ASYMPTOTIC BEHAVIOR OF A FOURTH ORDER MEAN FIELD EQUATION WITH DIRICHLET BOUNDARY CONDITION

Transcription

1 ASYMPTOTIC BEHAVIOR OF A FOURTH ORDER MEAN FIELD EQUATION WITH DIRICHLET BOUNDARY CONDITION FRÉDÉRIC ROBERT AND JUNCHENG WEI Abstract. We consider asymptotic behavior of the following fourth order equation e u u = ρ in, u = νu = 0 on R eu dx where is a smooth oriented bounded domain in R 4. Assuming that 0 < ρ C, we completely characterize the asymptotic behavior of the unbounded solutions.. Introduction In this paper, we study the asymptotic behavior of unbounded solutions for the following fourth order mean field equation under Dirichlet boundary condition { e u = ρ u in, (. R eu dx u = u ν = 0 on where ρ > 0 and R 4 is a smooth oriented bounded domain. In dimension two, the analogous problem { u = ρ e u in Σ, (. RΣ eu dx u = 0 on Σ where Σ is a smooth bounded domain in R, has been extensively studied by many authors. Let (u k, ρ k be a unbounded sequence of solutions to (. with ρ k C, max x Σ u k (x +. Then it has been proved that (P (no boundary bubbles u k is uniformly bounded near a neighborhood of Σ (Nagasaki-Suzuki [34], Ma-Wei [30]; (P (bubbles are simple ρ k 8mπ for some m and u k (x 8π m j= G(, x j in Cloc (Σ\{x,..., x m } (Brézis-Merle [8], Li-Shafrir [5], Nagasaki-Suzuki [34], Ma-Wei [30], where G is the Green function of with Dirichlet boundary condition. Furthermore, it holds that (.3 x R(x j, x j + i j x G(x i, x j = 0, j =,..., m where R(x, y = G(x, y π log x y is the regular part of G(x, y. Date: August 6th Mathematics Subject Classification. Primary 35B40, 35B45; Secondary 35J40. Key words and phrases. Asymptotic Behavior, Biharmonic Equations.

2 F.ROBERT AND J.WEI On the other hand, giving m points satisfying (.3, Baraket and Pacard [9] constructed multiple bubbling solutions to (. when the bubble points satisfy nondegeneracy condition. Del Pino, Kowalczyk and Musso [6] constructed multiple bubbling solutions to (. when the bubble points are topologically nontrivial. Li [4] initiated the computation of Leray-Schauder degree of the solutions to (.. He showed in [4] that the Leray-Schauder degree remains a constant for ρ (8π(m, 8πm and that the degree depends only on the Euler number of the domain. Chen and Lin [, 3] obtained the sharp estimates for the bubbling rate and the exact Leray-Schauder degree counting formula of all solutions to (. for all ρ 8πN. A related question connected to physics consists in adding Dirac masses to the nonlinear parts: we refer to Bartolucci-Chen-Lin-Tarantello [4] and to Tarantello [37] for results and asymptotics in this context. In [38], the second author considered the following fourth order equation under Navier boundary condition { e u = ρ u in, (.4 R eu dx u = u = 0 on where R 4 is a smooth and bounded domain. Assuming that is convex, the corresponding property (P and (P are established in [38]. Later, Lin and Wei [7] considered the attainment of least energy solution and removed the convexity assumption of [38]. Therefore, property (P and (P are established for (.4. Sharp estimates for the bubbles and the computation of topological degree are contained in [8] and [9]. The purpose of this paper is to establish the corresponding property (P and (P for equation (.: indeed, equation (. is more natural than (.4 from the viewpoint of the Adams inequality (see (. below. Our main result can be stated as follows. Theorem.. Assume that is a bounded smooth domain in R 4. Let (u k, ρ k be a sequence of solutions to (. such that (.5 0 < ρ k C, max x u k(x +. Then (a ρ k 64π m for some positive integer m. (b u k has m point blow up, i.e., there exists a set S = {x,..., x m } such that {u k } have a limit u 0 (x for x \S, where the limit function u 0 (x has the form m (.6 u 0 (x = 64π G(x, x j where G(x, y denotes the Green s function of under the Dirichlet condition, that is (.7 G(x, y = δ(x y in, G(x, y = ν G(x, y = 0 on. Furthermore, blow up points x j ( j m satisfy the following relation (.8 x R(x j, x j + l j x G(x j, x l = 0 ( j m i=

