Variational aspects of singular Liouville systems

Transcription

1 S.I.S.S.A. - Scuola Internazionale Superiore di Studi Avanzati Area of Mathematics Ph.D. Thesis in Mathematical Analysis Variational aspects of singular Liouville systems Supervisor: Prof. Andrea Malchiodi Candidate: Luca Battaglia Academic Year 04 5

2 Il presente lavoro costituisce la tesi presentata da Luca Battaglia, sotto la direzione del Prof. Andrea Malchiodi, al fine di ottenere l attestato di ricerca post-universitaria Doctor Philosophiae presso la SISSA, Curriculum in Analisi Matematica, Area di Matematica. Ai sensi dell art., comma 4, dello statuto della SISSA pubblicato sulla G.U. no. 36 del 3.0.0, il predetto attestato è equipollente al titolo di Dottore di ricerca in Matematica. Trieste, Anno Accademico

3 Contents Introduction Preliminaries 8. Notation Compactness results Analytical preliminaries and Moser-Trudinger inequalities Topological preliminaries, homology and Morse theory New Moser-Trudinger inequalities and minimizing solutions. Concentration-compactness theorem Pohožaev identity and quantization for the Toda system Proof of Theorem Proof of Theorem Existence and multiplicity of min-max solutions Topology of the space X Topology of the space X Test functions Macroscopic improved Moser-Trudinger inequalities Scale-invariant improved Moser-Trudinger inequalities Proof of Theorems 3., 3.3, 3.5, Non-existence of solutions Proof of Theorems 4., Proof of Theorem A Appendix 0 A. Proof of Theorem A. Proof of Theorem

4 Introduction This thesis is concerned with the study of singular Liouville systems on closed surfaces, that is systems of second-order elliptic partial differential equations with exponential nonlinearities, which arise in many problems in both physics and geometry. Such problems are attacked by a variational point of view, namely we consider solutions as critical points for a suitable energy functional defined on a suitable space. We will first discuss the existence of points of minima for the energy functional, which solve the problem. Then, in the cases when the energy cannot have global minimum points, we will look for critical points of other kind, the so-called min-max points. Finally, we will also give some nonexistence results for such problems. Let, g be a compact surface without boundary. We will consider the following system of PDEs: u i = a ij ρ j h j e uj 4π M m= α im δ pm, i =,..., N Here, = g is the Laplace-Beltrami operator with respect to the metric g and the other quantities have the following properties: A = a ij i,,...,n R N N is a positive definite symmetric N N matrix, ρ,..., ρ N R >0 are positive real parameters, h,..., h N C >0 are positive smooth functions, p,..., p M are given points, α im > for i =,..., N, m =,..., M. Recalling that udv g = 0 for any u H, by integrating both sides of in the whole surface we deduce h i e ui dv g = for all i =,..., N. Therefore, under the non-restrictive assumption that the surface area of of equals, can be re-written in the equivalent form h j e M uj u i = a ij ρ j h 4π α im δ je uj pm, i =,..., N. dv g To better describe the properties of such systems, it is convenient to perform a change of variables. Consider the Green s function G p of g centered at a point p, that is the solution of G p = δ p, G p dv g = 0 m=

5 and apply the following change of variable: u i u i + 4π M α im G pm. The newly-defined u i solve hj e uj u i = a ij ρ j h, j e uj dv g with h i := h i e 4π M m= αimgpm. 3 m= Basically, the new potentials h i absorbed the Dirac deltas appearing in. Since G p blows up around p like π log d, p, then h i will verify: hi C >0 \ {p,..., p M }, hi d, p m αim around p m. 4 Therefore, h i will tend to + at p m if and only if α im < 0 and it will tend to 0 at p m if and only if α im > 0. The form 3 is particularly useful because it admits a variational formulation. In fact, all and only its solutions are the critical points of the following energy functional defined on H N : J ρ u := a ij i, u i u j dv g i= ρ i log hi e ui dv g u i dv g. 5 Here, = g is the gradient given by the metric g, is the Riemannian scalar product and a ij are the entries of the inverse matrix A of A. Sometimes, to denote the dependence on the matrix A, we will denote the functional as J A,ρ. We will also denote as Q A u, or simply Qu, the quadratic expression a ij u i u j. i, The functional J ρ is well defined on the space H N because of the classical Moser-Trudinger inequalities by [73, 63, 38], which ensure exponential integrability in such a space. The system 3 is a natural generalization of the scalar Liouville equation u = ρ he u 4π M α m δ pm, which is equivalent, by manipulations similar to the ones described before, to he u u = ρ he. 6 u dv g m= Equation 6 arises in many well-known problem from different areas of mathematics. In statistical mechanics, it is a mean field equation for the Euler flow in the Onsager s theory see [7, 8, 46]. In theoretical physics, it is used in the description of abelian Chern-Simons vortices theory see [70, 76]. In geometry, 6 is the equation of Gaussian curvature prescription problem on surfaces with conical singularity see [, 3]. Here, each of the points p m will have a conical singularity of angle π + α m, whereas h is the Gaussian curvature of the new metric and the parameter ρ is determined by the Gauss-Bonnet theorem, that is by the Euler characteristic χ of. The scalar Liouville equations has been very widely studied in literature, with many results concerning existence and multiplicity of solutions, compactness properties, blow-up analysis et al., which have been summarized for instance in the surveys [7, 58].

