UNIFORMLY ELLIPTIC LIOUVILLE TYPE EQUATIONS PART II: POINTWISE ESTIMATES AND LOCATION OF BLOW UP POINTS

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1 UNIFORMLY ELLIPTI LIOUVILLE TYPE EQUATIONS PART II: POINTWISE ESTIMATES AND LOATION OF BLOW UP POINTS DANIELE BARTOLUI ) & LUIGI ORSINA ) Abstract. We refine the analysis, initiated in [], of the blow up phenomenon for solutions sequences of the following two dimensional uniformly elliptic Liouville type problem in divergence form, diva u) = µ Keu Ω Keu in Ω, u = 0 on Ω. We generalize a result of Y.Y.Li [3] to the case A I. To this end, in the same spirit of [], we obtain a sharp pointwise estimate for blow up solutions sequences. As a consequence of these estimates we are able to prove that if {p,, p N } is the blow up set corresponding to a given solutions sequence, then deta)p j ) = 0, j =,, N. This characterization of the blow up set yields an improvement of the a priori estimates already established in [].. Introduction In this paper we will refine the blow up analysis, initiated in [], of the following uniformly elliptic Liouville type problem diva u) = µ Keu Ω Keu in Ω, u = 0 on Ω,.) with µ > 0, K a positive continuous function on Ω, and Ω R any open, smooth, bounded domain. We assume A : Ω IM, to be a symmetric uniformly elliptic matrix, with 000 Mathematics Subject classification: 35J60, 35J99. ) Daniele Bartolucci, Department of Mathematics, University of Rome Tre, Largo San Leonardo Murialdo n., 0046, Rome, Italy. ) Luigi Orsina, Department of Mathematics, University of Rome La Sapienza, Piazzale A. Moro n.5, 0085 Roma, Italy.

2 D.B. & L.O. A = {A i,j }, i, j =, and define A x) = i,j=, A i,j x). The regularity properties we will assume for A, will be always thought to hold with respect to this norm. Moreover, we denote with.,.) the standard Euclidean scalar product on Ω, with. the corresponding norm and for p [, + ], let. p be the usual L p Ω)-norm. Throughout this paper we assume that A satisfies: there exist 0 < σ m σ M < +, such that.) and set σ m ξ Ax)ξ, ξ) σ M ξ, ξ R, x Ω, λ m = min Ω deta max deta = λm..3) Ω In particular, we will denote by σ x) σ x) the eigenvalues of Ax) and define λx) = detax). In this introduction, we will refer to.) also as problem P ) A. Let us recall some facts concerning the conformal case, that is A I. The existence theory for P ) I with µ 8π is a subtle problem: two crucial steps in its analysis are the concentrationcompacess [6] and mass-quantization results [4] for solutions sequences of u = Keu in Ω, Keu Ω where K is any strictly positive continuous function on Ω. We refer to [] and references quoted there for a detailed discussion on this point. As an immediate consequence of these results, one obtains uniform a priori estimates for solutions sequences of P ) I, with µ in any compact subset of R\8πN, see for example [3]. Then, by means of variational methods, it can be proved that there exist at least one solution for P ) I, with µ 8π, 6π), K a strictly positive continuous function and Ω any smooth and bounded domain with nontrivial topology, see []. All these results has been generalized to the case were K vanishes at some point, which is relevant for physical applications arising in field theory, see [3]. Another method to prove existence of solutions for P ) I, based on degree type techniques, has been proposed by Y.Y. Li in [3] and then successfully implemented by.s. Lin and.. hen, see [9],[0]. From this point of view, the first step in the analysis of P ) I has been obtained in [3], where the author, among other things, derive a

3 3 subtle pointwise estimate on the profile of blowing up solutions, see Theorem 0.c) in [3]. As a consequence of this delicate estimate, Y.Y. Li obtains a sharp mass-quantization result, see Theorem 0.e) in [3]. The method applied in [3] see also [4]), is difficult to apply in case A I. Hence, in [], we adopted another approach and, arguing as in [3], we generalized Theorem 0.e) in [3] to the uniformly elliptic case. To be more precise, let us recall the following extensions, as stated in Proposition 4. and Theorem 4. of [], of Theorem 0.e) in [3]. We assume that A n satisfies, A n A, in Ω ), as n +,.4) and there exist 0 < σ m σ M < +, such that.5) In particular,.5) implies 0 < σ m ξ A n x)ξ, ξ) σ M ξ, ξ R, x Ω, n N. 0 < λ m = min Ω detan max detan = λ M < +, n N..6) Ω Proposition.. Assume that A n satisfies.4),.5). Let Ω 0 Ω be any open and relatively compact subset and w n a solutions sequence for diva n w n ) = K n e w n in Ω 0.7) lim inf min n Ω 0 for suitable > 0. Suppose in addition that, for some p Ω 0, max w n min w n, n N,.8) Ω 0 Ω 0 K n 0, lim sup sup K n + K n ) < +,.9) n Ω 0 e w n, n N,.0) Ω 0 K n e w n βδ p, weakly in the sense of measures in Ω 0..) Then, lim inf n K np) > 0 and β = 8π detap)). We denote by G the Green s function uniquely defined in.) below.

