Concentration of ground states in stationary Mean-Field Games systems: part I

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1 Concentration of ground states in stationary Mean-Field Gaes systes: part I Annalisa Cesaroni and Marco Cirant June 9, 207 Abstract In this paper we provide the existence of classical solutions to soe stationary ean field gae systes in the whole space, with coercive potential and aggregating local coupling. This result is obtained under structural assuptions on the growth at infinity of the coupling ter and of the Hailtonian, by using a variational approach based on the analysis of the non-convex energy associated to the syste. Moreover, we analyse the liit of the syste as the diffusion vanishes, and we show that phenoena of concentration of ass appear. We also describe the asyptotic shape of the rescaled solutions in the vanishing viscosity liit, in particular proving the existence of ground states, i.e. classical solutions to ean field gae systes in the whole space without potential, and with aggregating coupling. AMS-Subject Classification. 35J50, 4970, 35J47, 9A3, 35B25 Keywords. Ergodic Mean-Field Gaes, Concentration-copactness ethod, Mass concentration, Seiclassical liit, Elliptic systes, Variational ethods. Contents Introduction 2 Soe preliinary regularity results 6 2. The Hailton-Jacobi-Bellan equation on the whole space A-priori estiates for the Kologorov equation Existence of solutions to the MFG syste for > A-priori estiates and energy bounds Existence of iniizers of E and solutions to for every > Convergence to ground states and concentration The rescaled proble A preliinary convergence result Concentration-copactness Convergence to ground states References 33 Introduction This paper is devoted to the analysis of ergodic Mean-Field Gaes systes with local decreasing coupling. The proble we consider is the following: given, M > 0, find a constant λ R for

2 which there exists a couple u, solving u + H u + λ = α + V x div H u = 0 on R, = M, where H : is a convex function, V : [0, + is a coercive potential, and α > 0. The second proble we address is the analysis of the asyptotic behaviour of the solutions as 0. Mean-Field Gaes MFG is a recent theory that odels the behaviour of a very large nuber of indistinguishable rational agents aiing at iniizing a coon cost. The theory was introduced in the seinal papers by Lasry, Lions [23, 24, 25, 26] and by Huang, Caines, Malhaé [9], and has been rapidly growing during the last decade due to its atheatical challenges and several potential applications fro econoics and finance, to engineering and odels of social systes. In the ergodic MFG setting, the dynaics of a typical agent is given by the controlled stochastic differential equation dx s = v s ds + 2 db s, where v s is the control and B s is a Brownian otion, and the cost of long-tie average for is given by T li T T E [Lv s + V X s α X s ]ds, 0 where the Lagrangian L is the Legendre transfor of H see 5 and x denotes the density of population of sall agents at a position x. In an equilibriu regie, a typical agent iniizes his own cost, his corresponding density is stable as tie s, and coincides with the population density. This equilibriu is encoded fro the PDE viewpoint in : a solution u of the Hailton-Jacobi-Bellan equation gives an optial control for the typical agent in feedback for H u, and the Kologorov equation provides the density of the agent playing in an optial way. The two key points of our setting are the following: firstly, the cost is onotonically decreasing with respect to the population distribution, naely agents are attracted toward congested areas. A large part of the MFG literature focuses on the study of systes with copetition, naely when the coupling in the cost is onotonically increasing; this assuption is essential if one seeks for uniqueness of equilibria, and it is in general crucial in any existence and regularity arguents, see, e.g [6], and references therein. On the other hand, odels with aggregation like have been considered in few cases, see [3, 4, 5]. Secondly, the state of a typical agent here is the whole euclidean space. Usually, the analysis of is carried out in the periodic setting, in order to avoid boundary issues and the non-copactness of. Few investigations are available in the truly non-periodic setting: see [30] for tie-dependent probles, [2] for the case of bounded controls, [7] for soe regularity results and [3] for the Linear-Quadratic fraework. We observe that the non-copact setting is even ore delicate for stationary ergodic probles like : a stable long-tie regie of a typical player is ensured if the Brownian otion is copensated by the optial velocity v s. In other words, if a force that drives players to bounded states is issing, dissipation eventually leads their distribution to vanish on the whole. This phenoenon is ipossible if the state space is copact. The ain issue here is that the behaviour of the optial velocity v s = H u is a-priori unknown, and depends in an iplicit way on V and the distribution itself. Let us ention that even without the coupling α, the ergodic control proble in unbounded doains has received a considerable attention, see e.g. [5, 20, 2] and references therein. The first assuption we ake is then the following: V C 0,η loc R for soe η 0, ], and C V ax{ x C V, 0} b V x C V + x b 2 2

3 for soe C V > 0 and b > 0. ote that V represents the spatial preference of a single agent; if it grows as x, it discourages agents to be far away fro the origin. At the PDE level, this will copensate the lack of copactness of. The requireent of V to be positive is just technical. Before stating the assuptions on H and α, let us coent on the viewpoint we will rely on. Since the early work [24], it is known that is associated to a control proble of PDE see also [28]. Let K,M be the set of, w L W,γ L γ satisfying ϕ dx = w ϕ dx ϕ C0, R 3 dx = M, 0 a.e.. We define the energy E, w := L w dx + V x dx α+ dx, 4 R R α + L is the Legendre transfor of H, i.e. Lq = H q = sup p [p q Hp], q. 5 ote that the expression L w/ has to be intended equal to zero if, w = 0, 0 and + if, w 0, 0 and 0: L / thus reads as the Legendre transfor of H. For any > 0, M > 0 fixed, we will look for solutions of as iniizers of E, that is we will consider the iniization proble e M = inf,w K,M E, w. 6 Once a iniizer, w is available, one can indeed construct a triple u, λ, solving. To coplete this progra, we will assue that H C 2 \ {0} is strictly convex, and there exist soe C H > 0, and γ > such that C H p γ C H Hp C H p γ p. 7 ote that fro 7 and 5, there exists C L > 0 such that C L q γ Lq C L q γ + C L, where = γ γ. We will also always assue that the power growth α of the coupling and the growth of the Lagrangian satisfy 0 < α < γ. 8 Moreover, we will require γ <, i.e. γ >. 9 Soe coents on 8 and 9 are in order. Condition 8 is necessary for 6 to be well-posed. Indeed, consider any 0, w 0 K,M such that 0 has copact support. An easy coputation shows that if α > /, then Eσ 0 σ, σ + w 0 σ as σ 0, so E is not bounded fro below on K,M. We show that 8 is indeed sufficient for e M to be finite, and allows to look for ground states of. This will be accoplished by a study of the Sobolev regularity of the Kologorov equation, see in particular Section 2.2. ote that the critical case α = / is ore delicate, and require additional analysis. We also ention that another critical exponent is intrinsic in : if α > /, one has to expect non-existence of solutions see [3]. 3