6 Liouville systems like 3 have several applications: in biology they appear in some models describing chemotaxis [7], in physics they arise in kinetic models of plasma [47, 45]. Particularly interesting are the cases where A is the Cartan matrix of a Lie algebra, such as A = A N = , which is the Cartan Matrix of SUN +. This particular system is known as the SUN + or the A N Toda system. The importance of the SU3 Toda system is due to its application in algebraic geometry, in the description of the holomorphic curves of CP N see e.g. [9, 5, 6], and in mathematical physics in the non-abelian Chern-Simon vortices theory see [37, 76, 70]. The singularities represent, respectively, the ramification points of the complex curves and the vortices of the wave functions. Two further important examples are given by the following systems B =, G = 3 which are known respectively as B and G Toda systems and can be seen as particular cases of the A 3 and A 6 Toda system, respectively. Just like the A Toda, their study is closely related to holomorphic curves in projective spaces. Although the matrices B, G are not symmetric, their associated Liouville system is equivalent to one with associated to a symmetric matrix and a re-scaled parameter, through the elementary substitution: 3 ρ = ρ 4 ρ 3 = ρ 3 6 Their energy functional will therefore be J B,ρu := Q B udv g ρ log h e u dv g J G,ρu := with Q B u = u Q G udv g ρ log + u u h e u dv g + u 4 ρ ρ ρ 3,. u dv g ρ log h e u dv g u dv g, u dv g ρ 3 log h e u dv g u dv g, Q G u = u + u u + u. 7 3 A first tool to attack variationally problem 6 is given by the Moser-Trudinger inequality, from the aforementioned references [73, 63, 38] and, in the singular case, by [4, 7]. Such an inequality basically state that the energy functional, which in the scalar case has the form I ρ u = u dv g ρ he u dv g udv g, 8 { } is bounded from below if and only if ρ 8π min, + min α m. Moreover, if ρ is strictly m=,...,m smaller than this threshold, then I ρ is weakly coercive, that is all of his sub-levels are bounded. Since I ρ is also lower semi-continuous, as can be easily verified, if this occurs then direct methods 3

7 from calculus of variations yield the existence of minimizers for I ρ, which solve 6. The first main goal of of this thesis is to prove Moser-Trudinger inequalities for singular Liouville system like 3, that is to establish sufficient and necessary conditions for the boundedness from below and for the coercivity of its energy functional J ρ. Such issues were addressed in the papers [] and [9]. The former considers the SU3 Toda system, that is the following system: h e u h e u u = ρ h ρ e u dv g h e u dv g ; 9 h e u h e u u = ρ h ρ e u dv g h e u dv g the latter studies the general case. Here, they are presented in Chapter. They are inspired by some results obtained for particular systems with no singularities [44, 74] and for similar problems on Euclidean domains [8, 9], on the sphere S [68] and on general compact manifolds [67]. The arguments used to this purpose are roughly the following. As a first thing, an easy application of the scalar Moser-Trudinger inequality gives boundedness from below and coercivity for small values of ρ. We then consider, for such values, the minimizing solutions u ρ of J ρ and perform a blow-up analysis. To this purpose, we first prove a concentration-compactness theorem for solutions of 3 and then show that compactness must occur under some algebraic conditions on ρ and α im, which are satisfied in particular as long as ρ is in the neighborhood of 0. Therefore, J ρ must be coercive for all ρ s which satisfy this condition. On the other hand, through suitable test functions, we can show that such conditions are indeed also sufficient for the coercivity. Next, we discuss whether J ρ can still be bounded from below when it is not coercive. In particular, we will consider the case of fully competitive systems, that is when a ij 0 for any i j hence also for the A N, B, G Toda system described before. The conditions for coercivity, which are in general pretty lengthy to state which will be done in Chapter, are much simpler under this assumption. In this case, coercivity occurs if and only if ρ i < 8π min {, + min m=,...,m α im }, a condition which is also very similar to the one in the a ii scalar case. This is basically due to the following fact: under the assumption of A being non-positive outside the diagonal, the blow-up of minimizing sequences u ρ is locally one-dimensional, that it, roughly speaking, each blowing-up component do not interact with any other. This means that a sharper blow-up analysis can be done using a local version of the scalar Moser-Trudinger inequality, thus enabling to prove that J ρ is bounded from below even in the borderline case ρ i = 8π min {, + min m=,...,m α im } for i =,..., N. a ii The next major problem considered in this work is the existence of non-minimizing solutions, in case the parameter ρ exceeds the range of parameters which gives coercivity. A first big issue which one encounters when looking for non-minimizing critical points is the lack of the Palais-Smale condition, which is needed to apply most of the standard min-max theorems. Actually, despite the Palais-Smale condition is not known to hold true for functionals like J ρ, we can exploit a monotonicity trick by Struwe [69], see also [34]. Basically, because of the specific structure of J ρ, and in particular the fact that t J tρ is non-increasing, we get the existence of t some converging Palais-Smale sequences at mountain-pass critical level. Due to this result, to apply standard min-max methods for a generic value of ρ we only need a compactness result for solutions of 3. In fact, if we had compactness of solutions, then we could 4

8 take ρ n ρ such that bounded Palais-Smale sequences exist for such values, getting a mountain n + pass solutions u ρ n and then, by compactness, considering u ρ = lim u ρn, which would solve 3. n + Non-compactness phenomena for the Liouville equation 6 have been pretty well understood. The only possible scenario is a blow-up around a finite number of points, with no residual mass see [6]. Local quantization values, that is the portions of the integral of he u which accumulate around each blow-up point, are also fully known: they equal 8π for blow-up at a regular point see [49] and 8π + α m in the case of blow-up at a singular point p m see [7, 5]. Therefore, the only values of ρ which could generate non-compactness are all the possible finite sum of such values. We get a discrete set on the positive half-line, outside of which we get compactness of solutions. Min-max methods can thus be applied for a generic choice of ρ. Concerning general Liouville systems, local quantization and blow-up analysis are still widely open problems. A classification of local blow-up values has been given only for very specific systems, namely the A Toda system 9 [43, 53] and, in the case of no singularities, the B and partially G Toda systems [5]: h e u h e u u = ρ h ρ e u dv g h e u dv g 0 h e u h e u u = ρ h ρ e u dv g h e u dv g h e u h e u u = ρ h ρ e u dv g h e u dv g. h e u h e u u = ρ h 3ρ e u dv g h e u dv g Anyway, the quantization results do not suffice, by themselves, to ensure a generic compactness results on ρ, due to the possibility of residual mass. Actually, in [30] it was proved that non-vanishing residual may indeed occur for the regular Toda system. The issue of residual has been rule out in the paper [4], where we showed that if it occurs, then it does only for one component of the A Toda. Similarly, for N N systems, there is at least one component which has not residual mass. This result is presented in Section.. Ruling out the chance of a double residual implies that, for blowing up sequences of solutions of the A, B, G Toda systems, at least one between ρ and ρ must be a finite combination of the local blow-up values. Therefore, the set of parameters to be excluded for the purpose of compactness is just a union of horizontal and vertical half-lines on the first quadrant so, similarly as before, conditions to apply min-max methods are satisfied for almost every ρ R >0. We are therefore allowed to search for min-max solutions, a goal to which Chapter 3 will be devoted. The strategy we will follow will be based on analysis of energy sub-levels and application of Morse theory, rather than usual mountain pass or linking theorems, as was usually done by many authors who studied similar problems. The reason of such a choice is that, whereas the two arguments are perfectly equivalent to prove existence of solutions, Morse theory gives also information about the number of solutions, provided the energy is a Morse functional, which a generic assumption in a sense which will be clarified later. We will actually show that a change of topology occurs between very high sub-levels of energy functional, which are contractible, and very low sub-levels. The compactness assumptions discussed in the previous paragraph ensure, thanks to [56], that a change of topology between sub-levels implies existence of solutions. Roughly speaking, if J ρ u is very negative, then the L -mass of h i e ui accumulates, for one or both i s, around a finite number of points, depending on the parameters ρ i and α im. This can be made rigorous by introducing a space X, a subspace of finitely-supported unit measures 5