3 ASYMPTOTIC BEHAVIOR 3 where (.9 R(x, y = G(x, y + log x y 8π. The main difficulty (and main difference between (. and (.4 is that for fourth order equations, Maximum Principle works for Navier boundary conditions but doesn t work for Dirichlet boundary conditions. More precisely, Green s function for the Navier boundary condition (.0 G(x, y = δ(x y in, G(x, y = G(x, y = 0 on is positive but the Green s function for Dirichlet boundary condition may become negative (see [5] and [0]. This poses a major difficulty in using the method of moving planes (as in [7] to exclude the boundary bubbles. We overcome this by using the Pohozaev identity and by proving strong pointwise estimates for blowingup solutions to (.. As an application of Theorem., we consider the following minimization problem (. J ρ (u = u dx ρ log e u dx, where is a bounded and smooth domain of R 4 and u H0 (. Here, H0 ( denotes the completion of Cc ( for the norm u u. Adams s version of the Moser-Trudinger inequality [] asserts that there exists C( > 0 such that (. e 3π u dx C( for all u H 0 ( such that u =. It follows from (. that J ρ is bounded from below if and only if ρ 64π (for the proof, see the appendix of [7]. Furthermore, if ρ < 64π, the minimizer of J ρ actually exists, that is, there exists a u ρ H 0 ( such that (.3 J ρ (u ρ := inf u H 0 ( J ρ (u := c ρ. For J 64π, it is an interesting question to ask whether the minimum c 64π can be attained or not. The Euler-Lagrange equation of J ρ is just (.. For the corresponding problem in two dimension, given Σ a smooth two-dimensional domain, we consider E ρ (u = ( u dx ρ log e u dx, u H0 (Σ Σ Σ where H0 (Σ denotes the completion of Cc (Σ for the norm u u. Again, by the Moser-Trudinger inequality, E ρ is bounded from below if and only if ρ 8π, and moreover, the minimum of E ρ is always attained if ρ < 8π. However, it has been noted that minimizers do not always exist for E 8π. Actually, it depends on the geometry of Σ in a very subtle way. For example, the minimum of E 8π is not attained if Σ is a ball in R, but, it is attained if Σ is a long and thin domain, see [0]. So, it is rather surprising to have the following claim. Theorem.. Let be a bounded C 4 domain in R 4, and u ρ denote a minimizer of J ρ for ρ < 64π. Assume that (.4 R (Q 0, Q 0 + 6π x R(Q 0, Q 0 > 0

4 4 F.ROBERT AND J.WEI for Q 0 such that R(Q 0, Q 0 = max P R(P, P, where R (x, P is defined by { R (x, P = 0 in, (.5 R (x, P = 4 4 x P, ν R (x, P = ν ( x P, on. Then u ρ is uniformly bounded in C 4 as ρ 64π. Consequently, the minimum of J 64π can be attained. As an example, when is a ball in R 4, J 64π is attained. It is a natural question to ask whether condition (.4 is satisfied for any domain of R 4 : indeed, this question is closely related to the comparison principle for the bi-harmonic operator with Dirichlet boundary condition. For instance, condition (.4 is satisfied if for any function u C 4 (, we have that { } (.6 u 0 in { u > 0 or u 0}. u 0 and ν u 0 on It is remarkable that the comparison principle (.6 does not hold on any domain: for instance, it is false on some annuli. It also follows from Grunau-Robert [9] that condition (.4 is satisfied on small perturbations of the ball. Semilinear equations involving exponential nonlinearity and fourth order elliptic operator appear naturally in conformal geometry and in particular in prescribing Q curvature on 4-dimensional Riemannian manifold M (see e.g. Chang-Yang [] (.7 P g w + Q g = Q gw e 4w where P g is the so-called Paneitz operator: ( P g = ( g + δ 3 R gi Ric g d, g w = e w g, Q g is Q curvature under the metric g, and Q gw is the Q-curvature under the new metric g w. Integrating (.7 over M, we obtain k g := Q g dv g = ( Q gw e 4w dv g = Q gw dv gw M M and k g is conformally invariant (here dv g denote the Riemannian element of volume. Thus, we can write (.7 as (.8 P g w + Q g = k g Qgw e 4w M Q gw e 4w dv g In the special case, where the manifold is the Euclidean space, P g =, and (.8 becomes (.9 h(xe 4w w = k g h(xe4w dx With u = w, ρ = 4k g, h, we arrive at equation (.. There is now an extensive litterature about this problem. For instance, we refer to Adimurthi- Robert-Struwe [], Baraket-Dammak-Ouni-Pacard [3], Druet [7], Druet-Robert [8], Hebey-Robert [], Hebey-Robert-Wen [], Malchiodi [3], Malchiodi-Struwe [3], Robert [35], Robert-Struwe [36] and the references therein. Note also that recently, Clapp-Muñoz-Musso [4] have proved the existence of blowing-up solutions to (.4 (that is (. with Navier boundary condition with arbitrary number of bubbles provided topological hypothesis on. M

