and assume (H1), (H2). If b > is sufficiently large, the sublevel I bλ is a retractof ¯ X and hence is contractible. Morse inequalities for I λ . The aim of this subsection is to prove a Morse-theoretical result for I λ , which will be crucialto get the multiplicity estimates of Theorem 2.5 and Theorem 2.6. Proposition 4.10.
Let ℓ ∈ { , . . . , m } and let us assume α , . . . , α ℓ > . If λ ∈ (8 π, + ∞ ) \ Λ , thenfor a generic choice of the function K , g (namely for ( K, g ) in an open and dense subset of K ℓ × M )there exists b = b ( K, g ) > such that • I bλ ≡ I bλ,K,g is a retract of ¯ X ≡ ¯ X K,g , • any solution u ∈ I bλ,K,g of ( ∗ ) λ is nondegenerate,where to emphasize the dependence on K and g we write ¯ X K,g = { u ∈ H g (Σ) : ˆ Σ u dV g = 0 , ˆ Σ Ke − h m e u dV g > } and I bλ,K,g = { u ∈ ¯ X K,g : 12 ˆ Σ |∇ u | dV g − λ log ˆ Σ Ke − h m e u dV g ≤ b } with h m defined in (1.5) .Proof. Let us fix ( ¯ K, ¯ g ) ∈ K ℓ × M .Next, we introduce the Banach space S of all C ,α symmetric matrices on Σ . The set M of all C ,α Riemannian metrics on Σ is an open subset of S .It is easy to verify that for small δ > , and any g ∈ G δ := { g ∈ S : k g k C ,α < δ } , ¯ g + g is aRiemannian metric and the sets H g + g (Σ) , L g + g (Σ) , L g + g (Σ) coincide respectively with H g (Σ) , L g (Σ) , L g (Σ) and the two norms are equivalent.Being ¯ K ∈ K ℓ , it satisfies (H1), (H2). Thus, it is not hard to see that for δ > small enough ¯ K + K satisfies (H1), (H2) for any K ∈ H δ := { h ∈ C ,α (Σ) : k h k C ,α (Σ) < δ } . Furthermore, by Theorem 2.1, it suffices to take a smaller δ > so that there exists R > such that for any ( K, g ) ∈ H δ × G δ all the critical points (with zero mean value) of I λ, ¯ K + K, ¯ g + g are contained in the ball B ( R ) ⊂ H g (Σ) .Taking a smaller δ > , if necessary, we have by Lemma 4.9 and Theorem 2.1 that there exists b > such that the sublevel I bλ, ¯ K + K, ¯ g + g is a retract of ¯ X ¯ K + K, ¯ g + g for any ( K, g ) ∈ H δ × G δ .Finally, for any u ∈ I bλ, ¯ K, ¯ g ´ Σ ¯ Ke − h m e u dV ¯ g ≥ e − bλ , so we can also assume that if u ∈ I bλ, ¯ K, ¯ g , then ´ Σ ( ¯ K + K ) e − h m e u dV ¯ g + g > for any ( K, g ) ∈ H δ × G δ .Once δ is fixed in this way it is possible to argue as in [28], where a transversality Theorem,obtained in [47], is applied to deduce that the following set is an open and dense subset of H δ × G δ ( ( K, g ) ∈ H δ × G δ : any u ∈ I bλ, ¯ K + K, ¯ g + g solution of the equation − ∆ ¯ g + g u = λ (cid:16) ( ¯ K + K ) e − hm e u ´ Σ ( ¯ K + K ) e − hm e u dV ¯ g + g − ´ Σ dV ¯ g + g (cid:17) is nondegenerate ) . Since this holds for any choice of ( ¯ K, ¯ g ) the thesis follows. (cid:3) As recalled in the previous subsection we do not know whether I λ satisfies the ( P S ) con-dition or not, thus Theorem 2.8 can not be directly applied. However, as already pointed outin [3], the ( P S ) -condition can be replaced by the request that appropriate deformation lemmashold for the functional.In particular a flow defined by Malchiodi in [44] allows to adapt to I λ the classical deforma-tion lemmas [19, Lemma 3.2 and Theorem 3.2] needed so that Theorem 2.8 can be applied for H = ¯ X and I = I λ . It is worth to point out that, even if the flow is defined for K positive,arguing as in Remark 4.8 it is not hard to check that the same construction applies also in thesign-changing case.In conclusion the following result holds true. Proposition 4.11.