4 4 D.B. & L.O. Theorem.. Assume that A n satisfies.4),.5), and let u n be a solutions sequence for diva n u n ) = µ K n e u n n Ω K ne u in Ω, n.) u n = 0 on Ω. Suppose in addition that A n x) σ n I, x B δ Ω), for suitable δ > 0, σ n [σ m, σ M ], n N,.3) lim inf min n Ω 0 lim inf µ n lim sup µ n < +,.4) n K n > 0, lim sup n n Then, there exist a subsequence, still denoted by u n, such that: either i) u n converges uniformly in Ω ), or max K n + K n ) < +..5) ii) there exist a non empty and finite set blow up set) S {p,, p N } Ω and N sequences x j,n p j, j =,, N, such that, as n +, where d = 4 min{min i j u n x j,n ) = Ω max u nx) +, i =,, N,.6) x x j,n d inf x j,n x i,n, min inf distx n N i i,n, Ω)}, n N u n x) N β j Gx, p j ), in locω\s),.7) j= µ n K n e u n Ω K ne u n in the sense of measures in Ω, with β j = 8π detap j )), j =,, N. In particular µ n µ = 8π N j= N β j δ pj,.8) j= detap j )), as n +..9) Notice that the assumption lim inf min K n > 0 in.5) is needed to obtain the uniform bound n Ω Ω eu n, n N. In particular, the function w n = u n log Ω K ne u n, satisfies.0) locally around any blow up point, and then, Proposition. yields.9). Using Theorem., we was

5 5 then able to prove, see Theorem. in [], the following result which extends the above mentioned uniform estimates to the uniformly elliptic case. Theorem.3. Assume that A Ω, IM, ) satisfies.) and Ax) σi, x B δ Ω), for suitable δ > 0, σ [σ m, σ M ]..0) Let K be a Lipschitz continuous function such that and denote with u µ any solution for.). min K > 0, K < +, Ω ) For any ε > 0 there exist a constant ε > 0, such that: u µ ε, µ 8πλ m ε. ) Define κ 0 := max{k N k )λ M < kλ m }, and assume that κ 0. For any k κ 0 and ε 0, 4πkλ m k )λ M )) there exist a constant ε,k > 0, such that: u µ ε,k, µ [8πk )λ M + ε, 8πkλ m ε]. At this point, we may explain the new results discussed in this paper. First of all, notice that the uniform estimates of Theorem.3 are much weaker then the corresponding ones for P ) I, which indeed hold for any µ U R\8πN. For example, let us mention that we don t have any a priori estimates for solutions of.) neither for µ [8πλ m, 8πλ M ], nor for any µ λ M in case λ m λ M. As a consequence, the degree methods of [3], [9], [0], cannot be applied. In this paper, we will make a first step to solve this problem. In particular, our aim is twofold. First of all, we wish to generalize the pointwise estimates of Theorem 0.c) in [3] to the uniformly elliptic case. Here we face a major problem, since, as mentioned above, the method of [3] is difficult to apply in the non conformal case. Instead, we will argue as in [], where the authors are able to generalize Theorem 0.c) of [3] to the case where A n I, n N and the weight function K n vanishes at some point. Applying the method of [], we obtain

6 6 D.B. & L.O. Theorem.4. Assume that all the hypothesis of Proposition. are satisfied, and set x n p such that w n x n ) = max u n +, and t n = e w nx n ) 0, as n +. Define ξ n = Knxn) 8. Ω 0 Then, there exist > 0, such that w nx) log t n + ξ n t n x x n ), A n x n )x x n )) ), x Ω 0, n N..) As a consequence of Theorem.4, we obtain a version of Theorem 0. in [3], suitable to be applied to.), see Theorem.5 below. Next, we will obtain further restrictions on the values of µ for which blow up may occur, see Theorem.6. To this end, we take our inspiration by the conformal case, where, assuming A n I, n N, the blow up points vector {p,, p N } must be a critical point for a given explicit function depending on the Green s and Robin s functions for Ω. Such a characterization of the location of the blow up points, follows by the analysis of the Pohozaev identity, to be used together with the version of Theorem. corresponding to the case A n I, n N, see for example [5]. In order to restrict the allowed blow up values for µ, we will argue in the same way. On the other side, it turns out that the analysis of the uniformly elliptic case is more delicate. Indeed, the Pohozaev identity involves a leading unbounded term depending on A n, whenever A n 0, see 3.38) below. In particular, the knowledge of the concentration phenomenon as provided by Theorem., is too rough to describe the exact asymptotic behavior of this divergent term. We will overcome this problem by means of two main estimates. The first one is also a crucial step in the proof of Theorem.4, and generalizes the corresponding result obtained in []. Indeed, under the assumptions of Theorem.4, let d > 0 small enough to guarantee B d p) Ω, and define β n := K n e un. Then, see Lemma 3.5 below, one can prove that β n 8π detap) B d p) log, for some uniform constant > 0. Another crucial point, concerns the singular behavior of the Green s function for uniformly elliptic equations. In particular, in Proposition.3, we will obtain a sharp version of an estimate already established in []. As a consequence of Lemma 3.5 and Proposition.3, we will be able to evaluate the exact contribution in the Pohozaev type identity due to the above mentioned divergent term. In particular, see Lemma 3.6, we will obtain an estimate on the gradient of a given blow up solutions sequence