5 ASYMPTOTIC BEHAVIOR 5 Our paper is organized as follows. In Section, we present two useful lemmas. Theorem. is proved in Section 3 and Theorem. is proved in Section 4. Notation: Throughout this paper, the constant C will denote various constants which are independent of ρ: the value of C might change from one line to the other, and even in the same line. The equality B = O(A means that there exists C > 0 such that B CA. All the convergence results are stated up to the extraction of a subsequence.. Some preliminaries We state two results in this section. The first one concerns the properties of the Green s function (.7. The second one is Pohozaev s identity. Recall that G(x, y is defined by (.7. As we remarked earlier, in general, G(x, y is not positive. We collect properties of G in the following lemma. Lemma.. There exists C > 0 such that for all x, y, x y, we have that ( (. G(x, y C log + x y (. i G(x, y C x y i, i Proof. These estimates are originally due to Krasovskiĭ [3]. Dall Acqua-Sweers [5] and Grunau-Robert [9]. We also refer to Next we state a Pohozaev identity for equation (.. Lemma.. Let u C 4 ( be a solution of u = f(u in. Then we have for any y R 4, 4 F (u dx = y, ν F (u dσ + x v u x y, ν dσ + ν v dσ ( v u + x y, Du + x y, Dv Dv, Du x y, ν dσ ν ν where F (u = u f(s ds, u = v and ν(x is the normal outward derivative of x 0 on. Proof. More general version of this formula can be seen, for example in [33]. In our case, integrating the identity on div((x y, v u + (x y, u v ( u, v(x y = (x y, v u + (x y, u v ( u, v for u, v C (, = x, and noting that and div((x yf (u = f(u(x y, u + 4F (u ( div v (x y + v u = v( v, x y + ( u, v if v = u, we get the desired formula.

6 6 F.ROBERT AND J.WEI 3. Proof of Theorem. Let u k be a family of solutions to problem (. such that there exists Λ > 0 such that (3. 0 < ρ k Λ. In this section, we study the asymptotic behavior of unbounded solutions and prove Theorem.. Let ( α k := log eu k dx and û k := u k α k. ρ k Theorem. is proved by a series of claims. We first claim that Claim : There exists C R such that α k C for all k N. Proof. Note that û k satisfies (3. û k = eûk in, û k = α k, ν û k = 0 on with eûk dx < C for all k. So (3.3 û k (x = and hence by (., û k (x dx C G(x, yeûk(y dy α k ( x G(x, y eûk(y dy dx ( x y eûk(y dy dx C. Similarly, integrating (3.3, we get that there exists C > 0 such that (3.4 û k + α k L ( C for all k N. It follows from Theorem. of [35] that there exists S, where S is at most finite, such that û k C(ω uniformly in ω for ω \S. Therefore, with (3.4, we get that (α k cannot go to when k +. This proves Claim. A consequence is the following proposition that concerns the case when u k is bounded from above: Lemma 3.. Let (u k, ρ k be a sequence of solutions to (. such that there exists Λ > 0 such that 0 < ρ k Λ. Assume that there exists C > 0 such that u k C for all k N. Then there exists u C 4 ( such that, up to a subsequence lim k + u k = u. Proof. It follows from the assumption of the lemma and Claim that û k C on. It then follows from (. and (3. that (u k is bounded in C 3 (. The conclusion follows from elliptic theory. In the sequel, we assume that (3.5 max x u k(x +. Our second claim is an upper bound on the L p norm of i û k :