4 As for condition 9, it guarantees Hölder regularity of iniizers. Basically, the integrability of a iniizing sequence n, w n of E increases as increases. By Corollary 2.9, iniizers are Hölder continuous if 9 is in force. In the case <, one has to expect just L p a-priori regularity of, and soe additional work is required to conclude the existence of a classical solution to. This study is atter of a work in progress. The ai of this work is two-fold. Firstly, for any fixed > 0, we prove the existence of classical ground states of. Secondly, we study the behavior of solutions in the vanishing viscosity liit 0. Copactness of iniizing sequences of E is guaranteed by the coercivity of V and Hölder regularity. Then, the existence of a solution u, λ of the HJB equation in is obtained by considering another functional with linearized coupling around the iniizer and the associated dual functional in the sense of Fenchel-Rockafellar as in [9]. One has to take care of the interplay between u and as x. The behavior of value function u is controlled by Bernstein estiates, while a control on the decay of coes fro the assuptions on V. ote that we will consider classical solutions to this syste with a slight abuse of terinology, that is u, C 2,β W,p, for soe β 0, and all p >. The existence result is stated as follows. Theore.. Suppose that 2, 7, 8 and 9 hold. Then, for every > 0 and M > 0, i There exists a iniizer, w K,M of E, that is E, w = inf E, w.,w K,M ii For any iniizer, w K,M classical solution to. of E, there exists u, λ such that u, λ, is a In the second part of the work, we analyze the behavior of the triple u, λ, coing fro a iniizer of E as 0. Fro the viewpoint of the odel, this aounts to reove the Brownian noise fro the agents dynaics. Heuristically, if the diffusion becoes negligible, one should observe aggregation of players induced by the decreasing onotonicity of coupling in the cost towards inia of the potential V, that are the preferred sites. Fro the PDE viewpoint, this phenoenon should appear in the concentration of the density around a finite nuber of points. In order to bring as uch as possible inforation to the liit, we consider an appropriate rescaling of, u, naely = α α +x, ũ = αγ α u α +x ux, 0 for all > 0. The rescaling is designed so that ũ, solves a MFG syste where the nonlinearities have the sae behavior of the original ones, i.e. H p γ as p, but the coefficient in front of the Laplacian is equal to one for all, see 73. Moreover, the couple ũ, is associated to a iniizer of a rescaled energy E, see 69. It turns out that in this rescaling process, the potential V becoes V = α α V α, and vanishes locally as 0. Therefore, as one passes to the liit, the potential cannot copensate anyore the lack of copactness of, and the convergence of in L has to be proven by other ethods. Heuristically, the aggregating force should be strong enough to overcoe the dissipation effect, but the clustering point can be hard to predict by lack of spatial preference. This is why we also have to translate in 0 by x. We will select x to be the iniu of u : heuristically, being u the value function, this is the point where ost of the players should be located. In order to recover copactness for the sequence, we ipleent soe ideas of the celebrated concentration-copactness ethod [27]. This principle states intuitively that if loss of 4

5 copactness occurs, splits in at least two parts which are going infinitely far away fro each other, that is χ BR 0 + χ \B 2R 0, with R, χ BR 0 a and χ \B 2R 0 M a for soe a 0, M a third possibility ight happen, but it is easily ruled out here by local estiates. This induces a splitting in the energy E, that is inf E inf E + inf E. 2 =M =a =M a One then exploits a special feature of E, which is called sub-additivity: inf E < =M inf E + inf E, =a =M a that akes 2 ipossible. While sub-additivity is easy to prove for E see Lea 4.5, the splitting 2 requires technical work, in particular due to the presence of the ter L w/ in E, that becoes increasingly singular as approaches zero a siple cut-off as in is not useful. The property 2 is proven in Theore 4.6. It relies on the Brezis-Lieb lea and a perturbation arguent. The L convergence of enables us to obtain the full convergence of ũ, to a liit MFG syste. By a unifor control of the decay of as x, that coes fro a Lyapunov function built upon ũ, and energy arguents, we are also able to keep track of x. In ters of the original density, x is the point around which ost of the ass is located. For technical reasons, we require additional hypotheses on V and H. We will assue that the potential V has a finite nuber of inia and polynoial behavior, that is for soe ˆb > 0, x j, j =,..., n, V x = hx n j= x x j ˆb, C V hx C V on. 3 note that nˆb = b, in assuption 2. Moreover, we will strengthen the assuptions on the Hailtonian of the syste: in particular we suppose that H C 0,γ loc 4 and we will need that either H is hoogeneous or the potential has a growth at infinitity controlled by γ: either Hp = C H p γ or b = nˆb < γ 2. 5 Actually, ost of the analysis can be carried over under the ore general assuptions 7, see the discussion at the beginning of Section 4. The second ain result of this work is stated as follows: Theore.2. Assue that 3, 7, 4 and 5 hold. Let u, λ, be as in Theore.. Then, there exist sequences 0 and x, such that for all η > 0 there exists R and 0 for which for all < 0, dx M η, 6 and for soe J =,..., n, C > 0, x x R α x x J C n α. 7 Moreover, α γ α u γ α +x, αγ α γ α +x converges respectively in Cloc R and in C loc L p for all p to a classical solution of u + H 0 u + λ = α div H 0 u = 0 8 = M, 5

6 where λ = li 0 α α λ and H 0 p = li 0 H p. Theore.2 describes the liiting behaviour of u, : in particular, concentrates around inia of V. Moreover, it states the existence of a solution to an ergodic MFG syste with decreasing coupling α, without coercive potential. In ters of the MFG odel, a stable longtie equilibriu is possible by the intrinsic aggregating force even without a spatial preference of the players. ote that in the very special case of quadratic Hailtonian, that is Hp = p 2 /2, the change of variables vx = C exp ux/2, x = v 2 x, transfors the syste into { 2 v + V x λv = v 2α+ v 2 = M. Concentration phenoena of positive ground states as 0 of the previous seilinear equation have been widely studied, in different settings. We just recall [8] where a siilar proble with ass constraint in diension = 2 and with α = is considered, see also references therein. We ake a final reark regarding soe results in [5], where a first-order one-diensional proble with aggregation is considered. In their odels, it happens that the focusing effect ay prevail the spatial preferences of the agents, naely the density has support far fro inia of V. It sees that such explicit solutions are not selected by liits of our viscous odels, that can be only singular easures. The paper is organized as follows. In Section 2 we provide soe regularity results on the Hailton-Jacobi-Bellan and Kologorov equation that will be used thoroughly. Mainly, they coe fro results in [5, 0, 3]. Section 3 is devoted to the proof of a solution to the MFG syste when > 0 is fixed. In Section 4, we study the vanishing viscosity liit 0: we prove copactness of the rescaled density and prove the full convergence of, ũ to ground states of a MFG syste which is potential-free. Acknowledgeents. The authors are partially supported by the Fondazione CaRiPaRo Project onlinear Partial Differential Equations: Asyptotic Probles and Mean-Field Gaes and PRAT CPDA57835 of University of Padova Mean-Field Gaes and onlinear PDEs. A.C. is partially supported by the IdAM-GAMPA project Tecniche EDP, dinaiche e probabilistiche per lo studio di problei asintotici. 2 Soe preliinary regularity results otations. For every p, p = p p will be the usual conjugate exponent of p. For all R > 0, x, B R x := {y : x y < R}. We will denote by ω := B 0. Finally, C, C, K, K,... denote positive constants we need not to specify. The assuptions on H guarantee the following see, e.g., [2, Proposition 2.] Proposition 2.. There exist C L, C, C 2 > 0 depending on C H and on γ such that p, q, i L C 2 \ {0} and it is strictly convex, ii 0 C L q γ Lq C L q γ +, iii Lq q Lq C q γ C, iv C q γ C Lq C q γ +. v C 2 p γ C 2 Hp C 2 p γ +. We will use the following standard result on Hölder functions vanishing at infinity. 6