7 7 in a neighborhood of any blow up point. Then, we will be able to prove that the blow up points vector must satisfy a constraint determined solely by the determinant of A. learly, this is in sharp contrast with the corresponding situation in the conformal case. We are in position to state our result in the following, Theorem.5. Let u n be a solutions sequence for.),.4),.5), satisfying b) of Theorem., that is,.6),.7),.8),.9) holds for u n in Ω. Denote with S {p,, p N } the corresponding blow up set. Then a) each x j,n p j, j =,, N is the unique absolute maximum point for u n in the ball x x j,n d; b) c) u nx j,n ) log Ω K n e u n, j =,..., N, n N; max u nx j,n ) u n x i,n ), n N; i,j=,...,n d) for any j =,..., N, n N, and x x j,n d, it holds e) u n x) log + µ n K nx j,n) 8 e unxj,n) e u nx j,n) ) x x j,n ), A n x j,n )x x j,n )) )..) deta)p j ) = 0, j =,, N..3) Using Theorem.5e), we can improve Theorem.3. Indeed, let us define Σ := {x Ω : deta)x) = 0}, Λ := {λ R + : detax) = λ, x Σ}, and assume that the critical values of detax) form a discrete set, q N : Λ) = q, Λ = {λ,, λ q }, λ < λ < < λ q. Note that λ m, λ M defined in.3) actually satisfy λ m min Λ = λ, λ M max Λ = λ q.

8 8 D.B. & L.O. Finally, for any i =,, q, let us define Σ i = {x Σ : detax) = λ i }, J i = N, if Σ i ) = +, J i = {,, Σ i )}, otherwise, and q Γ = {β R β = 8π n i λ i, n i J i, i =,, q}. i= We have the following improvement of Theorem.3. Theorem.6. Assume that A and K satisfy the assumptions of Theorem.3. Let u µ be any solution for.). For any compact subset I R + \ Γ, there exist a constant I > 0, such that: u µ I, µ I. We will omit the proof of Theorem.6, since it is a straightforward consequence of Theorem.5e) and.9). Notice that, in view of Theorem.6, and arguing as in [3], it can be shown that the Leray-Shauder degree for the resolvent operator associated to P ) A, say d µ, is a well defined, continuous function for any µ R \ Γ, see [3]. A concluding remark, concerning.0) and.3), is in order. Those hypothesis are needed to exclude blow up at the boundary. In particular, assuming.0) and.3), the moving plane method together with Kelvin s transform can be used to obtain that any critical point for a suitable solution u, must lie outside a neighborhood of Ω which does not depend by u, but solely by K and Ω, see for example [5]. Although we believe that an analogous result should hold even if we don t assume.0) and.3), it seems hard to make the moving plane method work in our situation as well. This paper is organized as follows. In Section, we will prove Proposition.3, which deals with a regularity property of the Green s function relative to uniformly elliptic equations. In Section 3, we will prove Theorem.4 and Theorem.5.

9 9. Sharp gradient estimates for the Green s function of uniformly elliptic equations In this section, we will prove Proposition.3, which will be used in the analysis of uniformly elliptic Liouville type equations, see Section 3 below. For any y Ω, consider the Green s functions Gx, y), G 0 x, y) as the unique solutions for divax) Gx, y)) = δ x=y x Ω, Gx, y) = 0 x Ω, divay) G 0 x, y)) = δ x=y x Ω, G 0 x, y) = 0 x Ω, respectively. For a proof of the following well known fact see for example []:.).) Lemma.. Fix y Ω, and let G 0 be the unique solution for.). Then G 0 x, y) = πλy) log A y)x y), x y)) + γx, y),.3) where λy) = detay) and γx, y) Ω Ω). In [], we obtained the following Proposition.. See also Lemma 4. in [7]) For any y Ω, consider the Green s functions Gx, y), G 0 x, y) as the unique solutions for.) and.) respectively. a) Assume that A is uniformly Hölder continuous. There exist a unique uniformly bounded function g L Ω Ω), such that Gx, y) = G 0 x, y) + gx, y), and sup g 0,.4) Ω Ω with 0 depending only on σ m, σ M, A 0,β Ω), β and Ω. b) Assume that A Ω, IM, ), fix y Ω, r 0 0, ) such that B r0 y) Ω, and set D r0 = B r0 y)\b r0/)y). Then g y x):=gx, y):x Ω R, satisfies g y W,p Ω),γ loc Ω\{y}), for any p [, ), and γ 0, ), and there exist g > 0 depending only on σ m, σ M, A Ω), p, Ω

10 0 D.B. & L.O. and r 0 such that g y D r0 ) g. Moreover, for any τ 0, ), there exist > 0 and > 0 depending on τ, r 0, σ m, σ M, A Ω) and g y D r0 ), such that g y x) r τ +, x B r y), r r 0..5) Note that standard elliptic regularity theory implies g y,γ loc Ω\{y}), for any γ 0, ). We are going to improve Proposition., and obtain the following sharp estimates for the gradient of the function g above. More exactly we have, Proposition.3. Assume that A Ω, IM, ), and let gx, y) be the function defined in Proposition.. Then, g y,γ loc Ω \ {y}) and there exist 3 > 0 and 4 > 0 depending on σ m, σ M, A Ω), such that gx, y) 3 log x y + 4, x, y) Ω Ω..6) Proof. By the proof of Proposition. in [], we know that g is the unique solution for divax) gx, y)) = hx, y) x Ω, gx, y) = 0 x Ω,.7) where hx, y) = div[ax) Ay)) G 0 x, y))]. Since A is twice continuously differentiable, there exist ψx, y) L Ω Ω) and ψ = ψ σ m, σ M, A Ω) ) depending only on σ m, σ M and A Ω), such that hx, y) = ψx, y), x Ω, y Ω and sup ψx, y) ψ..8) x y x Ω,y Ω We apply the Green s representation formula for g., y) in Ω gx, y) = Gx, z)hz, y) dz. Ω It is well known, see for example [], that there exist > 0, depending on σ m, σ M, Ω and A Ω), such that Gx, z) / x z for any x and z Ω. Hence, dz gx, y) ψ, x Ω, y Ω. x z z y Ω