7 ASYMPTOTIC BEHAVIOR 7 Claim : For all i =,, 3, p (, 4 i, there exists C = C(i, p such that i û k L p ( C. Proof. By Green s representation formula (3.3 and (., we have i û k (x i xg(x, y eûk(y dy Thus for any ϕ Cc (R 4, we have i û k (x ϕ dx ( C eûk C C x y i eûk dy. ( x y i ϕ(x dx i xg(x, y eûk(y dy ϕ(x dx dy eûk x y i L p ( ϕ L q ( dy C ϕ L q ( where p + q =. Here, we used that is bounded. By duality, we derive that i û k L p ( C. The third claim asserts that bubbles must have some distance from the boundary: Claim 3: Let (x k k N be such that u k (x k = max u k. Let µ k := e 4 ûk(x k. Then lim k + d(x k, µ k = +. Proof. Suppose otherwise, d(x k, = O(µ k. Let k := x k µ k. Then up to a rotation, we may assume that k (, t 0 R 3. Let ũ k (x := û k (x k + µ k x + 4 log µ k. Note that lim k + µ k = 0 (otherwise û k is bounded from above, and, as in the proof of Lemma 3., we get that (u k is bounded: a contradiction with (3.5. Let R > 0 and x B R (0 k, then we have by the representation formula (3.3 and (. i ũ k (x = µ i k i û k (x k + µ k x = µ i k i xg(x k + µ k x, yeûk(y dy ( Cµ i k x k + µ k x y i eûk(y dy B Rµk (x k + k \B Rµk (x k x k + µ k x y i eûk(y dy On k \B Rµk (x k, x k + µ k x y y x k µ k x Rµ k, eûk(y eûk(x k = µ 4 k. Hence i ũ k (x µ i 4 k B Rµk (x k dy x k + µ k x y i + C eûk dy C(R..

8 8 F.ROBERT AND J.WEI In particular, this implies that ũ k (x ũ k (0 C x for all x B R (0. Now let x k, we get û k (x k + α k C. This gives 4 log µ k + α k = O(. A contradiction with lim k + µ k = 0 and Claim. Thus d(x k, µ k +. Claim 4 concerns the first bubble: Claim 4: We have that lim ûk(x k + µ k x + 4 log µ k = 4 log ( + x k in C 4 loc (R4. Proof. By Claim 3, we have k R 4. Since ũ k (x = û k (x k + µ k x + 4 log µ k, ũ k (x ũ k (0 and ũ k = eũk in k. Note by Claim 3, i ũ k (x C(R, for all x B R (0. By standard regularity arguments, ũ k ũ in Cloc 4 (R4 where ũ satisfies (3.6 ũ = eũ, ũ(0 = 0, eũ dx < +. R 4 Note that solutions to (3.6 are nonunique. To characterize ũ, we compute ũ k (x = µ k x G(x k + µ k x, yeûk(y dy and for x B R (0, ( ũ k dx C eûk(y B R (0 CR results of [6] and [39] that ũ(x = 4 log B R dx and hence (0 eũk (3.7 lim lim µ k B R (0 eûk(y dy CR. That is, for any R > 0, we have B R (0 ũ dx CR. ( + x 8 6 R + k + eûk dx = 64π. B Rµk (x k dx x k + µ k x y dy It then follows from. Moreover, B Rµk (x k dx = eûk We say that the property H p holds if there exists (x k,,..., x k,p p such that, denoting µ k,i := e 4 ûk(x k,i, we have that x (i lim k,i x k,j k + µ k,i = +, i j, d(x (ii lim k,i, k + µ k,i = +, (iii lim k + (û k (x k,i + µ k,i x + 4 log µ k,i = 4 log( + x 8 6 in C4 loc (R4.