7 Lea 2.2. Suppose that 0, C 0,θ c h, for soe θ, c h > 0, and dx <. Then, x 0 as x. Moreover, if dx < η x R for soe η, R > 0, then where C > 0 depends only on c h,. θ ax x Cη θ+, 9 x R Proof. By contradiction, suppose that there exists δ > 0 and a sequence x n such that x n > δ for all n. We ay also assue that x n+ x n + for all n. By the Hölder regularity assuption, provided that x B r x n, and r θ for all n. Then, dx n x x n c h x x n θ δ 2, δ 2c h. Choose r = in{, B rx n dx n θ δ 2c h δ 2 B r0 = + }, so that B r x n B r x = that is ipossible. As for the second part, let M := ax x R x = x, x R note that such a axiu is achieved as a consequence of the first part of the lea. As before, x x c h x x θ M 2 /θ. M for all x B r x, where r = 2c h Therefore, and 9 follows. η > x R dx M 4 B r x = M /θ M 4 B 0, 2c h We recall the following well known result, proved in [8, Theore ]. Theore 2.3. Let f n f a.e. in and assue that f n L p C for all n and for soe p [, +. Then li n [ f n p L p f n f p L p ] = f p L p. Fro classical elliptic regularity, we have the following result. Proposition 2.4. Let p > and L p be such that ϕ dx K ϕ L p for all ϕ C0 for soe K > 0. Then, W,p and there exists C > 0 depending only on p, such that L p C K. 7

8 Proof. Fix any R >. Let ψ C 0 B 2 0, ϕrx := ψx so, ϕ C 0 B 2R 0 and vx := Rx on. Then, B 20 v ψ dx = R2 B 2R 0 ϕ dy KR2 ϕ p dy B 2R 0 = KR +/p ψ p dx B 20 /p /p KR /p ψ W,p B 20. Hence, by [, Theore 6.], v W,p B 0 and there exists a constant C, depending on p but not on R, such that Therefore, v L p B 0 v W,p B 0 CKR /p + v L p B 20. /p p dy = /p B R 0 B 0 v p dx /p C Letting R, we get that L p R n and the desired estiate. K + /p v p dx B 20 /p = CK + R L p B 2R The Hailton-Jacobi-Bellan equation on the whole space In this section we provide soe a-priori regularity estiates and existence results for Hailton- Jacobi-Bellan equations in the whole spaces of ergodic type. In particular we will consider failies of Hailton-Jacobi-Bellan equations u n + H n u n + λ n = F n x f n x on 20 where f n C 0 L, λ n R are equibounded in n, that is λ n λ and f n C f with C f independent of n. Moreover H n is for every n an Hailtonian which satisfies 7, with constants γ and C H independent of n; finally, there exists C F 0 and b 0 independent of n such that C F ax{ x C F, 0} b F n x C F + x b n and x. 2 ote that, differently fro assuption 2 for the potential V, the function F n can also be bounded, if b = 0. Theore 2.5. Let u n C be a sequence of viscosity solutions of the HJB equations 20. Then there exists a constant K > 0 depending on C H, C F, C f, γ,, λ such that u n x K + x b γ, 22 where b 0 is the growth of F n appearing in 2 and γ is the growth of H n appearing in 7. Proof. Without loss of generality we ay consider H n p = C H p γ for all n and p. Indeed, every v n solves u n + C H u n γ + λ n = F n x f n x + C H u n γ H n u n on, and since C H u n γ H n u n C H by 7, we can redefine f n to include C H u n γ H n u n, which then satisfies the bound f n C f + C H. ote that it is also sufficient to prove the theore for u n C 2, as the general case follows by approxiation. 8

9 We first clai that if v C 2 B 2 0 satisfies for soe k > 0, then we have for any r [, ], v + C H v γ k on B 2 0 v Lr B 0 C, 23 where C depends only on k, C H, γ,, r. If r [,, this is proven in [22, Theore A.]. The case r = follows by classical elliptic regularity, since if r in 23 is large enough, then v is bounded in L q B 3/2 0 for soe q >, and the stateent follows by Sobolev ebeddings. In view of these considerations, the gradient bound 22 easily follows if b = 0. For the case b > 0, fix x 0, and let δ = + x 0 b/γ. Let Then, v n solves Since δ, v n y := δ 2 γ γ un x 0 + δy on. v n + C H v n γ = δ γ F n x 0 + δy f n x 0 + δy λ n. δ γ F n x 0 + δy f n x 0 + δy λ n C F 3 + x 0 b + C f + λ + x 0 b C for all y B 2 0 by 2 and the bound on f n. Therefore, by the first clai, v n L B 0 C, for all n. In particular, choosing y = 0, and the desired estiate follows. u n x 0 = δ γ vn 0 C + x 0 b/γ, Moreover, we prove the following a-priori estiates on bounded fro below solutions to 20. Theore 2.6. Let u n C be a faily of uniforly bounded fro below viscosity solution to 20, that is for which there exists C > 0 such that u n C for every n. If b = 0 in 2, we oreover assue that there exists η > 0 and R > 0 independent of n such that F n x f n x λ n > η > 0, for all x > R. 24 Then there exists C > 0 such that u n x C x + b γ C, n, x, 25 where b 0 is the growth power appearing in 2 and γ is the growth power appearing in 7. Proof. The proof is based on the sae arguent as in [5, Proposition 3.4], we sketch it briefly for copleteness. Since u n is bounded fro below we can assue u n 0, up to adddition of constants. We assue by contradiction that 25 does not hold. Then there exist sequences x l and u nl, such that x l > 2R, x l +, and un l x l 0. Let a l = x l 2 and we define the function x l + b γ v l x = a + b γ l 9 u nl x l + a l x.