9 on, and by building two maps Φ, Ψ from very low sub-levels to the space X and vice-versa, such that their composition is homotopically equivalent to the identity on X. If X is not contractible, then low sub-levels of J ρ will also be non-contractible, hence we will deduce existence of solutions. Moreover, by estimating the ranks of the homology groups of X we will get an estimate on the multiplicity of solutions. Such a method has been introduced in [36] for a fourth-order elliptic problem and has been widely used to study the singular Liouville equation. Through this argument, general existence results have been proved for problem 6 in the case of no singularities [35] and in the case of positive singularities on surfaces of non-positive Euler-Poincaré characteristic [3], as well as partial existence results in the case of negative singularities [, 0] and of positive singularities on general surfaces [60, 4]. It has also been used in [59, 6, 4] to attack the regular SU3 Toda system in the cases when one or both between ρ, ρ are less then 8π, and even in similar problems with exponential nonlinearities, such as the Sinh-Gordon equation [77, 4] and the Nirenberg problem [55, 33]. Here, we will present the results obtained in the papers [, 0, 3, 8], the last of which is in preparation. In the first of such papers we study the SU3 Toda system 9. We assume to have non-positive Euler characteristic, that is neither homeomorphic to the sphere S nor to the projective plane RP, and we assume that coefficients α im to be non-negative. Here, following [3], we exploit the topology of to retract the surface on a bouquet of circles. By taking two of such retractions on disjoint curves we can by-pass a major issue which occurs in the study of Liouville systems of two or more equations, that is the interaction between the concentration of two or more components. In fact, through the push-forward of measures, we can restrict the study of u on a curve γ and of u on the other γ. Moreover, by choosing γ, γ not containing any of the singular points p m, we also avoid the issue of singularities. Performing such a retraction clearly causes a loss of topological information, but the partial characterization we give on sub-levels suffices to get a generic existence result. Furthermore, we can apply Morse inequalities to get an estimate from below on the number of solutions. This can be done for a generic choice of the potentials h, h and of the metric g, since J ρ is a Morse functional for such a generic choice, as was proved in [3] for the scalar case. In particular, if the characteristic of is greater or equal than, namely its Euler characteristic is negative, the number of solutions goes to + as either ρ or ρ goes to +. In the paper [9] we give a partial extension of the results from [] to the case of singularity of arbitrary sign. The main difference is that negatively-signed vortices actually affect the best constant in Moser-Trudinger inequality, as will be shown in detail in Chapter, therefore they cannot be simply ignored as was done before. On the contrary, we will have to take into account each point p m on γ i if α im < 0. This means that, since we need γ and γ to be disjoint, we have to assume max{α m, α m } 0 for any m, as well as the characteristic of to be non-positive. Moreover, we also need some algebraic condition on ρ and α im to let low sub-levels be not contractible, much like [0]. By Morse theory, we also get another generic multiplicity result similar to the one before []. In [3], we consider the singular SU3 system on arbitrary surfaces and we allow both α m and α m to be negative for the same p m. Here, the methods which were briefly described before cannot be applied anymore and we need a sharper analysis of sub-levels. To this purpose, we introduce a center of mass and scale of concentration, inspired by [6] but strongly adapted to take into account the presence of singularities. We basically show that, for functions with same center and scale, Moser-Trudinger inequality holds with a higher constant. In other words, we get a so-called improved Moser-Trudinger inequality. Such improved inequalities allow, for sufficiently small values of ρ, to give a precise characterization of sublevels, hence existence of min-max solutions also in this case. Finally, in [8] the results from [] are adapted to the regular B and G Toda systems 0,. We get similar general existence and multiplicity results for surfaces with non-positive Euler char- 6

10 acteristic. In Chapter 4 we give some non-existence result for singular Liouville systems 3, contained in the paper [3]. The first two results, inspired by the ones in [4] for the scalar equation, are for general systems defined on particular surfaces. The former holds on the unit Euclidean ball, equipped with the standard metric, with a unique singularity in its center; the latter holds on the standard unit sphere with two antipodal singularities. Concerning the result on the ball, we show, through a Pohožaev identity, that if a solution exists then the parameters ρ, α i must satisfy an algebraic relation. The argument used for the case of the sphere is similar. We exploit the stereographic projection to transform the solution of 3 on S in an entire solution on the plane. Then, we use another Pohožaev identity for entire solutions to get again necessary algebraic conditions for the existence of solutions. These result show that in the general existence results stated before the assumptions on χ is somehow sharp. The last result, inspired by [0], is given only for the SU3 Toda system, but it holds for any surfaces. It basically states that the system has no solutions if a couple of coefficients α m, α m is too close to. The result is proven by contradiction, using blow-up analysis. We assume that a solution exists for α m, α m arbitrarily close to and apply the Concentration-Compactness alternative from Chapter to this sequence. By ruling out all the alternatives we get a contradiction. By comparing this result with the ones in Chapter 3 we deduce that, to have such a general existence result, we need to assume all the coefficients α im to be positive. Before stating the main result of this thesis, we devote Chapter to some preliminaries. First of all, we introduce the notation we will use throughout the whole paper. Then, we present some known facts which will be used in the rest, mostly from analysis and topology, and some of their consequences which need very short proofs. We will postpone some proofs in an Appendix: the proof of a Pohožaev identity for entire solutions of singular Liouville systems and of the fact that being a Morse function is a generic condition for the energy functional. The reason of this choice is that such proofs are very similar to the ones for the scalar case, though quite lengthy. 7