Let ℓ ∈ { , . . . , m } and let us assume α , . . . , α ℓ > . If λ / ∈ Λ , a , b are regularvalues of I λ and all the critical points in { a ≤ I λ ≤ b } are nondegenerate, then { critical points of I λ in { a ≤ I λ ≤ b } } ≥ X q ≥ dim( H q ( I bλ , I aλ ; Z )) .
5. O
N THE HOMOLOGY GROUPS OF BARYCENTER SETS
In this section we compute the dimension of the homology groups of some spaces of formalbarycenters which have been introduced in the previous section.Keeping the notation of Proposition 4.1, we consider the space Z N,M = X N ∐ Y M , where X N = ∐ Ni =1 B g i and Y M = { y , . . . , y M } , with g i defined in (2.8). For k ∈ N , N, M ∈ N ∪ { } , with N + M ≥ , and q ∈ N ∪ { } we set(5.1) d q ( k, N, M ) = dim( ˜ H q ( Bar k ( Z N,M )); Z ) , with the convention that d q ( k, N, M ) = 0 if q < . HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 25
Proposition 5.1.
Let k ∈ N , N, M ∈ N ∪ { } , with N + M ≥ , and q ∈ N ∪ { } , thenif k + 1 − M ≤ N , d q ( k, N, M ) = (cid:16) N + M − N + M − p (cid:17) X a + . . . + a N = k − p + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − p (1 ≤ p ≤ k + 1) otherwise; if k + 1 − M ≥ N , d q ( k, N, M ) = (cid:16) N + M − N + M − p (cid:17) X a + . . . + a N = k − p + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − p (1 ≤ p ≤ N ) (cid:16) N + M − sM − s (cid:17) X a + . . . + a N = k − N − s + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − N − s (1 ≤ s ≤ M ) otherwise; where s a,g = (cid:0) a + g − g − (cid:1) and g i is defined in (2.8) .Moreover we adopt the following convention: if N = 0 X a + . . . + a N = ha i ≥ s a ,g . . . s a N ,g N = (cid:26) if h = 00 if h = 0 .Notice that if k + 1 − M = N the two formulas coincide.Proof.Step . The thesis holds true if k = 1 or N = 0 . If k = 1 , Bar ( Z N,M ) ∼ = Z N,M and so bydirect computation we have: d q (1 , N, M ) = N P i =1 g i (= N P i =1 s ,g i ) if q = 1 N + M − if q = 0 otherwise. If N = 0 , Bar k ( Z ,M ) is the ( k − -skeleton of a ( M − -symplex and so the followingformula holds d q ( k, , M ) = (cid:16) M − k (cid:17) if q = k − otherwise, where we adopt the convention that (cid:0) ab (cid:1) = 0 if a < b . Step . The thesis holds true if M = 0 for any k ≥ , N ≥ : that is, (5.2) d q ( k, N,
0) = (cid:16) N − N − p (cid:17) X a + . . . + a N = k − p + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − p (1 ≤ p ≤ min { k + 1 , N } ) otherwise. We will demonstrate (5.2) by induction on N , for any fixed k ≥ .If N = 0 , the formula holds by Step 1. Now, assume by induction that (5.2) holds true for acertain N and let us show its validity for N + 1 . Being X N +1 = X N ∐ B g N +1 , by Proposition 2.9, (2.11) and (2.13) we get d q ( k, N + 1 ,
0) = d q ( k, N,
0) + d q − ( k − , N,