10 By Theore 2.5, we get that u nl x K+ x b γ. Therefore, v l, v l are uniforly bounded. Moreover, v l is a solution to a b γ l v l + H nl a b γ l vl + λ nl = F nl x l + a l x f nl x l + a l x. In particular, recalling 7, we get that v l is a supersolution to a b γ b l v l + C H v l γ a b l λ nl + F nl x l + a l x f nl x l + a l x. ote that, for every l sufficiently large, by 2 and by 24 in the case b = 0 the right hand side of the equation a b l λ nl + F nl x l + a l x f nl x l + a l x > 0 for x such that x. Moreover, passing eventually to a subsequence, we get that v l v locally uniforly in n and 0. So v is a supersolution to C H v γ η > 0 in B0, with hoogeneous boundary conditions since v 0. By coparison, recalling the explicit forula of the solution to the eikonal equation f γ = C in B0, with hoogeneous boundary conditions, we conclude that vx C γ x for all x such that x. Moreover, by unifor convergence, we get that, eventually enlarging C and taking l sufficiently large, v l x C γ x for all x with x, in particular v l 0 C γ. Recalling the definition of v l, we get that v l 0 0, which yields a contradiction. a b γ b l Define λ n := sup{λ R : 20 has a solution u n C 2 }. Theore 2.7. Assue that for every n the function F n f n is locally Hölder continuous and bounded fro below uniforly in n. i λ n <, for every n, and there exists, for every n, a solution u n C 2 to 20 with λ n = λ n. Moreover λ n := sup{λ R : 20 has a subsolution u n C 2 }. ii If F n satisfies 2, with b > 0, then, for every n, the solution u n to 20 with λ n = λ n is unique up to addition of constants and satisfies 25. iii If F n 0, and there exists δ > 0 independent of n such that li sup f n x + λ n < δ < 0, 26 x + then for every n there exists a solution to 20 with λ n = λ n which satisfies 25 with b = 0. Proof. i. The proof of this result can be obtained by a straightforward adaptation of the proof of Theore 2. in [5], using the a-priori estiates on the gradient given in Theore 2.5. Observe that actually in [5] it is required a stronger assuption on F n f n, that is it is required it is locally Lipschitz continuous. This assuption is used to derive a-priori estiates on the gradient of solutions by using the so called Bernstein ethod see Appendix A in [5], which depends also on the L nor of F n f n. In our case we can weaken this assuption to just Hölder continuity so still ensuring classical elliptic regularity since we are using a-priori estiates on the gradient given in Theore 2.5, which depends only on the L nor of F n f n, and are obtained in [22] by the so called integral Bernstein ethod. ii. For the proof we refer to [20] see also [5] and [2]. In particular in [20], it is proved that u n is bounded fro below. By looking at the proof, it is easy to check that, due to the unifority in n of the nors of coefficients, the bound can be taken independent of n, then by Theore 2.6, we get the estiate on the growth. 0

11 iii. By adapting the arguent in [5, Theore 2.6], we get that there exists a bounded fro below solution to 20 with λ n = λ n, with bound unifor in n. Then using Theore 2.6, we get the estiate on the growth. We give a brief sketch of the proof of the existence of a bounded fro below solution. For every R > 0, we consider the ergodic proble { u R n + H n u R n + λ R n = f x < R u R 27 n x + x R. Using the result in [4], we get that for every R > 0 there exists a unique λ R n and a unique up to addition of constant solution u R n C 2 B R. First of all we clai that li R λ R n = λ n. It is easy to check that if R > R, then λ R n λ R n, and oreover that λ R n λ n. So, the sequence λ R n is converging as R + to soe λ n λ n. Moreover, by the sae arguent as in Theore 2.5, we get that for every copact K, there exists a constant C > 0 such that u R n C in K for every R sufficiently large and for all n. Without loss of generality we can assue that u R n 0 = 0 for every R. So, using the gradient bound, and elliptic regularity, we conclude that u R n is bounded in C 2 K by soe constant independent of R. Hence, by Ascoli-Arzelà Theore, and via a diagonalization procedure, we get that u R n converges locally in, with u n C 2. Moreover, u n is a solution to 20, with λ = λ n. Recalling the characterization of λ n and the fact that λ n λ n, we conclude that λ n = λ n. Then, we consider x R n B R such that u R n x R n = in x R u R n. Recalling that u R n is a solution to 27, we get by coputing the equation at x R n and by recalling that H n 0 0, that λ R n + fx R n H n 0 + λ R n + fx R n 0. Using condition 26, and recalling that λ R n λ n, we get that there exists a copact set K independent of R and of n and R 0 > 0 such that for all R > R 0, x R n K. Recalling that u R n 0 = 0 and u R n C in K with C independent of n, R, we conclude that u R n x R C for soe constant C independent of n, R. But, this iplies, since u R n x u R n x R n for every R, that passing to the liit u n x C, with C independent of n. 2.2 A-priori estiates for the Kologorov equation In this section we do not assue > and we provide general a-priori estiates for couples, w L W,q L such that x = M and + div w = 0 where q = { + <. Lea 2.8. Let β q q, for q <, and β < + for q. We define r β as follows r = + β. 29 Then, there exists a constant C, depending only on and β, such that W,r C γ CL C γ w L γ 28 dx + M γ L β 30 w dx + M γ L β, where C L is the constant appearing in Proposition 2.. We now assue that < β < + γ. 3

12 Then, there exists δ > 0 such that +δβ L β C M +δβ γ w γ dx where the constant C depends only on γ,, and β. CC L M +δβ γ L w dx, 32 Proof. Since W,q, by Sobolev ebedding and interpolation, we get that L β. Using + div w = 0, we get for all ϕ C0, ϕ dx = w ϕ dx. Using Holder inequality, recalling 29, we obtain w ϕ dx w ϕ dx R Therefore, we get that for all ϕ C 0, ϕ dx γ γ w γ dx w γ dx γ L β ϕ L r. γ L β ϕ r. We apply then Proposition 2.4 and we obtain that W,r and that there exists a constant C, depending only on r, such that L r C γ w γ dx γ L β. 33 Fro this inequality, using Proposition 2. and recalling that by interpolation, since L = M, L r γ L β M, we conclude the desired inequality 30. ow we fix η such that η = r = ote that, by a siple coputation using 29, we get η β = by 3, we conclude that that η > β. = M, we get + r. + β β γ, therefore, By Gagliardo irenberg inequality, and recalling that L η C + L r M Since η > β, by interpolation we get that there exists θ > such that θ L β L η M θ. Actually θ = +. β + β So, we substitute in 34 and 33 and we get, elevating both ters to +, + θγ L β C M θ + R w γ γ dx γ L β. 35 ow, since θ >, by 3, we get θ + γ γ = β β = β + β [ ] γ β + β > 0. 2