11 Chapter Preliminaries In this chapter we present some notation and some preliminary facts which will be useful in the following.. Notation The indicator function of a set Ω will be denoted as { if x Ω Ω x = 0 if x Ω Given two points x, y, we will indicate the metric distance on between them as dx, y. Similarly, for any Ω, Ω we will write: dx, Ω := inf{dx, y : x Ω}, dω, Ω := inf{dx, y : x Ω, y Ω }. The diameter of a set Ω will be indicated as diamω := sup{dx, y : x, y Ω}. We will indicate the open metric ball centered in p having radius r as Similarly, for Ω we will write B r x := {y : dx, y < r}. B r Ω := {y : dy, Ω < r}. For any subset of a topological space A X we indicate its closure as A and its interior part as Å. For r > r > 0 we denote the open annulus centered at p with radii r, r as A r,r p := {x : r < dx, p < r } = B r p \ B r p. If Ω has a smooth boundary, for any x Ω we will denote the outer normal at x as νx. If u C Ω we will indicate its normal derivative at x as ν ux := ux νx. Standard notation will be used for the usual numeric set, like N, R, R N. Here, N contains 0. A similar notation will be used for the space of N M matrices, which we will denote as R N M. The positive and negative part of real number t will be denoted respectively as t + := max{0, t} and t := max{0, t}. The usual functional spaces will be denoted as L p Ω, C, C N,.... A subscript will be added to indicate vector with positive component or almost everywhere positive functions, like 8

12 R >0, C >0. Subscript may be also added to stress the dependence on the metric g defined on. For a continuous map f : and a measure µ defined on, we define the push-forward of µ with respect to f as the measure defined by f µb = µ f B. If µ has finite support, its push-forward has a particularly simple form: K K µ = t k δ xk f µ = t k δ fxk. k= k= Given a function u L and a measurable Ω with positive measure, the average of u on Ω will be denoted as udv g = udv g. Ω Ω Ω In particular, since we assume =, we can write udv g = udv g. We will indicate the subset of H which contains the functions with zero average as { } H := u H : u = 0. Since the functional J ρ defined by 5 is invariant by addition of constants, as well as the system 3, it will not be restrictive to study both of them on H N rather than on H N. On the other hand, for a planar Euclidean domain Ω R or Ω with smooth boundary and a function u H Ω we will indicate with the symbol u Ω the trace of u on the boundary of Ω. The space of functions with zero trace will be denoted by H 0 Ω := { u H Ω : u Ω = 0 }.. The sub-levels of J ρ, which will play, as anticipated, an essential role throughout most of the paper, will be denoted as J a ρ = { u H N : J ρ u a }.. We will denote with the symbol X Y a homotopy equivalence between two topological spaces X and Y. The composition of two homotopy equivalences F : X [0, ] Y and F : Y [0, ] Z satisfying F, = F, 0 is the map F F : X [0, ] Z defined by F x, s if s F F : x, s F x, s if s >. The identity map on X will be denoted as Id X. We will denote the q th homology group with coefficient in Z of a topological space X as H q X. An isomorphism between two homology groups will be denoted just by equality sign. Reduced homology groups will be denoted as H q X, namely H 0 X = H 0 X Z, H q X = H q X if q. The q th Betti number of X, namely the dimension of its q th group of homology, will be indicated by b q X := rankh q X. The symbol b q X will stand for the q th reduced Betti number, namely the dimension of H q X, that is b0 X = b 0 X, bq X = b q X if q. 9

13 If J ρ is a Morse function, we will denote as C q a, b the number of critical points u of J ρ with Morse index q satisfying a J ρ u b. The total number of critical points of index q will be denoted as C q ; in other words, C q := C q +,. We will indicate with the letter C large constants which can vary among different lines and formulas. To underline the dependence of C on some parameter α, we indicate with C α and so on. We will denote as o α quantities which tend to 0 as α tends to 0 or to + and we will similarly indicate bounded quantities as O α, omitting in both cases the subscripts when it is evident from the context.. Compactness results First of all, we need two results from Brezis and Merle [6]. The first is a classical estimate about exponential integrability of solutions of some elliptic PDEs. Lemma.. [6], Theorem, Corollary Let Ω R be a smooth bounded domain, f L Ω be with f L Ω < 4π and u be the solution of { u = f in Ω u = 0 on Ω. 4π Then, for any q < there exists a constant C = C q,diamω such that e q ux dx C. f L Ω Moreover, e u L q Ω for any q < +. Ω The second result we need, which has been extended in [5, 6], is a concentration-compactness theorem for scalar Liouville-type equations, which can be seen as a particular case of the one which will be proved in Chapter : Theorem.. [6], Theorem 3; [6], Theorem 5; [5], Theorem. Let {u n } n N be a sequence of solutions of 6 with ρ n in C and S be defined by { S := x : x n R >0 and h n = V n h with V n n + x such that n + un x n log hn e un n + } +..3 n + Then, up to subsequences, one of the following occurs: Compactness If S =, then u n log hn e un dv g converges to some u in W,q. Concentration If S, then it is finite and u n log hn e un dv g in n + L loc\s. Let us now report the known local blow-up quantization results for the systems 6, 9, 0,. Theorem.3. [49]; [48], Theorem 0.; [6], Theorem 6; [5], Theorem.3 Let {u n } n N be a sequence of solutions of 6 with ρ = ρ n, let S be defined by.3 and let, for x S, σx be defined up to subsequences by h n B σx := lim lim rx e un dv g h. n e un dv g r 0 n + ρn If x {p,..., p M }, then σx = 8π, whereas σp m = 8π + α m. 0