0) + dim( ˜ H q ( Bar k ( B g N +1 ))) + dim( ˜ H q − ( Bar k − ( B g N +1 )))+ k − X ℓ =1 dim( ˜ H q ( Bar k − ℓ ( X N ) ∗ Bar ℓ ( B g N +1 ))) + k − X ℓ =2 dim( ˜ H q (Σ Bar k − ℓ ( X N ) ∗ Bar ℓ − ( B g N +1 ))) , (5.3)where the homology groups are intended with coefficient in Z . Let us compute all the termsin (5.3).The first two can be obtained using the inductive assumption. Next, again by the computationsin [3, Proposition 3.2], we know that(5.4) dim( ˜ H q ( Bar k ( B g N +1 ))) = (cid:26) s a N +1 ,g N +1 if q = 2 k − otherwise, and so(5.5) dim( ˜ H q − ( Bar k − ( B g N +1 ))) = (cid:26) s a N +1 ,g N +1 if q = 2 k − otherwise. Moreover, by (2.14), using (5.4) and the inductive assumption we have that k − X ℓ =1 dim( ˜ H q ( Bar k − ℓ ( X N ) ∗ Bar ℓ ( B g N +1 ))) == (cid:16) N − N − p (cid:17) X a + . . . + a N + ℓ = k − p + 1 a i ≥ , ℓ ≥ s a ,g . . . s a N ,g N s a N +1 ,ℓ if q = 2 k − p (1 ≤ p ≤ N ) otherwise, (5.6)and HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 27 k − X ℓ =2 dim( ˜ H q (Σ Bar k − ℓ ( X N ) ∗ Bar ℓ − ( B g N +1 ))) == (cid:16) N − N − p + 1 (cid:17) X a + . . . + a N + ℓ = k − p + 1 a i ≥ , ℓ ≥ s a ,g . . . s a N ,g N s a N +1 ,ℓ − if q = 2 k − p (2 ≤ p ≤ N + 1) otherwise. (5.7)In conclusion, combining (5.3), (5.4), (5.5), (5.6) and (5.7) we obtain that d q ( k, N +1 ,
0) = (cid:16) NN + 1 − p (cid:17) X a + . . . + a N +1 = k − p + 1 a i ≥ s a ,g . . . s a N +1 ,g N +1 if q = 2 k − p (1 ≤ p ≤ min { k + 1 , N + 1 } ) otherwise, so (5.2) holds true for N + 1 and this completes the proof of (5.2). Step . Conclusion.We will prove the formula by induction on M , with k ≥ and N ≥ fixed.If M = 0 the thesis is true by Step . Now, let us suppose that (5.2) holds for M and we provethat then it is also true for M + 1 .Being Z N,M +1 = Z N,M ∐ { y M +1 } , and ˜ H ∗ ( Bar k ( { y M +1 } )) = 0 , by (2.11) and (2.13) we get d q ( k, N, M + 1) = d q ( k, N, M ) + d q − ( k − , N, M ) . Hence by the inductive assumption we can compute d q ( k, N, M + 1) , obtaining thatif k + 1 − ( M + 1) ≤ Nd q ( k, N, M +1) = (cid:16) N + ( M + 1) − N + ( M + 1) − p (cid:17) X a + . . . + a N = k − p + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − p (1 ≤ p ≤ k + 1) otherwise; if k + 1 − ( M + 1) ≥ Nd q ( k, N, M +1) = (cid:16) N + ( M + 1) − N + ( M + 1) − p (cid:17) X a + . . . + a N = k − p + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − p (1 ≤ p ≤ N ) (cid:16) N + ( M + 1) − s ( M + 1) − s (cid:17) X a + . . . + a N = k − N − s + 1 a i ≥ s a ,g . . . s a N ,g N if q = 2 k − N − s (1 ≤ s ≤ M + 1) otherwise. So the formula holds for M + 1 and this concludes the proof. (cid:3) Lemma 5.2.
Let N , M ∈ N ∪ { } , N + M ≥ and let W N,M,J λ be the set defined in (4.7) , then dim( ˜ H q ( Bar ( W N,M,J λ ); Z )) = N + M − q = 0 P Ni =1 g i + | J λ | q = 10 otherwise.Proof. Being
Bar ( W N,M,J λ ) ∼ = W N,M,J λ it is immediate to see that dim( ˜ H q ( W N,M,J λ ; Z )) = N + M − q = 0 P Ni =1 ( g i + | J λ,A i | ) + P Mh =1 | J λ,C h | q = 10 otherwise,hence the thesis follows observing that N X i =1 | J λ,A i | + M X h =1 | J λ,C h | = | J λ | , where J λ,A i and J λ,C h are defined in (4.6). (cid:3)
6. C
ONCLUSION OF THE PROOFS OF THE MAIN RESULTS
In order to prove our main results the following two Propositions will be of use.
Proposition 6.1.
