Elliptic stability for stationary Schrödinger equations by Emmanuel Hebey. Part III/VI A priori blow-up theories March 2015

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1 Elliptic stability for stationary Schrödinger equations by Emmanuel Hebey Part III/VI A priori blow-up theories March 2015 Nonlinear analysis arising from geometry and physics Conference in honor of Professor Abbas Bahri

2 PART III. A PRIORI BLOW-UP THEORIES IN THE CRITICAL CASE OF THE STATIONARY SCHRÖDINGER EQUATION. III.1) Preliminary material. III.2) The general question. III.3) The L p -theory. III.4) The H 1 -theory. III.5) The C 0 -theory.

3 NOTE : The blue writing is what you have to write down to be able to follow the slides presentation.

4 PART III. A PRIORI BLOW-UP THEORIES IN THE CRITICAL CASE OF THE STATIONARY SCHRÖDINGER EQUATION. III.1) Preliminary material : The Sobolev inequality Ḣ1 L 2 in R n is written as u L 2 K n u L 2. The precise value of the sharp constant K n was computed by Aubin and Talenti and it was found that 4 K n = n(n 2)ω 2/n n where ω n is the volume of the unit n-sphere. The extremal functions are known. They form a (n + 1)-parameter family given by n 2 2 u Λ,x0 (x) = Λ Λ 2 + x x0 2 n(n 2) for Λ > 0 and x 0 R n. By an important result of Caffarelli-Gidas-Spruck (see also Obata), these extremals are the sole nonnegative solutions of the critical equation u = u 2 1 in R n (which has nontrivial solutions by opposition to its subcritical version u = u p 1, p < 2 ).,

5 III.2) The general question : Let (M, g) closed, n 3. Let (h α ) α be a converging sequence of functions (in a space to be defined according to the theory we deal with). We consider the following family of critical model equations g u + h α u = u 2 1, (E α ) and the goal we want to achieve in this third part is the following. Goal : describe the asymptotic behaviour of sequences (u α ) α of solutions of the (E α ) s in reasonnable spaces. There will be three reasonnable spaces : L 2, H 1, and C 0 (or C k, k 1) leading to the - L p -theory (L 2 -description of the asymptotics), - H 1 -theory (H 1 -description of the asymptotics), - C 0 -theory (pointiwse estimates). The two first theories are concerned with more general objects (Sequences in L p, Palais-Smale sequences in H 1 ) than sequences of solutions (there is room in L 2 and in H 1 to add small H 1 -terms leading to the notion of Palais-Smale sequences).

6 Solutions of (E α ) can be seen as (nonnegative) critical points of the free functionals I α : H 1 R given by I α (u) = 1 ( u 2 + h α u 2) dv g u 2 dv g. M M Definition : A sequence (u α ) α in H 1 is a Palais-Smale sequence(for short PS sequence) for the sequence (E α ) α of equations if : (i) ((I α (u α )) α is bounded in R, (ii) DI α (u α ) 0 in (H 1 ) as α +. The two equations I α (u α ) = O(1), DI α (u α ).(u α ) = o ( u α H 1) imply that Palais-Smale sequences are bounded in H 1 (Brézis-Nirenberg). Conversely, a sequence of solutions of (E α ), in the sense that each u α solves (E α ), which is bounded in H 1, is a Palais-Smale sequence. The Palais-Smale sequence notion extends the notion of H 1 -bounded sequences of solutions by relaxing the condition DI α (u α ) = 0 into DI α (u α ) 0.

7 III.3) The L p -theory : The L p -theory describes the asymptotical behaviour (the blow-up) in L 2 in terms of Dirac masses. The theory goes back to P.L.Lions and can be seen as an easy consequence of his concentration-compactness principle. Theorem : (Concentration-Compactness, P.L.Lions, 84) Let (u α ) α be a bounded sequence in H 1 such that u α u in H 1, and such that the measures µ α = u α 2 dv g µ and ν α = u α 2 dv g ν converge weakly in the sense of measures. Then there exist an at most countable set J, distinct points x j M for j J, and positive real numbers µ j, ν j > 0 for j J such that ν = u 2 dv g + j J µ u 2 dv g + j J ν j δ xj, µ j δ xj, 1 and such that K ν 2/2 n 2 j µ j, where K n is the sharp constant in the Sobolev inequality. In particular, j J ν2/2 j < +.

8 We assume here that the h α s converge in L. Let h be the limit of the h α s so that h α h in L as α +. Then the (E α ) s have, as a formal limit equation, our model equation The L p -theory theorem is stated as follows. Theorem : (L p -theory, P.L.Lions, 84) g u + h u = u 2 1. (E ) Let (h α ) α be a sequence in L such that h α h as α +, and (u α ) α a PS sequence of nonnegative functions for (E α ). There exists u H 1, a nonnegative solution of (E ), N N, x 1,..., x N M, and λ 1,..., λ N > 0 such that, up to a subsequence, u 2 α dv g u 2 dv g + N λ i δ xi (L p E) weakly in the sense of measures. Moreover, u α u in H 1. The x i s are referred to as the geometric blow-up points of the sequence (u α ) α. As a direct consequence of the theorem, since uα 2 u 2 outside the x i s, we get that (Brézis-Lieb) u α u in Hloc 1 (M\S), where S = {x i, i = 1,..., N}. i=1