14 Corollary.4. Let Γ = Γ α be defined by { Γ := 8π n + } + α m, n N, M {,..., M}. m M Then, the family of solutions {u ρ } ρ K H of 6 is uniformly bounded in W,q for some q > for any given K R >0 \ Γ. Proof. Take ρ n ρ K and apply Lemma. to n + un = u ρ n. If Concentration occurred, then we can easily see that ρ n hn e un h n e un dv g n + ρ = σx Γ, which is a contradiction since we assumed ρ K R >0 \ Γ. x S Therefore, we must have Compactness for u n log hn e un dv g. If u n were not bounded in W,q, then log hn e un dv g ±. n + Anyway, Jensen inequality gives log hn e un dv g log h n dv g C. Moreover, log + would imply inf un +, in contradiction with u n + n dv g = 0. x S σxδ x, hence hn e un dv g n + Definition.5. Let α, α be a couple of numbers greater than and let α,α R as the piece of ellipse defined by α,α = { σ, σ R 0 : σ σ σ + σ 4π + α σ 4π + α σ = 0 } We then define iteratively the finite set Ξ α,α α,α via the following rules: Ξ α,α contains the points 0, 0 4π + α, 0 0, 4π + α 4π + α + α, 4π + α 4π + α, 4π + α + α 4π + α + α, 4π + α + α..4 If σ, σ Ξ α,α, then any σ, σ α,α σ σ belongs to Ξ α,α. If σ, σ Ξ α,α, then any σ, σ α,α σ σ belongs to Ξ α,α. with σ = σ + 4πn for some n N and with σ = σ + 4πn for some n N and Theorem.6. [43], Proposition.4; [53], Theorem. Let {u n = u n, u n } n N be a sequence of solutions of 9 with ρ, ρ = ρ n, ρ n and let, for x, σx = σ x, σ x be defined up to subsequences by σ i x := lim lim r 0 n + ρn i If x {p,..., p M }, then σx Ξ 0,0, whereas σp m Ξ αm,α m. B rx h n i eun i dvg h n i eun i dv g..5

15 Remark.7. Notice that, if either α = α = 0 or they are both small enough, then Ξ α,α contains only the set Ξ α,α of six points defined in.4. The authors announced [75] that they refined the previous result by proving that σp m Ξ α m,α m if α m, α m C for some C > 0. They also conjectured that σp m Ξ α m,α m for any α m, α m. In [64], the authors prove that, in the regular case, all the values in Ξ 0,0 = Ξ 0,0 can be attained in case of blow-up see also [3, 30]. Theorem.8. [5] Let {u n = u n, u n } n N be a sequence of solutions of 0 with ρ, ρ = ρ n, ρ n and let σx be defined by.5. For any x : σx 4π{0, 0,, 0, 0,,, 3,,, 3, 3,, 4, 3, 4}. Let {u n = u n, u n } n N be a sequence of solutions of with ρ, ρ = ρ n, ρ n and let σx be defined by.5. If ρ n < 4π +, ρ n < 4π 5 + 7, then for any x : σx 4π{0, 0,, 0, 0,,, 4,,,, 6}. Let us now state a couple of Lemmas from [56] concerning deformations of sub-levels. Lemma.9. [56], Proposition. Let ρ R N >0, a, b R be given with a < b and let J a ρ, J b ρ be defined by.. Then, one of the following alternatives occurs: There exists a sequence {u n } n N of solutions of 3 with ρ n J a ρ is a deformation retract of J b ρ. ρ and a J ρ n + nun b, Corollary.0. Let ρ R N >0 be given and let J L ρ be defined by.. Then, one of the following alternatives occurs: There exists a sequence {u n } n N of solutions of 3 with ρ n ρ and J ρ n + nun +, n + There exists L > 0 such that J L ρ is a deformation retract of H N. In particular, J L ρ is contractible. In the scalar case the global compactness result.4 holds, hence for ρ Γ we can act as if Palais- Smale condition holds: Corollary.. Let Γ be as in Corollary.4, ρ R >0 \ Γ, a, b R be given with a < b and such that 6 has no solutions with a I ρ b. Then, Iρ a is a deformation retract of Iρ. b Moreover, there exists L > 0 such that Iρ L is contractible.

16 Proof. If ρ Γ then the second alternative must occur in Lemma.9, since the first alternative would give, by Corollary.4, u n u which solves 6 and satisfies a I ρu b. n + Moreover, by Corollary.4, we have u n H C for any solution un H of 6 with ρ n ρ, therefore by Jensen s inequality every solution of 6 verifies n + I ρ n u n Corollary.0 gives the last claim. u n dv g ρ log hdv g C ρ log hdv g =: L;.3 Analytical preliminaries and Moser-Trudinger inequalities To study the concentration phenomena of solutions of 3 we will use the following simple but useful calculus Lemma: Lemma.. [44], Lemma 4.4 Let {a n } n N and {b n } n N be two sequences of real numbers satisfying a n +, n + lim b n n + a n 0. Then, there exists a smooth function F : [0, + R which satisfies, up to subsequences, 0 < F t < t > 0, F t t + 0, F an b n n + +. Now we recall the Moser-Trudinger inequality for the scalar Liouville equation. Theorem.3. [38], Theorem.7; [63], Theorem ; [7], Corollary 9 There exists C > 0 such that for any u H log he u dv g udv g u dv g + C..6 6π min {, + min m α m } { In other words, the functional I ρ defined in 8 is bounded from below if and only if ρ 8π min, + min { } m and it is coercive on H if and only if ρ < 8π min, + min α m. m In particular, in the latter case I ρ has a global minimizer which solves 6. α m } We will also need a similar inequality by Adimurthi and Sandeep [], holding on Euclidean domains, and its straightforward corollary. Theorem.4. [], Theorem. Let r > 0, α, 0] be given. Then, there exists a constant C = C α,r such that for any u H0 B r 0 ux dx x α e 4π+αux dx C B r0 B r0 3