13 Therefore we deduce 32 fro 35 with δ = β [ γ + β ]. 36 Corollary 2.9. For every r < q, there exists C > 0 depending on, and r such that W,r C C L L R w dx + γ M. 37 γ Moreover, if > so q >, then C 0,θ and C 0,θ C C L L R w dx + γ M. 38 γ Proof. For q equivalently, we fix r < q and we choose β which satisfies 29 for such r. By Sobolev ebedding theore, W,r is continuously ebedded in L β. So, there exists C depending on and r such that L β C W,r. Using inequality 30, we get L β C R w γ dx + γ M. γ If we substitute again in 30 we get W,r C R w γ γ dx + γ M. In particular for q >, we can choose r > and by Sobolev ebedding theore we get that there exists θ = r and a constant C > 0 depending on and r such that C 0,θ C R w γ dx + γ M γ C C L L R w dx + γ M. γ For q <, we fix r < q, and choose the corresponding β in 29, that satisfies β < Hence we conclude again fro inequality 30.. Corollary 2.0. Assue that u, W,γ W,γ L solves the Kologorov equation in. For any β satisfying 3, there exists δ > 0 such that +δβ L β C u γ dx γ where C is a constant which depends only on, β, γ. Proof. We apply Lea 2.8 with w = H u. ote that w H u C H u γ +. So, since u L γ and L γ we obtain w L γ. 3

14 3 Existence of solutions to the MFG syste for > 0 3. A-priori estiates and energy bounds In this section, we provide bounds fro above and below for the energy E, assuring in particular that the iniu proble is well defined. Lea 3.. Let, w K,M. Then E, w K C α α 39 where C, K > 0 are constants depending only on, M, C L, γ, α, M, V. In particular there exists finite e M = inf E, w.,w K,M Proof. Recalling that V 0 and applying the estiate 32 with α = β, we get E, w L w dx α+ dx R α + C γ M +δ+α +α+δ L α+ α + +α L α+ + Cδ γ δ δ δ + α + where C is a constant depending only on, M, C L, γ, α and δ = [ ] γ α α. 40 Thefore, substituting in the energy, we get E, w C γ α α which gives the desired inequality. Lea 3.2. There holds α α α α, α + α inf E, w C 2 α K2 4,w K,M where C 2 > 0, K 2 are constants depending only on, M, C L, γ, α, V. Proof. We construct a couple, w K,M as follows. First of all we consider a sooth function φ : [0, + R which solves the following ordinary differential equation { φ r = φr + φr α φ0 = 2. Then, it is easy to check that 0 < φr 2 e r. We define x = Aφτ x, where A, τ are constants to be fixed, and wx = x. First of all we ipose M = xdx = A R τ φ y dy = A R τ C, 4

15 recalling that φ is exponentially decreasing. So A = Mτ C, where C = φ y dy. Observe also that α+ xdx = M α+ τ α C α+ φ α+ y dy = M α+ τ α C α+ C α 42 where C α = φ α+ y dy. We check, recalling that the growth condition 2, that the following holds y xv xdx = MC V φ y dy = C R τ τ b, 43 where K is a constant depending on, φ, C 0. Moreover, we copute, recalling that φ solves the ODE w γ = τ + γ M α C α α = γ τ γ γ + τ α M α C α α. 44 τ α We consider the energy at, w E, w = L w dx α+ dx + V xdx. R α + Using Proposition 2., and coputation 44 and 42, we get L w dx α+ dx R α + R C L w γ dx + C L M α+ dx R γ α + R = C L γ τ M γ + R M α C α τ α α+ dx + C L M α + τ = C L γ τ γ M + C L M α + γ γ α M α C α α+ dx = MC L + MCC α γ τ γ α + M α+ C α+ C α τ α + C L M. α+ We choose now τ such that τ = K α, where K is sufficiently large, in such a way that L w dx α+ dx C α α + CL M R α + where C is a constant depending on α, C L, M. Substituting this in the energy and recalling 43, we get the desired inequality. We get also a-priori bounds on iniizers and iniizing sequences. Proposition 3.3. Let, w K,M such that e M E, w η, for soe positive η. Then w γ dx C α α + K, 45 L α+ α C α + K, 46 α+ C 0,θ C 2 α + K, 47 for soe θ 0,, C, K positive constants which depends only on α,, V, C L. 5

16 Proof. ote that if, w K,M, then, θw K θ,m. Let, w K,M such that e M E, w + η, for soe positive η. Then we get, [ e M e θ M E, w + η E, θw = L w L θ w ] dx + η. 48 Recalling the properties of L in Proposition 2. we get [ L w L θ w ] dx MC L + C L θ γ We choose 0 < θ < so that θ γ > 0. Using 39 and 4, we get e M e θ M K + C θ α α γ α α = K K 2 + α α w γ dx. 49 K2 C 2 γ α α 50 α α C2 C θ γ = K + C α α, where, since C > C 2 and θ <, we get that C = C θ α α C2 > 0. Substituting 50 and 49 in 48, we conclude 45. Fro 45, using the estiate 32, with β = α + and δ + = γ α using 36, we get α+ L α+ C α w γ dx α C α C γ as it can be coputed α α + K α fro which we conclude 46. Finally, by 38 proved in Corollary 2.9, we deduce the a-priori bound on C 0,θ. 3.2 Existence of iniizers of E and solutions to for every > 0. We are now in the position to show existence of iniizers of the energy E in the class K,M for every, M > 0. Theore 3.4. For every > 0 and M > 0, there exists a iniizer, w K,M of E, that is E, w = inf E, w.,w K,M Moreover, for every iniizer, w K,M of E, there holds + x b L, w + x b/γ L, w L q for all q, 5 and there exist constants C > 0 and K, independent of, such that w dx + α+ L α+ α C α + K. 52 Proof. Let n, w n K,M be a iniizing sequence, that is E n, w n e M. This iplies that, choosing n sufficiently large, E n, w n e M +. Fro this we get n L w n dx + V x n dx E n, w n + n R α + α+ n dx e M + + α + α+ n. 53 6

17 Proceeding as in Proposition 3.3, we get that both w n γ dx and n C 0,θ C are γ n bounded by soe constant C depending on, e M, M,,. Eventually reducing θ and using Ascoli-Arzelá theore we get that, up to subsequences, n in C 0,θ. By interpolation between L and L, we obtain that n is also bounded uniforly in L α+, and hence, using again 53 and the assuptions on V, we have + x b L. Observe oreover that fro 53, we get that V x n dx C, where C is independent of n. Using the positivity of V and n and the assuption 2, we deduce fro that, for every R > C V, x R n xdx C in x R V x C C V R C V b. This iplies, by unifor convergence, that for every η there exists R such that xdx M η. B R 0 Using a diagonal arguent, we can prove that, eventually passing a subsequence, n in L and xdx = M. A consequence of the convergence of n is that there exists C > 0 such that n C on for all n. Therefore, C L w C γ n dx n L w n dx, R n that is bounded uniforly in n by 53 and Proposition 3.3. Thus, up to subsequences, w n w in L γ. ote that w dx w + x b/γ dx w γ /γ /γ dx + x b dx, and so we deduce that w L q for all q by interpolation and that w + x b/γ L. The convergence is strong enough to guarantee that, w K,M. Since E is lower seicontinuous on K,M,, w is a iniizer of E. Theore 3.5. Let, w K,M be a iniizer of E. Then, there exists u, λ such that u, λ, is a classical solution to u + H u + λ = α + V x div H u = 0 54 = M, and w = H u. Moreover, there exist constants K, K 2, C, C 2, C > 0, independent of, such that K C α α λ K 2 C 2 γ α α 55 u x C x b/γ+ C, u x C + x b/γ on. 56 7