9 ASYMPTOTIC BEHAVIOR 9 By Claim 4, H holds. Claim 5: Assume that H p holds. Then either H p+ holds, or there exists C > 0 such that (3.8 inf i=,...,p { x x k,i 4 }eûk(x C, x. Proof. Let w k (x := inf i=,...,p x x k,i 4eûk(x. Assume that w k L ( + when k +. Let y k be such that w k (y k = max w k and γ k := e 4 ûk(y k and v k (x := û k (y k + γ k x + 4 log γ k. Then v k satisfies v k = e v k. Note that y w k (y k = inf k x k,i 4 y i=,...,p +. Then lim k x k,i γk 4 k + γ k + for all i =,..., p. Assume that there exists i such that y k x k,i = O(µ k,i, Then y k = x k,i + µ k,i θ k,i and y k x k,i 4 eûk(y k = θ k,i 4 eûk(x k,i +µ k,i θ k,i +4 log µ k,i θ,i 4 ( + θ,i where θ,i = lim k + θ k,i. This implies that w k (y k = O(. A contradiction. Thus y k x k,i µ k,i + for all i =,..., p. Let x B R (0 and let ɛ (0,. Then w k (y k + γ k x w k (y k. That is, inf i=,...,p y k x k,i + γ k x 4eûk(y k +γ k x inf i=,...,p y k x k,i 4eûk(y k and so e v k(x inf i=,...,p y k x k,i 4 inf i=,...,p y k x k,i + γ k x 4. Let k k(r be such that y k x k,i γ k R ɛ for all i =,..., p, k k(r. Then for i =,..., p, we have y k x k,i + γ k x y k x k,i ( ɛ and inf i=,...,p y k x k,i + γ k x 4 inf i=,...,p y k x k,i 4 ( ɛ 4. This yields e v k(x ( ɛ 4, x B R(0, k k(r. Similar to Claim 3, we also have that d(y k, lim = + and lim k + γ v k(x = 4 log ( + x k k in C 4 loc (R4. Letting x k,p+ = y k, then H p+ holds. The claim is thus proved. Claim 6: There exists N such that H N holds and there exists C > 0 such that (3.9 inf i=,...,p x x k,i 4 eûk(x C, x. Proof. Otherwise, since H holds, then H p holds for all p. Given R > 0, we have B Rµk,i (x k,i B Rµk,j (x k,j = for all i j, k k(r. Then p ρ k = eûk dx = eûk(y dy 64π p + o( R i=,...,pb Rµk,i (x i B Rµk,i (x k,i i= where lim R + lim k + o( R = 0. Since ρ k Λ, we derive that p Λ/64π for all p: a contradiction. Hence Claim 6 holds. Claim 7: For p =,, 3, there exists C > 0 such that (3.0 inf i=,...,p x x k,i p p û k (x C, x.

10 0 F.ROBERT AND J.WEI Proof. By Green s representation formula, we have p û k (x = p xg(x, yeûk(y dy. Hence (3. p û k (x C x y p eûk(y dy. Let R k (x := inf i=,...,n x x k,i, k,i = {x : x x k,i = R k (x}. Then x y p eûk(y dy = x y p eûk(y dy k,i k,i B x xk,i (x k,i + x y p eûk(y dy. k,i \B x xk,i (x k,i Note that for y k,i \B x x k,i (x k,i, x y peûk(y C x y p y x k,i. Then Claim 4 6 and easy computations show that Thus (3. B R (0\B x xk,i (x k,i k,i \B x xk,i (x k,i x y p y x k,i 4 dy C x x k,i p. x y p eûk(y dy C x x k,i p. On the other hand, for y k,i B x x k,i (x k,i, we have x y x x k,i y x k,i x x k,i and hence (3.3 k,i B x xk,i (x k,i x y p eûk(y dy C x x k,i p. Combining (3. and (3.3, we obtain the desired estimates. Claim 8: Let x i := lim k + x k,i and S := {x i, i =,..., N}. Assume that lim k + α k = +. Then û k uniformly in \S. Proof. Let δ > 0 small such that δ := \ N i= B δ(x i is connected. Then û k (x C( δ for x δ by the representation formula (3.3. Let x δ δ, then we have û k (x = α k and hence û k (x + α k C for all x δ. This implies that û k uniformly. Claim 9: Assume that lim k + α k = +. Then there exists γ,..., γ N 64π such that N lim u k(x = γ i G(, x i in Cloc( \S. 4 k + i=

11 ASYMPTOTIC BEHAVIOR Proof. Since u k satisfies u k = e α k e u k and u k is bounded in Cloc 0 ( \S by Claim 8, by standard regularity arguments we deduce that u k ψ in C 4 ( \S, where ψ C 4 ( \S. Thus, for δ > 0 small enough, N u k (x = G(x, yeûk(y dy = G(x, yeûk(y dy + o(. i= B δ (x i Since G(x, is continuous in \{x}, we get that lim u k(x = k + N γ i G(x, x i where γ i := lim δ 0 lim k + B δ (x i eûk(y dy. By Claims 4 and 5, γ i 64π. Then ψ = N i= γ ig(x, x i. So we get the result. Claim 0: Let x i := lim k + x k,i and S := {x i, i =,..., N}. Assume that lim k + α k = α R. Then S and there exists u C 4 ( such that u = e α e u in, u = ν u = 0 in and i= lim u k = u in Cloc( 4 \ S. k + Proof. Indeed, with (3.4, we get that û k L ( C for all k N. It then follows from Theorem. of [35] that there exists û C 4 ( such that lim k + û k = û in Cloc 3 (. Therefore S. It then follows from Claims 6 and 7 and standard elliptic theory that there exists u C 4 ( \ S such that lim k + u k = u in C 4 loc( \ S. Moreover, passing to the limit k + in Claim 7, we get that inf x x i u(x C for all x \ S. i=,...,n We are left with proving that u can be smoothly extended to S. We fix x 0 S and we let δ > 0 small enough such that x x 0 u(x C for all x B δ (x 0 \ {x 0 }. Therefore, there exists C > 0 such that for all x, y B δ (x 0 \ {x 0 } such that x x 0 = y x 0, we have that u(x u(y C. Taking y, we then get u(x C for all x B δ (x 0 \ {x 0 }. Proceeding similarly for all the points of S, we get that there exists C > 0 such that u(x C for all x \ S. We let w H0 ( such that w = e α e u. (Since u C, we may simply put e u = when x = x 0. It follows from standard theory that w C 3 ( and that w(x = G(x, ye α e u(y dy