11 hoose R > diamω) and set v = x y, ρ = x y v. Then Ω dz x z z y B R 0) dz v z z = π R 0 0 drdθ ρ ρr cos θ + r = π π = π log ρ log cos θ)dθ + log R ρ cos θ + ρ ρr cos θ + R )dθ. Since R > diamω) > ρ, it follows that 0 0 gx, y) 3 log x y + 4, x, y) Ω Ω, with 3 = π ψ, and suitable 4 > 0, as desired. The result of Proposition.3 cannot be improved in general, as the following example shows Example Let Ax) = ar)i, where I is the identity matrix, r = x and a any R ) function such that A satisfies.). Fix y = 0 and denote with Gx, 0) the unique solution for.) in Ω = B 0). learly G Gr) is radial, and an explicit calculation yields Gr) = ar) a0) log r + πa0) πa0)ar) log r + r log t ȧt) a t) dt. It is readily seen that gx, 0) gr) = Gr) + πa0) log r satisfies.6). 3. Sharp pointwise estimates In this section, we will extend to the Euclidean uniformly elliptic case the pointwise estimates for Liouville type equations on two dimensional compact manifolds due to Y.Y. Li, see Theorem 0.c) and Theorem 0.3 in [3]. In this situation, the loss of the conformal structure creates a major problem, since it seems hard to apply the moving planes method as in [3]. Instead, we will obtain pointwise estimates by a different argument. We will follow the approach of [], where the authors extend Y.Y. Li s result to the case where singular data are included in the Liouville type equations. Then, we will be able to obtain a complete generalization of Theorem 0. in [3] to the uniformly elliptic case, see Theorem.5 below. First of all we will prove Theorem.4, which extends Theorem 0.3 in [3].

12 D.B. & L.O. Proof of Theorem.4. Observe that if w n x) is a solutions sequence for.7),.8) in Ω 0, with K n x) satisfying.9), then for any t > 0, w t) n y) = w n x n + ty) + log t is a solutions sequence for.7),.8) in Ω t := {y R x n + ty Ω 0 }, with K n t) y) = K n ty) satisfying.9). Hence, we may always assume that x n = 0, n N, and B 0) Ω. Next, using.8) and.) as in Lemma. of [3], we conclude that for any r [, ], there exist > 0, such that max Ω 0\B rx n) w n min Ω 0\B rx n) w n, n N. As a consequence, we juseed to prove.) in B 0). To achieve this result, we will closely follow the proof of Theorem. in []. It will be clear during the proof that we can assume A n x) Ax), n N, with A any symmetric matrix satisfying.). For any ζ R, we set Q ζ y) = y, A ζ)y), y R, 3.) and let Gx, y) be the unique solution for.). To simplify notations, we will also denote with > 0, various constants which may change, even from line to line, during the proof. By our hypothesis, we have w n 0) = max w n +, and t n = e w n0) 0, as n +. Ω Passing to a subsequence if necessary, and using Proposition., we can assume that K n K, in 0 locω 0 ), K0) > 0, and ξ n ξ := K0), as n +. 8 In particular, we see that.) holds in Ω 0, with β = 8πλ0). Define Ω n = { x R : x Ω 0, n N}, and We have the following v n x) = w n x) w n 0), x Ω n. 3.) Lemma 3.. The function v n defined in 3.) satisfies v n x) v x) = log + ξ Q 0 x)), in locr ),

13 3 and K0) R e v = 8πλ0). 3.3) Proof. Applying the Brezis-Merle type theorems for uniformly elliptic equations, see either Theorem 4. in [7] or Theorem 3. in [], together with standard elliptic regularity theory as in [4] pg. 58, we conclude that, passing to a subsequence if necessary, v n v in loc R ), where v is a solution for diva0) v) = K0)e v in R, e v, R v0) = 0, v x) 0, x R. Passing to the new coordinates x = T x, where T = A 0), that is T t A0)T = I, we conclude that v x) = vt x), satisfies v = K0)e v in R,, R v0) = 0, v x) 0, x R. 3.4) Using the well known result of hen-li [8], we have v x) = log + ξ x ). 3.5) Going back to the original coordinates, we see that the extracted subsequence converges to v satisfying 3.3). The same argument shows that any convergent subsequence relative to v n converges to v, and the conclusion follows. Define β n = K n e w n β = 8πλ0), as n ) B 0) Using Propositions. and.3 we will extend to the uniformly elliptic case some estimates concerning the decay of v n and v n, see for example [], pg 55. In the following, we will denote with O) various quantities bounded uniformly with respect to n and x.