18 Proof. Let, w be a iniizer of E. Define the space of test functions { A = A b,γ := ψ C 2 ψx ψx : li sup <, li sup x x b/γ x x b ote that we also have, for all ψ A, We clai that li sup x ψx <. x b/γ+ < }. 57 ψ dx = w ψ dx ψ A. 58 Indeed, consider a radial sooth cutoff function χx which is identically equal to one in B 0 and identically zero in \B 2 0. Set χ R x := χx/r; we have χ R C R and χ R C R 2 on for soe positive constant C. Since the equality = divw holds in the weak sense on, we ay ultiply it by χ R ψ with ψ A and integrate by parts to obtain χ R ψ + 2 ψ χ R + ψ χ R dx = w χ R ψ + ψ χ R. dx 59 B 2R B 2R ote that for soe positive C, w ψ dx C w + x b/γ dx <, by the integrability properties 5. Moreover, R x 2R ψ dx C + x b dx < + x b/γ+ ψ χ R dx C R x 2R R 2 dx C + x b/γ dx 0 as R, R x 2R because b/γ b. Reasoning in a siilar way, we also have that R x 2R ψ χ R and R x 2R w ψ χ R converge to zero as R. Equality 58 then follows by passing to the liit in 59. Therefore, the proble of iniizing E on K,M is equivalent to iniize E on K, where K := {w, L W,γ L γ : w, satisfies 5, 58, 0 and = M} As in [9, Proposition 3.], convexity of L iplies that, w is also a iniizer of the following convex functional on K: J, w = Ψ, w + V x α dx. We now ai to prove that sup{λm : ψ + H ψ + λ V x α on for soe ψ A} = in J, w. 60 w, K We proceed as in [, Theore 3.5]: setting L, w, λ, ψ := J, w + ψ + w ψ λ dx + λm, 8

19 we have in J, w = in,w K,w sup λ,ψ R A L, w, λ, ψ, where the iniu in the right hand side has to be intended aong couples, w L W,γ L γ satisfying 5. ote that L,, λ, ψ is convex, and L, w,, is linear. Moreover, since L,, λ, ψ is weak-* lower sei-continuous, we can use the in-ax theore see [7, Theore 2.3.7], to get in sup,w λ,ψ R A sup λ,ψ R A,w sup λ,ψ R A L, w, λ, ψ = in L R w in L,w R sup λ,ψ R A,w in L, w, λ, ψ = + V x α + ψ + w ψ λ dx + λm = w + V x α + ψ + w ψ λ dx + λm, where the interchange of the in and the integration is possible by standard results in convex optiisation. By coputation, in,w R L w + V x α + ψ + w ψ λ is zero whenever ψ H ψ λ + V x α is positive, and it is otherwise. Therefore, we have proven 60. By Theore 2.7, i, ii, there exists u C 2 such that u + H u + λ = V x α on, 6 and which satisfies the first estiate in 56. Moreover, by the gradient estiates in Theore 2.5 with F = V and f = α we have that and u x C + x b/γ+, u x C + x b/γ on u x H u x + λ + V x + α x C + x b on by 7, so u A. Thus, the supreu in the left hand side of 60 is achieved by λ, and it holds true that λ M = J, w. 62 This gives in particular 55, using Leata 3., 3.2, and 52, recalling Proposition 3.3. We now use 62, 6 and 58 with ψ = u to get 0 = that iplies L w + V x α λ dx = = L L w w w = H u on the set { > 0}. u + H u dx + H u + u w dx, Hence, the Kologorov equation + div H u = 0 holds in the weak sense, and by elliptic regularity we conclude that u, λ, is a classical solution to. We conclude with the proof of the first ain result of this paper. Proof of Theore.. Follows by Theores 3.4 and

20 4 Convergence to ground states and concentration In the second part of the work we analyse the behaviour of the syste as 0. Our ai is to show that concentration phenoena happen, and also to describe the asyptotic shape of the liits in ters of ground states of the syste. As stated in the introduction, we will suppose that fro now on 3, 7 and 5 hold. We ention that Proposition 4.3 and Theore 4.6 could be proved under the ore general assuptions of Theore.. On the other hand, if only 7 is satisfied, soe ore work is required to prove the estiates in Proposition 4.0 and then to conclude the full convergence result. So, we decided to pass to the stronger assuptions 7, 5 in order not to increase too uch the technicalities in the paper. 4. The rescaled proble Given x to be chosen later see 72 below, we consider the following rescaling y := α α y + x, wy := +α α w α y + x ũy := α γ α u α y + x ux λ := αγ α λ. 63 ote that, w K,M if and only if, w K,M =: K M, that is 0, dx = M and ϕ dx = w ϕ dx ϕ C0. We introduce the rescaled potential note that we do not translate by x here V y = α α h and the rescaled Lagrangian and Hailtonian L q := α α L By 7, there exists C > 0 such that So, we get that α α q, α y n j= α y xj ˆb, 64 H p = αγ α H α α p C L q γ L q C L q γ + C L α α, 66 C q γ C α α L q C q γ + C αγ α, C H p γ α α CH H p C H p γ, 67 C p γ α α C H p C p γ + C α α. li L q = L 0 q := C L q γ, 0 li H p = H 0 p := C H p γ uniforly in Moreover, 4 iplies that H is locally bounded in C 0,γ, and therefore that also H p H 0 p = C H γ p γ 2 p locally uniforly. We rescale the energy using 63: E, w = α α L w + V y + y α + α+ dy =: α α E, w, 69 20

21 where and the iniization proble 6 is therefore equivalent to ẽ M = y = α x, 70 inf E, w., w K M Recalling Leata 3., 3.2, we get that there exists C, C 2 > 0 and K, K 2, independent of, such that C K α α inf E, w C 2 K 2 α α. 7, w K M Let, w be a iniizer of E and u, λ, be the associated solution to 54, as defined in Theore 3.5. Then by 56, there exists x such that u x = in u x. 72 We fix such x in 63 and we define the corresponding y according to 70. An iediate corollary of Theore 3.4 is the following. Corollary 4.. For every > 0 there exist a iniizer, w K M to E. Moreover, there exists ũ, λ such that ũ 0 = ũ 0 and ũ, λ, is a classical solution to ũ + H ũ + λ = α + V y + y div H ũ = 0 73 = M, and w = H ũ. Finally there exist θ 0,, C, C, C 2 > 0 independent of such that C λ C 2, γ w dx + α+ L α+ C, C 0,θ C. 74 Proof. The corollary directly follows by Theore 3.4 and Theore 3.5, taking into account the previous rescalings 63, with the choice of x defined in 72. Estiate 74 is a consequence of 55 and 52. Finally the Hölder estiate is obtained by applying the sae arguents as in Corollary 2.9 and Lea A preliinary convergence result In this section, we provide soe preliinary convergence results, where we are not preventing possible loss of ass in the liit. First of all we need soe a-priori estiates on the solution to 73. Lea 4.2. Let, w K M be a iniizer to E as in Corollary 4., and let ũ, λ, be the associated classical solution to 73. Then, there exist constants K, C, K > 0, independent of, such that xdx K, R > and ũ y C + y nˆb γ, 75 B R 0 α+nˆb α y nˆb C and V y + y K α+nˆb α y nˆb