Let λ ∈ (8 kπ, k + 1) π ) , k ∈ N , and assume (H1), (H2).If b > is such that I bλ is contractible and Σ + ≃ Z N,M , where Z N,M is defined in (4.1) , then there exists
L > sufficiently large so that dim( H q +1 ( I bλ , I − Lλ ; Z )) ≥ dim( ˜ H q ( Bar k ( Z N,M ); Z )) for any q ≥ .Proof. By assumption I bλ is contractible thus, from the exactness of the homology sequence, . . . → ˜ H q +1 ( I − Lλ ; Z ) → ˜ H q +1 ( I bλ ; Z ) → H q +1 ( I bλ , I − Lλ ; Z ) → ˜ H q ( I − Lλ ; Z ) → . . . we derive that H q +1 ( I bλ , I − Lλ ; Z ) ∼ = ˜ H q ( I − Lλ ; Z ) , for any q ≥ ,H ( I bλ , I − Lλ ; Z ) = 0 . Let us consider the continuous projection Ψ introduced in Proposition 4.1 and the map j : Bar k ( Z N,M ) −→ I − Lλ σ = P ki =1 t i δ x i ϕ µ,σ , which is well defined by Lemma 4.3 (i) applied with Z = Z N,M .Then Ψ ◦ j is homotopically equivalent to the identity on Bar k ( Z N,M ) . This fact follows fromProposition 4.1 and Lemma 4.3 (ii) .Hence, Ψ ∗ ◦ j ∗ = Id | H ∗ ( Bar k ( Z N,M )) and so the desired conclusion follows by dim( ˜ H q ( I − Lλ ; Z )) ≥ dim( ˜ H q ( Bar k ( Z N,M ); Z )) . (cid:3) HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 29
Proposition 6.2.
Let λ ∈ (8 π, π ) and assume (H1), (H2).If b > is such that I bλ is contractible and Σ + ≃ Z N,M , where Z N,M is defined in (4.1) , then there exists
L > sufficiently large so that dim( H q +1 ( I bλ , I − Lλ ; Z )) ≥ dim( ˜ H q ( Bar ( W N,M,J λ ); Z )) for any q ≥ ,where W N,M,J λ is defined in (4.7) .Proof. The proof is completely analogous to the one of the previous proposition, where W N,M,J λ and ϕ µ,p,α play the role of Z N,M and ϕ µ,σ respectively, while Proposition 4.4 and Lemma 4.6must be applied instead of Proposition 4.1 and Lemma 4.3. (cid:3) Proof of Theorem 2.2.
By Lemma 4.9 there exists b > so that the sublevel I bλ is contractible, thenwe are in position to apply Proposition 6.1 and so for L > sufficiently large dim( H q +1 ( I bλ , I − Lλ ; Z )) ≥ dim( ˜ H q ( Bar k ( Z N,M ); Z )) for any q ≥ .Hence, by virtue of (H3), dim( H q +1 ( I bλ , I − Lλ ; Z )) > for some q , as it can be directly checkedapplying Theorem 5.1 and recalling that N + = N + M . Therefore, I Lλ is not a retract of I bλ , sothe conclusion follows from Lemma 4.7. (cid:3) Proof of Theorem 2.3.
We argue as in the proof of Theorem 2.2, indeed applying in this caseLemma 4.9 and Proposition 6.2 we have that for b and L sufficiently large positive dim( H ( I bλ , I − Lλ ; Z )) ≥ dim( H ( Bar ( W N,M,J λ ); Z )) . Lemma 5.2 combined with (H4) allows to see that dim( H ( I bλ , I − Lλ ; Z )) > , so we concludeagain by Lemma 4.7. (cid:3) Proof of Theorem 2.5.
By virtue of Proposition 4.10, we can fix ( K, g ) ∈ K ℓ × M , such that anysolution u ∈ I bλ of ( ∗ ) λ is nondegenerate. Next, combining Proposition 4.11 (with a = − L ) andProposition 6.1, we get that for L > sufficiently large { solutions to ( ∗ ) λ } ≥ X q ≥ { critical points in {− L ≤ I λ ≤ b } with index q }≥ X q ≥ dim( ˜ H q ( Bar k ( Z N,M ); Z )) . We conclude by Proposition 5.1. (cid:3)
Proof of Theorem 2.6.
The proof is completely analogous to the one of Theorem 2.5, where W N,M,J λ plays the role of Z N,M and Proposition 6.2 and Lemma 5.2 are applied in place of Proposition6.1 and Proposition 5.1. (cid:3)
Acknowledgements.