12 F.ROBERT AND J.WEI for all x. For δ > 0 small enough and x \S, (3.4 u k (x = G(x, yeûk(y dy = ( S N i= B δ(x i c G(x, yeûk(y dy + O(δ. Passing to the limit (first in k and then in δ in (3.4 and noting that u C, we get that u(x = G(x, ye α e u(y dy for all x \S. Therefore, u w in \S and u can be extended smoothly as a C 3 function on. Coming back to the definition of w, we get that w is C 4 and then u C 4 (. This ends the proof of Claim 0. As a remark, let us note that if the concentration points were isolated (that is x i x j for all i j, the argument above would prove that (u k is bounded uniformly near the boundary, which would immediately exclude boundary blow-up. Now, we exclude the boundary blow-up in case lim k + α k = + : Claim : Assume that lim k + α k = +. Let x 0. Then lim lim dx = 0. r 0 k + eûk B r(x 0 In particular, S =. Proof. We argue by contradiction and we let x 0 S. Then (3.7 yields lim lim r 0 k + eûk dx 64π. B r(x 0 Thus for all δ > 0, we have that (3.5 dx 3π eûk B δ (x 0 for all k N large enough. Furthermore, we may assume that S B δ (x 0 = {x 0 }. Let y k := x 0 + ρ k,r ν(x 0 with B (3.6 ρ k,r = (x x r(x 0 0, ν( u k dx B (ν(x r(x 0 0, ν( u k dx where r << r such that (ν(x 0 ν for x B r (x 0. Here ν(x is the outer normal vector to T x0 at x. Then it is easy to see that ρ k,r r and (3.7 (x y k, ν( u k dx = 0. B r(x 0 Now applying the Pohozaev s identity in B r (x 0 with y = y k, f(u = e α k e u k and F (u = e α k (e u k, and using Dirichlet boundary condition and (3.7, we obtain that 4 (eûk e α k dx = x y k, ν (e α k e u k e α k dσ B r(x 0 B r(x 0 u k B r(x 0 ν u k dσ

13 ASYMPTOTIC BEHAVIOR 3 [ ] + B r(x 0 x y k, ν ( u k + ( u k < x y k, u k > dσ ν [ ] + B r(x 0 ν u k x y k, u k + < u k, u k >< x y k, ν > dσ. Note that u k Ψ = N i= γ ig(x, x i in C 3 ( \S, where G(x, x 0 = 0. Thus we obtain that all the terms in the last three integrals are of the form [ ] lim O( dx = O(r 3 k + B r(x 0 while lim (x y k, ν(e α k e u k e α k dσ = O(r 4. k + B r(x 0 Since lim k + α k = +, we thus obtain that (3.8 eûk dx Cr3 for k N large enough. Therefore, lim B r(x 0 lim r 0 k + dx = 0. eûk B r(x 0 A contradiction with (3.5. This proves Claim. Claim : We have that lim α k = +. k + Proof. We argue by contradiction and assume that, up to extracting a subsequence, lim k + α k = α R. We let x 0 S (this follows from Claim 0. Arguing as in Claim, we get that 4 (eûk e α k dx = x y k, ν (e α k e u k e α k dσ B r(x 0 B r(x 0 u k + + B r(x 0 B r(x 0 B r(x 0 ν u k dσ [ ] x y k, ν ( u k + ( u k < x y k, u k > dσ ν [ ] ν u k x y k, u k + < u k, u k >< x y k, ν > dσ.