22 Proof. By the rescaling 63 and the choice of x in 72, we have that 0 = ũ 0 = in ũ. Since ũ is a classical solution to 73, we get, coputing the equation in 0, H 0 + λ α 0 + V y. Recalling that H 0 0 and V 0, we get that α 0 λ C 2 > 0, where C 2 is the constant appearing in 74. Since is uniforly bounded in C 0,θ, as stated in 74, and 0, there exists a constant K depending on θ, C, C 2 as in 74, such that for all R > we have xdx K > B R 0 Up to passing to a subsequence we get that locally uniforly, where 0 is in C 0,θ, with θ < θ and with Hölder nor bounded. Moreover, due to 77, by Fatou lea and unifor convergence in copact sets, we obtain M dx K > 0. Using the fact that is a iniizer of E, we obtain that there exists a constant, independent of, such that V x + y xdx C. This iplies, using 64 and 77, that for all R >, we have, letting K x i for every i, K α α C B R 0 V x + y xdx K α α n α y γ α R xj j= ˆb inf n x B R 0 j= K α α α x + α y x j ˆb α y γ α R K nˆb K α+nˆb α y nˆb K αγ α. 78 Therefore, 78 iplies that there exists a constant C > 0 such that α+nˆb α y nˆb C. Recalling 64 we get that there exist constants K 0, K, independent of, such that 0 V y + x K 0 α+nˆb α y nˆb + K 0 α+nˆb α x nˆb + K 0 αγ α K α+nˆb α x nˆb +, which gives 76. Finally, the gradient estiate in 75 coes fro 76 and Theore 2.5. Proposition 4.3. Let, w K M be a iniizer to E as in Corollary 4., and let ũ, λ, be the associated classical solution to 73. Up to subsequences, we get that λ λ, and ũ ū,, ũ ū, H ũ H 0 ū 79 locally uniforly, where ū 0 = ū0, and ū, λ, is a classical solution to { ū + H 0 ū + λ = α + gx div H 0 ū = 0 80 for a continuous function g such that 0 gx C on for soe C > 0. Moreover, there exists a 0, M] such that R n dx = a, 8 22

23 and C, K, κ > 0 such that ūx C x C, ū K on, e κ x xdx < Proof. First of all observe that, since V is a locally Hölder continuous function, then 76 iplies that, up to subsequence, V x + y gx, locally uniforly as 0, where g is a continuous function such that 0 gx C, for soe C > 0. Using the a-priori estiate 75, 74 and 76, and recalling that ũ is a classical solution to 73, by classical elliptic regularity theory we obtain that ũ is locally bounded in C,α in every copact set, uniforly with respect to. So, up to extracting a subsequence via a diagonalization procedure, we get that ũ ū, ũ ū, locally uniforly, and λ λ. Moreover, also H ũ H 0 ū locally uniforly. So, we can pass to the liit in 73 and obtain that ū, λ, ū is a solution to 80, which is classical by elliptic regularity theory. Observe that ū is a solution to ū + H 0 ū + λ = α + gx. By Theore 2.5, we get that there exists a constant K depending on sup g and λ such that ū K. Moreover, by construction ū 0. Since is Hölder continuous, and such that dx = a 0, M], by Lea 2.2, we get that 0 as x +. Therefore, we get that li inf x + α x + gx λ H 0 0 λ > 0. So, by Theore 2.6, recalling that by construction ū0 = 0 ūy, we get that ū satisfies ūx C x C 83 for soe C > 0. To conclude, consider the function Φx = e κūx. We clai that we can choose κ > 0 such that there exist R > 0 and δ > 0 with Φ + H 0 ū Φ > δφ x > R. 84 Indeed, since ū solves the first equation in 80, we get Φ + H 0 ū Φ κ λ κ ū 2 α Φ. Using 83 and 0 as x +, we obtain the clai. Reasoning as in Proposition 4.3 in [2], we get that e κū dx < +, which concludes the estiate 82. Reark 4.4. With estiates 82 in force, the pointwise bounds stated in [29, Theore 6.] hold, naely there exist positive constants c, c 2, such that 4.3 Concentration-copactness x c e c2 x on. In this section we show that actually there is no loss of ass when passing to the liit as in Proposition 4.3. In order to do so, we apply a kind of concentration-copactness arguent. First of all we show that the functional E, w enjoys the following subadditivity property. Let us denote ẽ M = in,w KM E, w. 23

24 Lea 4.5. For all a 0, M, there exists a constant C = Ca, M 0 depending only on a, M and the data not on, such that CM, M = 0 = C0, M, Ca, M > 0 for 0 < a < M and ẽ M ẽ a + ẽ M a Ca, M. 85 Proof. Let c > 0. For all adissible couples, w K M we have E c, cw = cl w cα+ α + α+ + cv x + y dx If, in addition, c >, we obtain for all, w K M ẽ cm = = ce, w ccα α + inf E, w E c, cw ce, w ccα,w K cm α + α+ dx 86 α+ dx. 87 Let now n, w n be a iniizing sequence of E in K M, such that E n, w n ẽ M + C2M 4 where C 2 M is the constant appearing in 7, which depends on M and on the data of the proble. Recalling that V 0 and L 0, and using 7, we get, for all sufficiently sall, α + α+ n dx 3C 2 R 4 + K 2 α α C 2 M 2 So, this estiate in particular holds for a iniizer of E. Therefore in 87 we get, taking, w to be a iniizer of E which exists by Corollary 4. ẽ cm E c, cw ce, w ccα α+ dx cẽ M cc α C 2M. α + R 2 Hence, we conclude that > 0. ẽ cm cẽ M cc α C 2 2 < cẽ M. 88 Using 88 twice with c = M/a and c = a/m a yields if a > M/2, otherwise it suffices to replace a with M a ẽ M < M a ẽa M a [ α ] M C2 a a 2 = ẽ a + M a ẽ a M a a < ẽ a + ẽ M a M a [ α ] M C2 a a 2 α ] C2 a. 2 [ M a Theore 4.6. Let ũ, λ, and ū, λ, be as in Proposition 4.3. Then, R n dx = M. 89 Proof. Let c > 0 be such that ce x such c exists by Reark 4.4. For R sufficiently large to be chosen later, we define { ce R x R ν R x = ce x x > R. 90 So in particular x ν R x for x > R. 24