F. D. M. has been supported by PRIN
FYK7_ and Fondi Avvioalla Ricerca - Sapienza 2015, whereas R .L.-S. and D. R. have been supported by the Feder-Mineco Grant MTM2015-68210-P and by J. Andalucia (FQM116). During the preparation ofthis work R. L.-S. was hosted by University of Rome La Sapienza, and he wishes to thank thisinstitution for the kind hospitality and F .D.M. for the invitation. The authors are grateful to W.Chen, C. Li and S. Kallel for their suggestions and discussions concerning the subject. Finally, they want to express their gratitude to the referees for their careful reading and their valuablecomments on the manuscript. R
EFERENCES [1] M.O. Ahmedou, S. Kallel, C.B. Ndiaye, The resonant boundary Q-curvature problem and boundary-weightedbarycenters, preprint arXiv:1604.03745.[2] A. Ancona, Elliptic operators, conormal derivatives and positive parts of functions. With an appendix by HaïmBrezis. J. Funct. Anal. 257 (2009), no.7, 2124–2158.[3] D. Bartolucci, F. De Marchis, A. Malchiodi, Supercritical conformal metrics with conical singularities, Int.Math. Res. Not. IMRN (2011), no. 24, 5625–5643.[4] D. Bartolucci, C. S. Lin, Sharp existence results for mean field equations with singular data, J. DifferentialEquations 252 (2012), no. 7, 4115-4137.[5] D. Bartolucci, C. S. Lin, Uniqueness results for mean field equations with singular data, Comm. Partial Differ-ential Equations 34 (2009), no. 7-9, 676-702.[6] D. Bartolucci, C. S. Lin, G. Tarantello, Profile of blow-up solutions to mean field equations with singular data,Comm. Partial Differential Equations 29 (2004), no. 7-8, 1241-1265.[7] D. Bartolucci, A. Malchiodi, An improved geometric inequality via vanishing moments, with applications tosingular Liouville equations, Comm. Math. Phys. 322 (2013), no. 2, 415–452.[8] D. Bartolucci, E. Montefusco, Blow-up analysis, existence and qualitative properties of solutions for the two-dimensional Emden-Fowler equation with singular potential, Math. Methods Appl. Sci. 30 (2007), no. 18, 2309–2327.[9] D. Bartolucci, G. Tarantello, Liouville type equations with singular data and their applications to periodicmultivortices for the electroweak theory, Comm. Math. Phys. 229 (2002), no. 1, 3–47.[10] D. Bartolucci, G. Tarantello, Asymptotic blow-up analysis for singular Liouville type equations with applica-tions, J. Differential Equations 262 (2017), no. 7, 3887–3931.[11] L. Battaglia, A. Jevnikar, A. Malchiodi, D. Ruiz,
A general existence result for the Toda system on compact surfaces,
Adv. Math. 285 (2015), 937–979.[12] M. Berger, Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. DifferentialGeom., 5 (1971), no. 3-4, 325–332.[13] F. Borer, L. Galimberti, M. Struwe, "Large" conformal metrics of prescribed Gauss curvature on surfaces ofhigher genus, Comment. Math. Helv. 90 (2015), no. 2, 407-428.[14] G. Bredon,
Topology and geometry, volume 139 of Graduate Texts in Mathematics. Springer-Verlag, New York,1993.[15] H. Brezis,
Functional analysis, Sobolev spaces and partial differential equations,
Universitext. Springer, New York,(2011).[16] H. Brezis, F. Merle, Uniform estimates and blow-up behavior for solutions of − ∆ u = V ( x ) e u in two dimen-sions, Comm. Partial Differential Equations 16 (1991), no. 8-9, 1223–1253.[17] A. Carlotto, On the solvability of singular Liouville equations on compact surfaces of arbitrary genus, Trans.Amer. Math. Soc. 366 (2014), no. 3, 1237–1256.[18] A. Carlotto, A. Malchiodi, Weighted barycentric sets and singular Liouville equations on compact surfaces, J.Funct. Anal. 262 (2012), no. 2, 409–450.[19] K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, PNLDE 6, Birkhäuser, Boston,1993.[20] W. Chen, C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math. (2) 145 (1997), no.3, 547–564.[21] W. Chen, C. Li, A priori estimate for the Nirenberg problem, Discrete Contin. Dyn. Syst. Ser. S 1 (2008), no. 2,225–233.[22] W. Chen, C. Li, Prescribing Gaussian curvatures on surfaces with conical singularities, J. Geom. Anal. 1 (1991),no.4, 359–372.