17 Corollary.5. Let r > 0, α, 0] be given. Then, there exists a constant C = C α,r such that for any u H0 B r 0 + α log B r0 x α e ux dx 6π B r0 ux dx + C Proof. By the elementary inequality u θu + 4θ with θ = 4π + α Ω uy dy we get + α log + α log 6π 6π Ω Ω Ω Ω x α e ux dx x α e θux + 4θ dx uy dy + + α log x α 4π+α e Ω uy dy + C. ux Ω uy dy dx Let us now state the Moser-Trudinger inequality for the regular SU3 Toda system: Theorem.6. [44], Theorem.3 There exists C > 0 such that for any u = u, u H : i= log e ui dv g u i dv g Q A udv g + C. 4π In other words, the functional J ρ defined by 5, in the case A = A, α im 0, is bounded from below on H if and only if ρ, ρ 4π and it is coercive on H if and only if ρ, ρ < 4π. In particular, in the latter case I ρ has a global minimizer which solves 9. From this result, we deduce a Moser-Trudinger inequality for the SU3 Toda system on domains with boundary. This can be seen as a generalization of the well-known scalar Moser-Trudinger inequalities on Euclidean domains from [3], which we report: Theorem.7. Let Ω R be a smooth simply connected domain. Then, there exists C > 0 such that, for any u H Ω, log Ω e ux dx uxdx Ω 8π Ω ux dx + C..7 Before stating the inequality for the SU3 Toda, we introduce a class of smooth open subset of which satisfy an exterior and interior sphere condition with radius δ > 0: { } A δ := Ω : x Ω x Ω, x \ Ω : x = B δ x Ω = B δ x Ω.8 4

18 Theorem.8. There exists C > 0 such that, for any u H B 0, log e uix dx u i xdx B 0 B 0 i= Q A uxdx + C. π B 0 The same result holds if B 0 is replaced with a simply connected domain belonging to A δ for some δ > 0, with the constant C is replaced with some C δ > 0. Sketch of the proof. Consider a conformal diffeomorphism from B 0 to the unit upper half-sphere and reflect the image of u through the equator. Now, apply the Moser-Trudinger inequality to the reflected u, which is defined on S. The Dirichlet integral of u will be twice the one of u on B 0, while the average and the integral of e u will be the same, up to the conformal factor. Therefore the constant 4π is halved to π. Starting from a simply connected domain, one can exploit the Riemann mapping theorem to map it conformally on the unit ball and repeat the same argument. The exterior and interior sphere condition ensures the boundedness of the conformal factor. In Chapter 3, we will need to combine different type of Moser-Trudinger inequalities. To do this, we will need the following technical estimates concerning averages of functions on balls and their boundary: Lemma.9. There exists C > 0 such that for any u H, x, r > 0 one has udv g udv g C u dv g. Brx B rx B rx Moreover, for any R > there exists C = C R such that udv g udv g C u dv g. Brx B Rr x B rx The same inequalities hold if B r x is replaced by a domain Ω B Rr x such that Ω A δr for some δ > 0, with the constants C and C R replaced by some C δ, C R,δ > 0, respectively. The proof of the above Lemma follows by the Poincaré-Wirtinger and trace inequalities, which are invariant by dilation. Details can be found, for instance, in [39]. We will also need the following estimate on harmonic liftings. Lemma.0. Let r > r > 0, f H B r 0 with Then, there exists C = C r r > 0 such that A r,r 0 B r 0 fxdx = 0 be given and u be the solution of u = 0 in A r,r 0 u = f on B r 0 u = 0 on B r 0 ux dx C fx dx A r,r 0. 5

19 Finally, we give a result concerning entire solution of singular Liouville systems, which will be used to prove the non-existence results in Chapter 4. Unlike the previously stated results, this one has not been published, up to our knowledge, nor it follows straightforwardly from any known results. Anyway, it can be proved similarly as in the scalar case [5], Theorem. As anticipated in the introduction, the proof will be postponed to the Appendix. Theorem.. Let H,..., H N C loc R \ {0} be such that, for suitable a, c 0, b >, C > 0, x a C H ix C x b x B 0 \ {0} 0 < H i x C x c x R \ B 0; let U = U,..., U N be a solution of U = a ij H j e Uj in R x b + x c e Uix dx < + R, i =,..., N.9 and define ρ i := H i xe Uix dx, R τ i := x H i xe Uix dx, R i =,. Then, a ij ρ i ρ j 4π ρ i + τ i = 0..0 i, i=.4 Topological preliminaries, homology and Morse theory We start with a simple fact from general topology, which anyway will be essential in most of Chapter 3: Lemma.. { } Let be a compact surface with χ 0 and p 0,..., p 0M, 0 p,..., p M, p,..., p M be given points of. [ ] χ Then, there exist two curves γ, γ, each of which is homeomorphic to a bouquet of + circles and two global projections Π i : γ i such that: γ γ =. p im γ i for all m =,..., M i, i =,. p 0m γ i for all m =,..., M 0, i =,. Sketch of the proof. If = T g is a g-torus, two retractions on disjoint bouquets of g circles can be easily built. For instance, as in [3], can be assumed to be embedded in R 3 in such a way that each hole contains a line parallel to the x 3 axis and that the projection P i on each plane { x 3 = i+} is a disk with 6