9 Proof of the theorem : Since (u α ) α is bounded in H 1 we can assume that, up to a subsequence, u α u in H 1, u α u in L 2, and u α u a.e. It is easily seen (Yamabe) that u solves (E ). Let µ α = u α 2 dv g and ν α = u α 2 dv g. By the weak compactness of measures, µ α µ and ν α ν as α +. Let ϕ C. By the PS property, DI α (u α ).(ϕu α ) = o(1), and thus ( u α (ϕu α )) + h α ϕuα 2 = u α 2 ϕ + o(1). (1) We compute ( u α (ϕu α )) = µ(ϕ) + u ( g u )ϕ u 2 ϕ + o(1), and h α ϕuα 2 = h ϕu 2 + o(1). There holds u α 2 ϕ = ν(ϕ) + o(1). By CCP µ u 2 dv g + j J µ jδ xj and ν = u 2 dv g + j J ν jδ xj. Since u solve (E ), we get from (1) that µ j ϕ(x j ) ν j ϕ(x j ) (2) j J j J for all ϕ C. The series µ j and ν j converge and thus µ j ν j for 1 all j. By CCP, K ν 2/2 n 2 j µ j. Thus ν j Kn 2 and J has to be finite. This proves (L p E). Q.E.D.

10 III.4) The H 1 -theory : The question now is to understand what kind of objects are hidden behind the Dirac masses of the L p -theory, namely to understand the α-dynamics of the formation of Dirac masses in the L p -theory. The key notion there is that of a bubble (or sphere singularity). The definition is as follows. Definition : (Bubble) A bubble is a sequence (B α ) α of functions, B α : M R, given by B α (x) = µ 2 α + µ α dg (xα,x)2 n(n 2) for all x M and all α, where d g is the Riemannian distance, (x α ) α is a converging sequence of points in M, and (µ α ) α, µ α 0, is a sequence of positive real numbers converging to zero as α +. The x α s are the centers of the bubble. The µ α s are the weights of the bubble. n 2 2

11 Up to changing the Riemannian distance d g by the Euclidean distance in the definition of a bubble, we recognize in this definition the extremal functions u Λ,x0 for the sharp Euclidean Sobolev inequality. These, as already mentioned, are also (Caffarelli-Gidas-Spruck) the sole nonnegative solutions of u = u 2 1 in R n (which is the limit equation we get by blowing-up the (E α ) s as in the Gidas-Spruck baby blow-up argument). Let x 0 = lim α x α. Then B α 0 in L loc (M\{x 0}). We can compute that, actually, B α 0 in L in M\B xα (r α ) when r α µ α. On the other hand, since B α (x α ) = 1, we get that B α (x α ) + as α +. µ (n 2)/2 α There holds Bα 2 dv g 1 K δ n n x0, where K n is the sharp constant in the Euclidean Sobolev inequality. Bubbles are perfect candidates to be hidden behind the Dirac masses of the L p -theory.

12 The following theorem is stated in Struwe s form. Related references are by Sacks-Uhlenbeck, Wente, Brézis-Coron, Bahri-Coron. Theorem : (H 1 -theory, M. Struwe, 84) Let (h α ) α be a sequence in L such that h α h as α +, and (u α ) α be a PS sequence of nonnegative functions for (E α ). There exist u H 1, a nonnegative solution of (E ), k N, and k bubbles (B i α) α, i = 1,..., k, such that, up to a subsequence, u α = u + k Bα i + R α, (H 1 E) i=1 where (R α ) α is a sequence in H 1 such that R α 0 in H 1 as α +.

13 The bubbles in (H 1 E) do not interact one with another (at the H 1 -level) and (H 1 E) comes with an important equation (that we refer to as the structure equation) which implies that the H 1 -scalar product between two bubbles (B i α) α and (B j α) α in (H 1 E) tends to zero as α +. The structure equation is written as µ i,α µ j,α + µ j,α µ i,α + d g (x i,α, x j,α ) 2 µ i,α µ j,α + (SE) as α +, for all i j, where the x i,α s and µ i,α s are the centers and weights of the bubbles (Bα) i α. In particular, the 2 H and 1 2 L norms 2 of the u α s respect the decomposition (H 1 E). We can check (H 1 E) (L p E), where N in (L p E) is the number of distinct limits we get by the convergence of the centers x i,α of the bubbles (there may be bubbles which accumulate one on another, and thus that k > N), where the x i s, i = 1,..., N, in (L p E) are the limits of the x j,α s, j = 1,..., k, where the λ i s in (L p E) are given by λ i = n i Kn n for all i = 1,..., N, and where n i is the number of x j,α s, j = 1,..., k, which converge to x i so that N i=1 n i = k (grape decomposition).