14 4 F.ROBERT AND J.WEI Letting k +, we then get with Claim 0 that 4 3π 4 e α k dx + x y, ν (e u α e α dσ + + B r(x 0 B r(x 0 B r(x 0 B r(x 0 B r(x 0 u u dσ ν [ ] x y, ν ( u + ( u < x y, u > dσ ν [ ] ν u x y, u + < u, u >< x y, ν > dσ for all r > 0 small enough, where y := lim k + y k depends on r with y x 0 r. With Claim 0, we know that u C 4 (. Passing to the limit r 0 above, we get that the RHS goes to zero. A contradiction. Then lim k + α k = +, and Claim is proved. Claim 3: γ i = 64π, i =,..., N. Proof. Since x i, the same proof as in Lemma 3.5 of Lin-Wei [38] gives the claim. We also refer to Druet-Robert [8]. Claim 4: The identity (.8 holds. Proof. The proof is exactly the same as that of Theorem. of Lin-Wei [38] and as in Druet-Robert [8]. Theorem. follows form Claims Proof of Theorem. By Theorem., there are no boundary bubbles for (.. The proof of Theorem. follows along the lines of Sections 3 and 4 of [7]: we just need to change the Navier boundary condition to Dirichlet boundary condition. Let us sketch the changes. We first choose a good approximate function: fix P and let (4. U ɛ,p (x := log γɛ 4 (ɛ + x P 4, where γ := 3 7 = 384. We consider the projection of U ɛ,p : { (4. P U ɛ,p e U ɛ,p = 0 in, P U ɛ,p = ν P U ɛ,p = 0 on. Set (4.3 P U ɛ,p = U ɛ,p ϕ ɛ,p Then ϕ ɛ,p satisfies (4.4 On, we have for ɛ sufficiently small { ϕ ɛ,p = 0 in, ϕ ɛ,p = U ɛ,p, ν ϕ ɛ,p = ν U ɛ,p on. U ɛ,p (x = log(γɛ 4 8 log x P 4ɛ x P + O(ɛ4

15 ASYMPTOTIC BEHAVIOR 5 uniformly in C 4 (. Comparing (4.4 with (.9 and (.5, we have (4.5 ϕ ɛ,p = log(γɛ 4 64π R(x, P ɛ R (x, P + O(ɛ 4, in. We now use P U ɛ,p as a test function to compute an upper bound for c 64π. Let Q 0 be such that R(Q 0, Q 0 = max Q R(Q, Q. Similar computations in [page 799,[7]] yield J 64π [P U ɛ,q0 ] = A 0 (64π max R(P, P P [ ] ɛ 64π R (Q 0, Q 0 + (64π x R(Q 0, Q 0 + o(ɛ 4 where A 0 is a generic constant. By our assumption (.4, we have (4.6 c 64π < A 0 (64π max R(P, P. P On the other hand, let u ρ be a minimizer of J ρ for ρ < 64π. If u ρ blows up as ρ 64π, then a lower bound can be obtained by following exactly the same computation in [7]: (4.7 c 64π A 0 (64π max R(P, P. P From (4.6 and (4.7, we deduce that blow-up does not occur. Then u ρ is uniformly bounded from above. It then follows from Lemma 3. that u ρ converges to a minimizer of J 64π when ρ 64π. Finally, when is a ball, (without loss of generality, we may take = B (0, by the result of Berchio, Gazzola and Weth [5], u is radially symmetric and strictly decreasing. Here Q 0 = 0. Now, by the so-called Boggio s formula [6], we have G(x, y = [x,y] x y 8π for x, y B (0. Thus and hence G(x, 0 = 8π (v v 3 dv, where [x, y] = x y + ( x ( y, ( log x + x, R(x, 0 = ( x 8π, x R(0, 0 = π > 0. It is easy to compute R (x, 0 = 4( x and hence R (0, 0 = 8 > 0. This shows that condition (.4 is satisfied. Theorem. is thus proved. ACKNOWLEDGMENTS The research of the second author is partially supported by a Competitive Earmarked Grant from RGC of HK. This work was carried out while the first author was visiting the Chinese University of Hong-Kong: he thanks this institution for its support and its hospitality.