14 4 D.B. & L.O. Lemma 3.. For any small δ > 0, there exist R δ >, δ > 0 and n δ N, such that ) βn v n x) πλ0) δ log Q0 x) + δ, R x, R > R δ, n > n δ, 3.7) v n x) δ, R x, R > R δ, n > n δ. 3.8) x In particular, for any < τ <, there exist τ > 0, such that sup ỹ τ K n ỹ)e v nỹ) dỹ τ. 3.9) Proof. n N ỹ Notice that 3.9) is a straightforward consequence of 3.7) and Lemma 3.. We apply the Green s representation formula for w n in Ω, and use.8),.) and Proposition. to obtain w n x) min w n = Gx, y)k n y)e w ny) dy + O) = B 0) y y 4πλy) log Q y x y) K ny)e w ny) dy + O), x Ω, n N. We scale the coordinates according to x = x, y = ỹ and obtain v n x) = 4πλ ỹ) log t n Q ỹ x ỹ) K n ỹ)e v nỹ) dỹ ỹ ỹ + log + min B 0) w n + O), x Ω n, n N. Using 3.), we see that, n N, v n 0) = 0 = 4πλ ỹ) log t n Q K n ỹ)e v nỹ) dỹ + log + min w n + O), ỹỹ) B 0) and then, v n x) = ỹ 4πλ ỹ) log Q ỹỹ) Q ỹ x ỹ) K n ỹ)e v nỹ) dỹ + O), x Ωn, n N. 3.0) In view of Lemma 3. and 3.6), for any ε 0 > 0, we can choose R 0 >, and ν 0 N, such that K n ỹ)e v nỹ) dỹ βn ε 0, and K n ỹ)e v nỹ) ε0, R > R 0, n > ν 0. ỹ R R ỹ 3.)

15 5 Next, by 3.) and.5), we see that, x R > R 0, and ỹ R, there exist ν N, such that 4πλ ỹ) log Q ỹỹ) Q ỹ x ỹ) 4πλ0) ε 0) log Q 0 x ỹ) + O), n > ν. Setting 0 = 6πλ M +, we conclude that for any ε 0 > 0 small enough, and for any 4πλ m n > max{ν 0, ν }, it holds β n v n x) πλ0) 0ε 0 ) log Q0 x) + R ỹ 4πλ ỹ) log Q ỹỹ) Q ỹ x ỹ) K n ỹ)e v nỹ) dỹ+o), 3.) x R > R 0. Setting D n = {R ỹ } \ { ỹ x x } and using the uniform ellipticity condition.5), together with 3.), we obtain ỹ x x ỹ x x R ỹ R ỹ 4πλ ỹ) log Q ỹỹ) Q ỹ x ỹ) K n ỹ)e v nỹ) dỹ = 3.3) πλ ỹ) log ỹ x ỹ K n ỹ)e v nỹ) dỹ + O) = πλ ỹ) log ỹ x ỹ K n ỹ)e v nỹ) dỹ + D n πλ ỹ) log πλ ỹ) log ỹ x ỹ K n ỹ)e v nỹ) dỹ + O) ỹ x x log From 3.) and 3.3), 3.4) we have v x) hoosing ε 0 > 0 such that ỹ x x ỹ x ỹ K n ỹ)e v nỹ) dỹ +O) πλ ỹ) log ỹ K n ỹ)e v nỹ) dỹ+ x ỹ dỹ + O) ε 0 πλ m log x + O). 3.4) ) βn πλ0) 0ε 0 log Q0 x) + ε 0 log x + O) = πλ m [ βn πλ0) 0 + πλ m ) ] ε 0 log Q0 x) + O). 6πλ M + 5 ) ε 0 < δ we conclude that 3.7) holds. 4πλ m

16 6 D.B. & L.O. We are left with the proof of 3.8). Using.8),.), Proposition.3 and the explicit expression of G 0 as given in.3), we have [ v n x) = πλ ỹ) and then, v n x) ỹ ỹ [ A ỹ) x ỹ) Q ỹ x ỹ) ] + g x, ỹ) K n ỹ)e v nỹ) dỹ + O ), 3.5) ] x ỹ + t n log K n ỹ)e v nỹ) dỹ + O ), x. 3.6) x ỹ β From 3.6), we may find ν N, such that for any n > ν, it holds n δ > 4 δ. Moreover, πλ0) observe that r + r log r r, 0 < r c, c > 0. Then, using 3.7), and arguing as in 3.3), 3.4), we may estimate the integral in 3.6) as follows, [ ] x ỹ + t n log K n ỹ)e v nỹ) dỹ 3.7) x ỹ ỹ B x x) [ x ỹ + log x 4 δ Hence, 3.8) follows by 3.6). ] [ dỹ x ỹ ỹ 4 δ + x + log B x x) dỹ x ỹ + K n ỹ)e v nỹ) dỹ x D n ] K n ỹ)e v nỹ) dỹ x D n x 3 δ + x δ x, x > R δ. 3.8) Remark 3.3. Using 3.7), 3.9), and arguing as in 3.3), 3.4) and 3.7), 3.8), we can prove other useful estimates. In particular, only minor changes are needed to show that ỹ x ỹ K n ỹ)e v nỹ) dỹ x, R δ < x, 3.9) ỹ and, using Proposition.3, g x, ỹ) K n ỹ)e v nỹ) dỹ ỹ ỹ log x ỹ K n ỹ)e v nỹ) dỹ + 3.0)

17 ỹ log x ỹ K n ỹ)e v nỹ) dỹ log x, x > R δ. 3.) The next Lemma provides a crucial estimate on the decay of v n x) in the region x log. In particular, we will need an estimate on v n better than 3.8), see 3.3). 7 Lemma 3.4. There exist ν N and > 0, such that, for any n > ν v n x) + β n πλ0) log Q 0 x), 3.) for any x log, and v n x) + for any x R and R > R δ. β n πλ0) A 0) x Q 0 x) x + log x, 3.3) Proof. Using 3.0), 3.) and.5), and arguing as in [], one can prove that 3.) holds, for any x log, and πλ0) ỹ A 0) x Q 0 x) A 0) x ỹ) Q 0 x ỹ) K n ỹ)e v nỹ) dỹ x, 3.4) holds, for any x > R δ. We refer to [] pg for details. Next, observe that A ỹ) x ỹ) A 0) x ỹ) πλ ỹ) Q ỹ x ỹ) πλ0) Q 0 x ỹ) K n ỹ)e v nỹ) dỹ 3.5) ỹ ỹ πλ0) πλ0) ỹ ỹ A ỹ) x ỹ) Q ỹ x ỹ) A ỹ) x ỹ) Q 0 x ỹ) K n ỹ)e v nỹ) dỹ+ A ỹ) x ỹ) A 0) x ỹ) Q 0 x ỹ) K n ỹ)e v nỹ) dỹ+ ỹ λ ỹ) λ0) K n ỹ)e v nỹ) dỹ x ỹ [ x ỹ Q 0 x ỹ) Q ỹ x ỹ) x ỹ 4 + A ỹ) A 0) x ỹ ] K n ỹ)e v nỹ) dỹ+