20 p p 03 p p γ γ p 0 p 0 p Figure.: The curves γ, γ g holes. Then, there exists two bouquets of circles γ i such that P i γi are homeomorphisms and there exists retractions r i : P i P i γ i, and we suffice to define Π i := P i γi r i P i. One can argue similarly with a connected sum = P k of an even number of copies of the projective plane, since this is homeomorphic to a connected sum of a T k and a Klein bottle, which in turn retracts on a circle; therefore, P k retracts on two disjoint bouquets of k circles. If instead is a connected sum of an odd number of projective planes, one can argue as before setting the retractions constant on the last copy of P. In all these cases, g = + χtg, k = + χ P k [ ] χ P k+ = +. Finally, with a small deformation, the curves γ i can be assumed to contain all the points p im and they will not contain any of the other singular points. We can apply those deformations to γ without intersecting γ or vice versa because \ γ is pathwise connected. See Figure. for an example. In Chapter 3 we will often have to deal with the space M of Radon measures defined on, especially unit measures. Such a space will be endowed with the Lip topology, that is with the norm of the dual space of Lipschitz functions: µ Lip := sup φdµ.. φ Lip, φ Lip As a choice of this motivation, notice that by choosing, in., φ = d, y, one gets d Lip δ x, δ y dx, y for any x, y. This means that M contains a homeomorphic copy of. Moreover, one can see immediately that L >0 embeds into M. Concerning u H N, there is a natural way to associate to any u, through the system 3, a N-tuple of positive normalized L functions, that is N elements of the space { } A := f L : f > 0 a.e. and fdv g =.. Precisely, we define u,..., u N h e u h e u dv g,..., hn e u N h N e u N dvg =: f,u,..., f N,u..3 Lemma.3. Let A be defined by. and f i,u be defined by.3. Then, the map u f i,u is weakly continuous for i =,..., N. 7

21 Proof. It will suffice to give the proof for the index i =, which we will omit. We will just need to prove the continuity of the map u he u, since dividing a non-zero element by his norm is a continuous operation in any normed space. Let u n H be converging weakly, and strongly in any L p, to u, and fix q > in such a way that qα m > m. From the elementary inequality e t t e t we get, by Lemma.3 and Hölder s inequality, u he n dv g he u dv g u he e un u dv g he u u n u e un u dv g hq e qu q dv g u n u q q q dvg e q q un u q dv g Ce q udvg+ q 6π min{,+q minm αm} u q dv g u n u q q q dvg Ce q q un u dv g+ q 4πq un u q dv g C u n u q q q dvg 0. n + From the proof of the previous Lemma we deduce the following useful Corollary: Corollary.4. The functional J ρ : H N R defined in 5 is of class C and weakly lower semi-continuous. Speaking about unit measures, a fundamental role will be played by the so-called K-barycenters, that is unit measures supported in at most K points of, for some given K. They will be used in Chapter 3 to express the fact that f i,u concentrates around at most K points. For a subset X, we define: { K } K X K := t k δ xk : x k X, t k 0, t k =..4 k= k= If we choose X to be homeomorphic to a bouquet of g circles, such as for instance the curves γ, γ defined in Lemma., the homology of the K- barycenters on X is well-known: Proposition.5. [3], Proposition 3. Let γ be a bouquet of g circles and let γ K be defined by.4. Then, its homology groups are the following: Z if q = 0 H q γ K = Z K+g g if q = K 0 if q 0, K. 8

22 Remark.6. In Proposition.5, when g = we get not only a homology equivalence but a homotopy equivalence between S K and SK. We will also need a similar definition, which extends the K-barycenters defined before. We still consider unit measure with finite support, but we do not give a constraint on the number of the points, but rather on a weight defined on such points. Given a set X, a finite number of points p,..., p M X and a multi-index α = α,..., α M with < α m < 0 for any m =,..., M, we define the weighted cardinality ω α as { + αm if x = p ω α {x} := m if x {p,..., p M }, ω α {x k } := ω α {x k }. We then define the weighted barycenters on X with respect to the a parameter ρ > 0 and the multi-index α as { } X ρ,α := t k δ xk : x k X, t k 0, t k =, 4πω α J < ρ..5 x k J As a motivation for this weight, introduced in [] to study the singular Liouville equation, consider inequality.6. In the case of no singularities, the constant multiplying u dv g is 6π, and in case of one singular point p m with α m < 0, that constant is 6π + α m. Roughly speaking, the weight of each point represents how much that point affects the Moser- Trudinger inequality.6. The space of weighted barycenters can be in general more complicated than the non-weighted barycenters, which are a particular case given by defining ω α {x} = for any x X and K as the largest integer strictly smaller than ρ 4π. Anyway, both the weighted and the non-weighted barycenters are stratified set, that is, roughly speaking, union of manifolds of different dimensions with possibly non-smooth gluings. For this reason, they have the fundamental property of being a Euclidean Neighborhood Retract, namely a deformation retract of an open neighborhood of theirs. x k J x k J x k J Lemma.7. [], Lemma 3. Let, for ρ R >0, α = α,..., α M, ρ,α be defined as in.5. Then, there exists ε 0 > 0 and a continuous retraction ψ ρ,α : { } µ M : d Lip µ, ρ,α < ε0 ρ,α In particular, if µ n σ for some σ ρ,α, then ψ ρ,α µ n σ. n n Another tool which we will take from general topology is the join between two spaces X and Y, defined by X Y [0, ] X Y :=.6 where is the identification given by x, y, 0 x, y, 0 x X, y, y Y, x, y, x, y, x, x X, y Y. Basically, the join expresses a non-exclusive alternative between X and Y : if t = 0 we only see X and not Y, if t = we see only Y and if 0 < t < we see both X and Y for more details, see [40], page 9. This will be used in Chapter 3 as a model for the alternative between concentration of f,u and f,u. The homology of X Y depends from the homology of X and Y through the following: 9

23 Proposition.8. [40], Theorem 3. Let X and Y be two CW -complexes and X Y be their join as in.6. Then, its homology group are defined by H q X Y = q q =0 H q X H q q Y. In particular, and + q bq X Y = bq X b q q Y bq X Y = q =0 + bq X + q=0 q =0 q =0 bq Y. Remark.9. By taking, in the previous Proposition, two wedge sum of spheres X = S D N and Y = S D N, we find that X Y has the homology of another wedge sum of spheres S D+D+ N N. In the same book [40] it is shown that actually a homotopically equivalence S D N S D N S D +D + N N holds. In particular, from Remark.6, S S S K+ S K+ K K S K+K. Remark.30. Proposition.8 shows, in particular, that if both X and Y have some non-trivial homology, then the same is true for X Y. It is easy to see that a partial converse holds, concerning contractibility rather than homology: if either X or Y is contractible, that X Y is also contractible. In fact, if F is a homotopy equivalence between X and a point, then x, y, t, s F x, s, y, t is a homotopical equivalence between X Y and the cone based on Y, which is contractible. Morse inequalities yield the following estimate on the number of solutions, through the Betti numbers of low sub-levels: Lemma.3. Let ρ R >0 be such that J A,ρ is a Morse functional and u H H of 9. Then, there exists L > 0 such that # solutions of 9 The same result holds if A is replace by B or G. + q=0 bq J L A,ρ. C for any solution u Proof. By Corollary.4, L < J ρ u L for some L > 0. In particular, L is a regular value for J ρ, hence the exactness of the sequence H q J L ρ Hq J L ρ Hq J L ρ, Jρ L 0 Hq J L ρ Hq J L ρ...