16 6 F.ROBERT AND J.WEI References [] D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math., 8, (988, [] Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liuville equations in dimension four, J. Eur. Math. Soc., 8, (006, [3] S. Baraket, M. Dammak, T. Ouni, and F. Pacard, Singular limits for 4-dimensional semilinear elliptic problems with exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear. [4] D. Bartolucci, C.-C. Chen, C.-S. Lin, G. Tarantello, Profile of blow-up solutions to mean field equations with singular data, Comm. Partial Differential Equations, 9, (004, [5] E. Berchio, F. Gazzola, and T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math., to appear. [6] T. Boggio, Sull equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei, 0, (90, [7] H. Brézis, Y.Y. Li and I. Shafrir, A sup + inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal., 5, (993, [8] H. Brézis and F. Merle, Uniform estimate and blow up behavior for solutions of V = V (xe u in two dimensions, Comm. Partial Differential Equations, 6, (99, [9] S. Baraket and F. Pacard, Construction of singular limits for a semilinear elliptic equation in dimension, Cal. Var. PDE, 6, (998, -38. [0] S-Y A. Chang, C.C Chen and C.S. Lin, Extremal functions for a mean field equation in two dimension, Lectures on partial differential equations, 6-93, New Stu. Adv. Math., Int. press, MA 003. [] S-Y A. Chang and P.C. Yang, Extremal metrics of zeta function determinants on 4-manifolds, Ann. Math., 4, (995, 7-. [] C.C Chen and C.S. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surface, Comm. Pure Appl. Math., 55, (00, [3] C.C. Chen and C.S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56, (003, [4] M. Clapp, C. Muñoz and M. Musso. Singular limits for the bi-laplacian operator with exponential nonlinearity in R N, Preprint (007. [5] A. Dall Acqua and G. Sweers, Estimates for Green function and Poisson kernels of higherorder Dirichlet boundary value problems, J. Diff. Eqns., 05, (004, [6] M. Del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations. Calc. Var. Partial Differential Equations, 4, (005, [7] O.Druet, Multibumps analysis in dimension - Quantification of blow up levels. Duke Math.J., 3, (006, [8] O. Druet and F. Robert, Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth, Proc. Amer. Math. Soc., 34, (006, [9] H.C. Grunau and F.Robert, Stability of the positivity of biharmonic Green s functions under perturbations of the domain, Preprint, (007. [0] H.C. Grunau and G. Sweeers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann, 307, (997, [] E. Hebey and F. Robert, Coercivity and Struwe s compactness for Paneitz type operators with constant coefficients, Cal. Var. PDE, 3, (00, [] E. Hebey, F. Robert and Y. Wen, Compactness and global estimates for a fourth order equation with critical Sobolev growth arising from conformal geometry, Commun. Contemp. Math., 8, (006, [3] Ju.P. Krasovskiĭ, Isolation of singularities of the Green s function (Russian, Izv. Akad. Nauk SSSR Ser. Mat., 3, (967, English translation in: Math. USSR, Izv.,, (967. [4] Y.Y. Li, Harnack inequality: the method of moving planes, Comm. Math. Phy., 00, (999, [5] Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of u = V e u in dimension two, Indiana Univ. Math. J., 43, (994, [6] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in R 4, Comment. Math. Helv., 73, (998, 06-3.

17 ASYMPTOTIC BEHAVIOR 7 [7] C.S. Lin and J.-C. Wei, Locating the peaks of solutions via the maximum principle. II. A local version of the method of moving planes, Comm. Pure Appl. Math., 56, (003, [8] C.S. Lin and J.-C. Wei, Sharp estimates for bubbling solutions of 4-dimensional mean field equations, Preprint, (007. [9] C.S. Lin, L. Wang and J.-C. Wei, Topological degree for 4-dimensional mean field equations, Preprint, (007. [30] L. Ma and J.-C. Wei, Convergence for a Liouville equation, Comm. Math. Helv., 76, (00, [3] A. Malchiodi, Compactness of solutions to some geometric fourth-order equations, J. Reine Angew. Math., 594, (006, [3] A. Malchiodi and M. Struwe, Q-curvature flow on S 4. J. Differential Geom., 73, (006, -44. [33] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 8, (993, 5-5. [34] K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearity, Asymptotic Analysis, 3, (990, [35] F. Robert, Quantization effects for a fourth order equation of exponential growth in dimension four, The Royal Society of Edinburgh, Proceedings A, to appear. [36] F.Robert and M.Struwe, Asymptotic profile for a fourth order pde with critical exponential growth in dimension four, Advanced Nonlinear Studies, 4, (004, [37] G.Tarantello, A quantization property for blow-up solutions of singular Liouville-type equations. J. Funct. Anal., 9, (005, [38] J.-C. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. Partial Differential Equations,, (996, [39] J.-C. Wei and X.W. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 33, (999, F. Robert Laboratoire J.A.Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, 0608 Nice Cedex, France address: frobert@math.unice.fr J. Wei Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong address: wei@math.cuhk.hk