18 8 D.B. & L.O. ỹ ỹ λ ỹ) λ0) K n ỹ)e v nỹ) dỹ x ỹ [ A ỹ) A 0) + λ ỹ) λ0) ỹ x ỹ ] K n ỹ)e v nỹ) dỹ ỹ x ỹ K n ỹ)e v nỹ) dỹ x, R δ < x. 3.6) where we used frequently.5) and.6) and the last inequality follows by 3.9). Next, we recall the representation formula for v n 3.5), that is v n x) = ỹ [ πλ ỹ) A ỹ) x ỹ) Q ỹ x ỹ) Hence, using together 3.4), 3.5), 3.6), and 3.0), 3.), we have v n x) + ỹ ỹ β n πλ0) A 0) x Q 0 x) πλ ỹ) πλ0) A ỹ) x ỹ) Q ỹ x ỹ) ỹ ] + g x, ỹ) K n ỹ)e v nỹ) dỹ + O ). A 0) x Q 0 x) πλ0) A 0) x ỹ) Q 0 x ỹ) A 0) x ỹ) Q 0 x ỹ) K n ỹ)e v nỹ) dỹ+ K n ỹ)e v nỹ) dỹ+ g x, ỹ) K n ỹ)e v [ nỹ) dỹ + O ) x + t ] [ ] n + log x x + O) x + t n log x, x > R δ. Using the above Lemmas, we can generalize a subtle estimate, concerning the convergence rate of β n to 8πλ0), already obtained in [] in case A I. Indeed, we have the following Lemma 3.5. There exist a constant > 0, and ν N, such that, n > ν, β n 8πλ0) log ). 3.7)

19 9 Proof. We use Lemma 3.4 and and scale back to w n. Then, and w n x) = β n πλ0) log x + β n w n x) = πλ0) [ β n πλ0) A 0)x Q 0 x) ] log ) + O), 3.8) + O x + log ) x 3.9) for any log x. We will need also the estimates in Lemmas 3. and 3.. Indeed, taking the gradient of vx) in 3.5) and scaling back to w n, we have [ t ] n x w n x) + ξ t n x +, x Rt n, 3.30) for any n > n 0 large enough, while, by 3.8), scaling back to w n, we have w n x) δ x, x R, R > R δ, n > n δ. 3.3) We are going to analyze the Pohozaev identity on the shrinking ball Bn := { x log }. Indeed, multiplying the equation in.7) by x, w n x)), after some integration by parts, we obtain the following identity, [ ] x, ν) w n x), A n x) w n x)) ν, A n x) w n x))x, w n x)) dσx)+ B n w n x), [x A n x)] w n x))dx = B n x, ν)ψ n x)dσx) B n 3.3) [ψ n x) + x, log K n x))ψ n x)]dx, B n where ν = x x and ψ n = K n e w n. Using 3.9), we have [ x, ν) βn x 3 w n x), A n x) w n x))dσx) = B n 4π λ 0) Q 0 x) + O β n 4π λ 0) π 0 dθ νθ), A 0)νθ)) + O B n and ν, A n x) w n x))x, A n x) w n x))dσx) = B n log ) = B n β n πλ0) + O [ βn x 3 4π λ 0) Q 0 x) + O x )] dσx) = log ), x )] dσx) =

20 0 D.B. & L.O. β n 4π λ 0) π 0 dθ νθ), A 0)νθ)) + O log ) = β n πλ0) + O log ). Moreover, by 3.30) and 3.3), a straightforward evaluation shows that w n x), [x A n x)] w n x))dx = O log ). B n Using 3.8), we have [ψ n x) + x, log K n x))ψ n x)]dx + B n x, ν)ψ n x)dσx) = β n + B n O) ψ n x) + O log ) β n πλ0) = βn + O log ) β n πλ0). B d \ B n β n Since 4, as n +, we see that the desired conclusion follows substituting all the πλ0) estimate above in 3.3). As a consequence of Lemma 3.5, we will be able to conclude the proof of Theorem.4. On the other side, before accomplishing this task, we will derive another subtle consequence of Lemma 3.5. Indeed, for later use, see 3.43) below, we will need a slightly improved version of 3.3). This improved result is a consequence of 3.7). In particular, we obtain the following Lemma 3.6. v n x) + 4ξn A 0) x + ξnq 0 x) for any R x d and R > R δ. Proof. We apply the estimate 3.3) in Lemma 3.4, that is v n x) + β n A 0) x πλ0) Q 0 x) x + log ) x + log x + log x, 3.33) x, 3.34) for any R δ < x d. Notice that β n A 0) x 4ξ A 0) x n πλ0) Q 0 x) + ξnq 0 x) 4 A 0) x 4ξ A 0) x n Q 0 x) + ξnq 0 x) + β n πλ0) 4 A 0) x Q 0 x) x 3 + log ) x,