[23] W. Chen, C. Li, Moving planes, moving spheres, and a priori estimates, J. Differential Equations 195 (2003),no. 1, 1–13.[24] C. C. Chen, C. S. Lin, Mean field equation of Liouville type with singular data: topological degree, Comm.Pure Appl. Math. 68 (2015), no. 6, 887–947. HE SINGULAR MEAN FIELD PROBLEM WITH SIGN-CHANGING POTENTIALS 31 [25] C. C. Chen, C. S. Lin, Mean field equations of Liouville type with singular data: sharper estimates. DiscreteContin. Dyn. Syst. 28 (2010), no. 3, 1237-1272.[26] T. D’Aprile, F. De Marchis, I. Ianni, Prescribed Gauss curvature problem on singular surfaces, preprint.[27] F. De Marchis, Multiplicity result for a scalar field equation on compact surfaces, Comm. Partial DifferentialEquations 33 (2008), no. 10-12, 2208–2224.[28] F. De Marchis, Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal 259 (2010),no. 8, 2165–2192.[29] F. De Marchis, R. López-Soriano, Existence and non existence results for the singular Nirenberg problem, Calc.Var. Partial Differential Equations, 55 (2016), no. 2, paper no. 36, 35 pp.[30] M. Del Pino, C. Román, Large conformal metrics with prescribed sign-changing Gauss curvature, Calc. Var.Partial Differential Equations 54 (2015), no. 1, 763–789.[31] Z. Djadli, A. Malchiodi, Existence of conformal metrics with constant Q -curvature, Ann. of Math. 168 (2008),no. 3, 813–858.[32] Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genuses, Commun. Contemp.Math. 10 (2008), no. 2, 205-220.[33] G. Dunne, Self-dual Chern-Simons Theories, Lecture Notes in Physics (1995).[34] B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and Related Properties via the Maximum Principle, Comm. Math.Phys. 68 (1979), no. 3, 209-243.[35] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathema-tischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, (1977).[36] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.[37] J.L. Kazdan, F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. (2) 99 (1974), 14–47.[38] T. J. Kuo, C. S. Lin, Estimates of the mean field equations with integer singular sources: non-simple blowup, J.Differential Geom. 103 (2016), no. 3, 377-424.[39] C.H. Lai, Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions, World Scientific Singa-pore, 1981.[40] P. Li, R. Schoen, L p and mean value properties of subharmonic functions on Riemannian manifolds, ActaMath. 153 (1984), no. 3-4, 279-301.[41] Y.Y. Li, I. Shafrir, Blow-up analysis for solutions of − ∆ u = V e u in dimension two, Indiana Univ. Math. J. 43(1994), no. 4, 1255-1270.[42] R. López-Soriano, D. Ruiz, Prescribing the Gaussian curvature in a subdomain of S with Neumann boundarycondition, J. Geom. Anal. 26 (2016), no. 1, 630-644.[43] M. Lucia, A mountain pass theorem without Palais-Smale condition, C. R. Math. Acad. Sci. Paris 341 (2005),no. 5, 287–291.[44] A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations 13(2008), no.11-12, 1109–1129.[45] A. Malchiodi, D. Ruiz, New improved Moser-Trudinger inequalities and singular Liouville equations on com-pact surfaces, Geom. Funct. Anal. 21 (2011), no. 5, 1196–1217.[46] G. Mondello, D. Panov, Spherical metrics with conical singularities on a 2-sphere: angle constraints, Int. Math.Res. Not. IMRN 2016, no. 16, 4937–4995.[47] J.C. Saut, R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial DifferentialEquations 4 (1979), no. 3, 293-319.[48] M. Struwe, On the evolution of harmonic mappings of Riemmanian surfaces, Comment. Math. Helv. 60 (1985),no.4, 558–581.[49] G. Tarantello, Self-Dual Gauge Field Vortices: An Analytical Approach, PNLDE 72, Birkhäuser Boston, Inc.,Boston, MA, 2007.[50] G. Tarantello, Analytical aspects of Liouville-type equations with singular sources, Stationary partial differ-ential equations. Vol I, 491–592, Handb. Differ. Equ., North-Holland, Amsterdam, (2004)[51] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc.324 (1991), no. 2, 793–821.[52] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer-Verlag, 2001.F RANCESCA D E M ARCHIS , D
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