25 We observe that as R + ν R xdx R n = Cω e R + Ce x dx 0. \B R 9 Since and H ũ H 0 ū locally uniforly, there exists 0 = 0 R such that for all 0, We observe that for all 0, + H ũ H 0 ū ce R x R ν R ν R x x. 93 Indeed, if x > R, then + 2ν R + ν R ν R, since ν R. On the other hand, if x R, then by ν R ce R + 2 ce R = ce R = ν R. Fro 93 we deduce that + 2ν R. 94 Moreover, since a.e. by Theore 2.3, recalling that dx = M, = a and R n using 9 and 94, we have that + 2ν R dx = M a + 2 ν R dx M a as R +, 95 R li α+ dx = α+ dx + li α+ dx R α+ dx + li + 2ν R α+ dx. 0 We consider the function, w L w. This is a convex function in, w. copute w L w = L w, so in particular by 66 we get We w C L γ C L α w L w C L w γ + C L α. 97 Moreover, L w = L w + w L w, therefore, again by 66 we get w C L γ C L α L w α We clai that, if we define w = H 0 ū, then C L w γ + C L α α. 98 E, w E, w + E + 2ν R, w w + 2 ν R + o + o R, 99 where o is an error such that li 0 o = 0. ote that V y+y dx = V y+y dx+ V y+y +2ν R dx 2 V y+y ν R dx. Recalling 65, the estiate 76 and the definition of ν R, we have 2 V y + y ν R dx CR nˆb+ e R. Hence we obtain V y+y dx V y+y dx+ V y+y +2ν R dx CR nˆb+ e R

26 By 96 we get α+ dx α + R α + α+ dx α + + 2ν R α+ dx + o 0 Finally, we estiate the kinetic ters in the energy. Splitting L w dx = L w dx + B R \B R L w dx, we proceed by estiating separately the two ters. Estiates in \ B R. First of all, note that by 82, 67 and 66, we get that L w = L H 0 ū C for coe constant C > 0, just depending on the data. Moreover, recalling that ce x, we get that, eventually enlarging C, \B R L By convexity of the function, w L w, we get that \B R L w + \B R + w \B R dx w dx C e x dx = o R. 02 x >R + 2ν R L \B R + 2ν R L w w + 2 ν R + 2ν R [ + 2ν R L w w + 2 ν R + 2ν R w w + 2 ν R + 2ν R dx 2ν R dx 03 ] w 2 ν R dx. 04 We recall that w = H 0 ū C by 82 and ν R Cν R by definition. Moreover, by 75 and 67, w = H ũ C [ + x nˆb γ ] γ C + x nˆb. Using the triangular inequality we get the following, where the constant C can change fro line to line, w w + 2 ν R + 2ν R H ũ + H 0 ū Cν R ν R + 2ν R + 2ν R C + x nˆb + 2ν R + C + 2ν R + Cν R C + x nˆb + 2ν R on \ B R 0, where we used respectively the fact that + 2ν R, ν R and that + 2ν R ν R. ow, using 98 and 05, we can estiate 03, and by 97 and 05 we can estiate 04. Indeed, we get + 2ν R L w w + 2 ν R 2ν R dx C + x nˆbν R xdx + 2ν R \B R \B R and \B R w [ + 2ν R L w ] w + 2 ν R w +2 ν R dx C + 2ν R + x nˆb γ νr xdx, \B R 26

27 because w C on. Therefore, we ay conclude, possibly enlarging C, that \B R L w dx + 2ν R L \B R w w + 2 ν R dx C + x nˆb γ νr xdx ν R \B R Finally, putting together 02 and 06, we have, choosing C suffficiently large \B R L w dx + + 2ν R L \B R \B R L w dx w w + 2 ν R + 2ν R dx C + x nˆb γ νr xdx. 07 \B R Estiates in B R. Again by convexity of the function, w L w, we get that B R L w + dx B R B R L L w w dx [ dx + w L w ] w w dx. 08 B R We now estiate 08. We recall that w H 0 ū K and also H ũ K by 82, for all 0 R. Then, using this fact and 97 and 98 and recalling 92, we get w L dx = L H 0 ū dx Ce R B R and B R B R w [ L H 0 ū] H u + H ũ H 0 ū dx Ce R. This iplies that for all 0 R L B R ow we observe that by 66, B R +2ν R L w dx B R L w [ w + 2 ν R dx C + 2ν R B R w dx Ce R. 09 w w + 2 ν R + 2ν R ] γ + +2ν R dx. By 94 we get that + 2ν R + 2ν R Ce R, eventually enalarging C. Moreover, reasoning as in 05, we get w w + 2 ν R + 2ν R H ũ + H ũ H 0 ū C + 2ν R + 2ν R where we used that ν R = 0 for x < R, that H ũ K, that by 94 H ũ H 0 ū +2ν R C by 93 and 92. So, we conclude that B R + 2ν R L +2ν R, w w + 2 ν R dx Ce R ν R 27

28 Putting together 09 and 0 we get, choosing C suffficiently large and for all 0 R, B R L w dx L w dx B R + + 2ν R L B R Therefore, suing up, 07, 00 and 0, we conclude w w + 2 ν R dx C e R. + 2ν R E, w E, w + E + 2ν R, w w + 2 ν R 2 +o CR nˆb+ e R C e R C + x nˆb γ e x dx. x >R M a Let now c R = M a+2 ν R dx. We have that c R as R + and c R <. In particular, c R + 2ν R, c R w w + 2 ν R K M a. By the coputation 86 we get c R E + 2ν R, w w + 2 ν R 3 c α R = E c R + 2ν R, c R w w + 2 ν R + c R + 2ν R α+ dx. α + Observe that by 74 there exists C independent of such that 0 + 2ν R α+ dx α+ + α+ + 2ν R α+ α+ C. Therefore, 3 reads E c R + 2ν R, c R w w + 2 ν R Cc R c α R α + c RE + 2ν R, w w + 2 ν R. Using this inequality, and recalling that E, w = ẽ M, we obtain fro 2 ẽ M ẽ a + ẽ M a + c R E + 2ν R, w w + 2 ν R 4 c α R +Cc R α + + o CR nˆb+ e R C e R C + x nˆb γ e x dx. By Lea 4.5, we get that x >R ẽ M ẽ a + ẽ M a Ca, M, where Ca, M > 0 for a < M and CM, M = 0. This iplies in particular that 0 > Ca, M c R E + 2ν R, w w + 2 ν R + C cα R α + + o CR nˆb+ e R C e R C + x nˆb γ e x dx. x >R By 7 we get that there exist K = KM a > 0 such that E + 2ν R, w w + 2 ν R K. So we obtain 0 > Ca, M c R K + C cα R α + +o CR nˆb+ e R C e R C + x nˆb γ e x dx. x >R If a < M, this gives a contradiction, choosing R sufficiently large and < 0 R. 28