21 for all x > R δ, where we used Lemma 3.5. learly we have v n x) + 4ξn A 0) x + ξnq 0 x) v n x) + β n πλ0) A 0) x Q 0 x) and the conclusion follows by the last estimate and 3.34). + β n πλ0) A 0) x Q 0 x) 4ξn A 0) x + ξnq 0 x). With the aid of Lemma 3.5, we may conclude the proof of Theorem.4. Indeed, by 3.7) and 3.) in Lemma 3.4 we have v n x) + 4 log Q 0 x), n > ν, 3.35) for x log any log and ν N large enough. Scaling back to w n, we conclude that.) holds for x. As observed at the very beginning of the proof, we are left to show that such an estimate remains valid in the region R x log, for suitable R > 0, that is we need to prove that 3.35) holds for any R x log. Hence, in view of 3.7), fix R 0 > and n 0 N so large that the following estimates holds true: K n x)e v n x), x R, R R x 7 0, n > n 0. Since v n x) converges to v x) see Lemma 3.) uniformly in x R and v x) + 4 log Q 0 x) in x R, by taking n 0 larger if necessary, we find v n x) + 4 log Q 0 x) ) for x = R and n n 0. We construct ϕ ± n x) as follows: ϕ ± n x) = 8πλ0)G x, 0) + 4 log ± ζ x)), where G has been defined in.), and ζ is the unique solution for diva n x) ζ x)) = x 5 ζ x) = x in {R < x < log }, on {R < x < log }. 3.37) Then div A x) ϕ ± n ) = c x 5, {R < x < log }.

22 D.B. & L.O. Since ζ is uniformly bounded, by Lemma. and Proposition., we have and, in particular, ϕ ± n x) = 4 log Q 0 x) + O), {R x log }, v n x) ϕ + n x) = v n x) + 4 log Q 0 x) γ x, 0) g x, 0) ζ x)) + + g ) 0, x = R, x = log, R whenever R+ g ) R. On the other side, choosing larger if necessary, we have div A x) v n ϕ ± n ) ) x 7 x 5 0, R < x < log. The same argument, with the same choice of, works for v n ϕ n Then, by the maximum principle with reversed inequalities. and the conclusion of Theorem. follows. ϕ n x) v n x) ϕ + n x), R x log, As a consequence of Theorem.4, we can prove the first part of Theorem.5, which is a generalization of Theorem 0. in [3]. Moreover, using Lemma 3.6, we will be able to analyze the second Pohozaev identity in a neighborhood of the blow up set, and obtain.3). Proof of Theorem.5. The proof of a) is worked out as in [3]. We refer to [3] pg for details. Observe moreover that c) follows immediately from b). Proofs of b) and d). To prove b), define w n = u n log Ω K ne u n. Using the Dirichlet boundary conditions for u n and.5), we conclude that w n is a solutions sequence for.7),.8) and.0) in Ω. Let Ũ r := {x Ω : x p j < r}, and r d, with d chosen as in Theorem.. Using.8) j=,,n and.8) as in Lemma. of [3], we conclude that for any compact subset ω Ω \ Ũr, there exist ω > 0, such that for any j =,, N max Ω\Ũr w n min Ω\Ũr w n ω, n N.

23 It follows that w n satisfies all the hypothesis of Theorem.4 in a suitable neighborhood of any given blow up point. Next, define ξ i,n = µ n Knxi,n) 8, c n := log Ω K ne u n, and notice that, j =,, N, u n x j,n ) c n +, as n +. Hence, for any d x x j,n d, e w nx j,n ) w n x) log ) = + ξj,n ew nx j,n ) Qxj,n x x j,n ) u n x) c n u n x j,n ) + c n c n + log e c n + ξj,ne u ) nx j,n ) Qxj,n x x j,n ) = u n x) u n x j,n ) c n + u n x j,n ) + log e c ) n u n x j,n ) + ξ j,n Q xj,n x x j,n ) = 3 u n x) + u n x j,n ) c n + O). At this point, using.7) and.) we obtain b). In particular, it follows from b) that log Ω K ne u n = u nx j,n ) + O), j =,, N and then.) implies d). Proof of e). For any given blow up point p i S and after a translation if necessary, we can assume that the corresponding unique local maximum point, as given by a), satisfies x i,n 0, for any n N. Define t n := t i,n = e w n0) 0, as n +, and ξ n := ξ i,n = µ n K n0) 8. We analyze the second Pohozaev identity in a suitable spherical neighborhood B r := {x Ω x r}, r d, of this local maxima. Multiplying the equation in.7) by w n x), after some integration by parts, we obtain the following identity, [ ] ν, A n x) w n x)) w n x) w nx), A n x) w n x))ν dσx) B r w n x), [ A n x)] w n x))dx = ψ n x)ν dσx) + log K n x))ψ n x)dx, B r B r B r where ν = x x and ψ n = K n e w n. As a consequence of Theorem., we have w n x) u n x) 3.38) N β j Gx, p j ), in locω\s), 3.39) j= and ψ n x) := K n x)e w nx) N β j δ pj, j=