24 reduces to Therefore, Morse inequalities give: # solutions of 0 = + q=0 H q+ J L ρ, Jρ L C q J ρ = + q=0 = Hq J L ρ. C q J ρ ; L, L + q=0 b q J L ρ, Jρ L = + q=0 bq J L ρ. Finally, we need a density result for J ρ, given in [3] for I ρ, which will be proved in the Appendix. Theorem.3. Let M be the space of Riemannian metrics on, equipped with the C norm, and M its subspaces of the metrics g satisfying dv g =. Then, there exists dense open set D M C >0 C >0, D M C >0 C >0 such that for any g, h, h D D the three of J A,ρ, J B,ρ, J G,ρ are all Morse functions from H to R.

25 Chapter New Moser-Trudinger inequalities and minimizing solutions This chapter will be devoted to proving two Moser-Trudinger inequalities for systems 3, namely to give conditions for the energy functional J ρ to be coercive and bounded from below. The first result gives a characterization of the values of ρ which yield coercivity for J ρ and some necessary conditions for boundedness from below: Theorem.. Define, for ρ R N >0, x and i I {,..., N}: { αim if x = p α i x = m 0 otherwise. Λ I,x ρ := 8π + α i xρ i a ij ρ i ρ j, i I i,j I. Λρ := min I,xρ I {,...,N},x..3 Then, J ρ is bounded from below if Λρ > 0 and it is unbounded from below if Λρ < 0. Moreover, J ρ is coercive in H N if and only if Λρ > 0. In particular, if this occurs, then it has a minimizer u which solves 3. By this theorem, the values of ρ which yield coercivity belong to a region of the positive orthant which is delimited by hyperplanes and hypersurfaces, whose role will be clearer in the blow-up analysis which will be done in this chapter. The coercivity region is shown in Figure.: Theorem. leaves an open question about what happens when Λρ = 0. In this case one encounters blow-up phenomena which are not yet fully known for general systems. Anyway, we can say something more precise if we assume the matrix A to be non-positive outside its main diagonal. First of all, it is not hard to see that notice that in this case where α i := Λρ = min i {,...,N} 8π + αi ρ i a ii ρ i, { } min α ix = min 0, min α im..4 m {,...,M},x m {,...,M} In particular, only the negative α im play a role in the coercivity of J ρ, like for the scalar case and unlike the general case in Theorem..

26 Figure.: The set Λρ > 0, in the case N =. Therefore, under these assumptions, the coercivity region is actually a rectangle and the sufficient condition in Theorem. is equivalent to assuming ρ i < 8π + α i a ii for any i: Figure.: The set Λρ > 0, in the case N =, a 0. With this assumption, the blow-up analysis needed to study what happens when Λρ = 0 is locally one-dimensional, hence can be treated by using well-known scalar inequalities like Lemma.5. Therefore, we get the following sharp result: Theorem.. Let Λρ as in.3, α i as in.4 and suppose a ij 0 for any i, j =,..., N with i j. Then, J ρ is bounded from below on H N if and only if Λρ 0, namely if and only if ρ i 8π + α i for any i =,..., N. In other words, there exists C > 0 such that a ii + α i log hi e ui dv g u i dv g QudV g + C.5 a ii 8π i= Remark.3. We remark that assuming A to be positive definite is necessary. If A is invertible but not symmetric 3

27 definite, then J ρ is unbounded from below for any ρ. In fact, suppose there exists v R N such that a ij v i v j θ v for some θ > 0. Then, consider the family of functions u λ x := λv x. By Jensen s inequality we get: J ρ u λ a ij i, u λ i u λ j dv g i, ρ i i= log h i dv g θ λ v + C The proofs of Theorems. and. will be given respectively in Sections.3 and.4.. Concentration-compactness theorem. n + The aim of this section is to prove a result which describes the concentration phenomena for the solutions of 3, extending what was done for the two-dimensional Toda system in [, 57]. We actually have to normalize such solutions to bypass the issue of the invariance by translation by constants and to have the parameter ρ multiplying only the constant term. In fact, for any solution u of 3 the functions v i := u i log hi e ui dv g + log ρ i.6 solve v i = a ij hj e vj ρ j, i =,..., N..7 hi e vi dv g = ρ i Moreover, we can rewrite in a shorter way the local blow-up masses defined in.5 as σ i x = lim lim hn i e vn i dvg. r 0 n + B rx For such functions, we get the following concentration-compactness alternative: Theorem.4. Let {u n } n N be a sequence of solutions of 3 with ρ n n + ρ RN >0 and h n i = Vi n h i Vi n n + C N, {v n } n N be defined as in.6 and S i be defined, for i =,..., N, by S i := { x : x n x such that n + vn i x n Then, up to subsequences, one of the following occurs: Compactness If S := N i=s i =, then v n v which solves.7. Concentration If S, then it is finite and hn i e vn i σ i xδ x + f i n + + n + with }..8 v in W,q N for some q > and some n + x S as measures, with σ i x defined as in.8 and some f i L. In this case, for any given i, either vi n n + in L loc \ S and f i 0, or v n i in W,q loc \ S for some q > and some suitable v i and f i = h i e vi > 0 a.e. on. n + v i 4