24 4 D.B. & L.O. in the sense of measures in Ω. Hence, for any 0 < r d, as n +, [ ] ν, A n x) u n x)) u n x) u nx), A n x) u n x))ν dσx) β i B r B r and using also.5), [ ] ν, Ax) Gx, p i )) Gx, p i ) Gx, p i), Ax) Gx, p i ))ν dσx), 3.40) ψ n x)ν dσx) 0, B r 3.4) log K n x))ψ n x)dx ψ n x)dx = β i. 3.4) B r B r As mentioned in the introduction, the problem here is to control the term B r w n x), [ A n x)] w n x))dx, in the second Pohozaev identity above. Notice that, in contrast with the first Pohozaev identity 3.3), we miss a factor x in this integral. In particular, it is not difficult to see that this term must be unbounded. On the other side, using 3.33), we will prove the following laim lim lim r 0 n + Proof of the laim. To prove the laim, define log rξ ) n w n x), [ A n x)] w n x))dx = 3π deta) )0). 3.43) B r φ n x) = 4ξn A 0) x + ξnq 0 x), x + and observe that, by 3.33) and Lemma 3., we have v n = φ n + h n, with h n x) ) log x + log x, for any R x d, and suitable R > and h n x) 0, uniformly in x R. Setting L n := A n 0), we obtain the following identity w n x), L n w n x))dx = B r ỹ r ỹ r v n ỹ), L n v n ỹ))dỹ = ỹ r φ n ỹ), L n h n ỹ))dỹ + h n ỹ), L n h n ỹ))dỹ. ỹ r φ n ỹ), L n φ n ỹ))dỹ+

25 5 Using the explicit expression of φ n, the properties of h n and Lemma 3., one can see that the last two integrals are uniformly bounded. On the other side, we have ỹ r φ n ỹ), L n φ n ỹ))dỹ = 6ξ 4 n ỹ r Then, setting z = ξ n ỹ and s n = rξn +, as n +, we obtain lim log rξ ) n 6 φ n ỹ), L n φ n ỹ))dỹ = lim n + ỹ r n + log s n 6 lim s A 0) z, L n A 0) z) n n + z =s n + Q 0 z)) dσ z) = 6 lim n + s4 n 6 π 0 Finally, observe that 6 lim n + s4 n π 0 A 0)νθ), L n A 0)νθ)) s 4 nq 0 νθ)) A 0)ỹ, L n A 0)ỹ) + ξ nq 0 ỹ)) dỹ. π 0 z s n A 0) z, L n A 0) z) + Q 0 z)) d z = A 0)νθ), L n A 0)νθ)) + s nq 0 νθ))) dθ = s 4 nq 0νθ)) + s nq 0 νθ))) dθ = A 0)νθ), [ A0)]A 0)νθ)) 6π Q 0 νθ)) = deta)0) = 3π deta)0). deta0) and, arguing as above, w n x), [ A n x)] w n x))dx = w n x), L n w n x))dx+ B r B r lim sup r 0 lim sup r 0 lim n + B d w n x), [ A n x) L n ] w n x))dx, log rξ ) n w n x), [ A n x) L n ] w n x))dx B r lim sup n + A n x) L n x B r lim sup r 0 log rξ n ) sup Ax) A0) = 0. x B r B r w n x) dx Using the claim, we will obtain.3). Indeed, to conclude the proof, divide 3.38) by log rξn use 3.40), 3.4), 3.4) and 3.43). and

26 6 D.B. & L.O. References [] D. Bartolucci,.. hen,.s. Lin & G. Tarantello, Profile of Blow Up Solutions To Mean Field Equations with Singular Data, omm. in P. D. E., 97-8) 004), [] D. Bartolucci & L. Orsina, Uniformly elliptic Liouville type equations: concentration compacess and a priori estimates, omm. Pure and Appl. Analysis, 43) 005), [3] D. Bartolucci & G. Tarantello, Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, omm. Math. Phys., 9) 00), [4] D. Bartolucci & G. Tarantello, The Liouville equations with singular data: a concentration-compacess principle via a local representation formula, Jour. of Diff. Equations, 85), 00), [5] J. Bebernes & D. Eberly, Mathematical Problems from ombustion Theory, A. M. S. 83, Springer-Verlag New York 989). [6] H. Brezis, F. Merle, Uniform estimates and blow-up behaviour for solutions of u = V x)e u in two dimensions, omm. in P.D.E., 68,9) 99), [7] S. hanillo & Y.Y. Li, ontinuity of solutions of uniformly elliptic equations, Manuscr. Math., 77 99), [8] W. hen &. Li, lassification of solutions of some nonlinear elliptic equations, Duke Math. J., 633) 99), [9].. hen &. S. Lin, Sharp Estimates for Solutions of Multi-bubbles in ompact Riemann Surfaces, omm. Pure Appl. Math., 55 00), [0].. hen &. S. Lin, Topological Degree for a Mean Field Equation on Riemann Surfaces, omm. Pure Appl. Math., 56) 003), [] W. Ding, J. Jost, J. Li & G.Wang, Existence results for mean field equations, Ann.Inst.H.Poincarè Anal. Non Lin., 6 999), [] M. Grűter & K.O. Widman, The Green function for uniformly elliptic equations, Manuscr. Math., 37 98), [3] Y.Y. Li, Harnack type inequality: the method of moving planes, omm. Math. Phys., ), [4] Y.Y. Li & I.Shafrir, Blow-up analysis for Solutions of u = V x)e u in dimension two, Ind. Univ. Math. J., 434) 994), [5] L. Ma & J.. Wei, onvergences for a Liouville equation, omm. Math. Helv., 76 00), address, Daniele Bartolucci: bartolu@mat.uniroma3.it, bartoluc@mat.uniroma.it address, Luigi Orsina: orsina@mat.uniroma.it