14 Brief sketch of proof of the theorem : There are two preliminary lemmas (easy), one inductive lemma (difficult), and one concluding lemma (easy). The two preliminary lemmas are as follows : (L1) (Brézis-Nirenberg, Yamabe) PS sequences (u α ) α are bounded in H 1 and, up to a subsequence, they converge weakly in H 1, strongly in L 2, and a.e. to some u which solves (E ). and (L2) If (u α ) α is a PS sequence for (E α ), and u α u in H 1, then v α = u α u is a PS sequence for the free functional I 0 (u) = 1 u u 2, there holds that v α 0 in H 1, and for all α, where I = lim I α. I 0 (v α ) = I α (u α ) I (u ) + o(1) Roughly speaking (L2) states that we can get rid of the limit profile u and the potentials h α which, both, can be set to zero.

15 The key inductive lemma is as follows. (L ind ) Let (v α ) α be a PS sequence of nonnegative functions for I 0 such that v α 0 in H 1 but v α 0 in H 1 as α +. There exist a bubble (B α ) α, and a PS sequence (w α ) α of nonnegative functions for I 0 such that, up to a subsequence, w α = v α B α + R α for all α, where R α 0 in H 1 as α +, and 1 for all α (the constant nkn n I 0 (B α ) of the bubble). I 0 (w α ) = I 0 (v α ) 1 nk n n At last the concluding lemma is as follows. + o(1) being precisely, up to o(1), the free energy (L3) (Aubin, Brézis-Nirenberg) Let (v α ) α be a PS sequence of nonnegative functions for I 0 such that v α 0 in H 1 and I 0 (v α ) c as α +. If c < 1 nk, then v n n α 0 in H 1 as α +. With these lemmas (L1), (L2), (L3), and (L ind ) we can prove the theorem.

16 Brief sketch of proof of the theorem (continued) : Let (u α ) α be a PS sequence of nonnegative functions for (E α ). By (L1) and (L2), the u α s are bounded in H 1, u α u in H 1, u α u in L 2, u α u a.e., where u 0 solves (E ). Moreover, v α = u α u is a PS sequence for I 0 such that v α 0 in H 1 and I 0 (v α ) = I α (u α ) I (u ) + o(1). A nice (tricky though easy) argument shows that we can assume that v α 0 for all α (up to adding a R α 0 in H 1 to the v α s). We let w 0 α = v α. In case w 0 α 0 in H 1, we have the theorem with k = 0. If not the case w 0 α 0 in H 1 and by (L ind ) there exist a bubble (B 1 α) α, and a PS sequence (w 1 α) α of nonnegative functions for I 0 such that, up to a subsequence, wα 1 = wα 0 Bα 1 + R α, and I 0 (wα) 1 = I 0 (wα) 0 1 nkn n + o(1) for all α, where R α 0 in H 1. Clearly, w 1 α 0 in H 1. Here again, either w 1 α 0 in H 1, and we get the theorem with k = 1, or w 1 α 0 in H 1 and we can apply again (L ind ). We go on with this procedure.

17 At the k th step we get k bubbles (B i α) α, and a PS sequence (w k α) α of nonnegative functions for I 0 such that, up to a subsequence, w k α = w 0 α k Bα i + R α, i=1 I 0 (w k α) = I 0 (w 0 α) k nk n n + o(1) for all α, where R α 0 in H 1. By (L3), wα k 0 in H 1 if I 0 (wα) k c with c < 1 nk. Obviously this implies that the process has to step at some n n stage since, at each step, we substract a fix amount of energy (1/nKn n ) to the initial energy I α (u α ) I (u ). When the process stops, w 0 α k Bα i = R α i=1 and this is precisely (H 1 E) since wα 0 = u α u. The theorem is proved. Q.E.D.

18 III.5) The C 0 -theory : At this stage we would like to get sharper estimates involving pointwise asymptotics. For this we need to drop dealing with PS sequences (since PS sequences are stable by the addition of R α s when R α 0 in H 1 ). The C 0 -theory has to do with H 1 -bounded sequences of solutions of (E α ). Namely with sequences (u α ) α of nonnegative functions such that g u α + h α u α = u 2 1 α (E α ) and u α H 1 = O(1) for all α. Two remarks are in order. Rk1 : (In general, R α 0 in L ). The naive idea stating that the C 0 -theory is just the H 1 -theory with the rest (R α ) α converging to zero in L is false. The solutions u α of the Yamabe equation on the sphere, which can be made to blow up, have an H 1 -decomposition with one bubble like u α = B α + R α, R α 0 in H 1, but (as we can compute) R α 0 in L when n 6, while we even have that R α L + when n 7. In other words, we will have to come with something which is slightly more subtle than the sole convergence of the rest to zero in L. The second remark is even more important.

19 Rk2 : (Bubbles in the H 1 -decomposition may interact at the C 0 -level). As already mentionned, because of the structure equation (SE), bubbles do not interact one with another at the H 1 -level. The point here is that a bubble (B α ) α, with centers and weights x α and µ α, live in the H 1 -world essentially in the balls B xα (r α ) for r α µ α. In particular, it H 1 -dies outside such balls : lim lim B α 2 dv g = 0. R + α + M\B xα (Rµ α) On the other hand, the B α s live up to µ α in the C 0 -world : for any R > 0, there exists ε R > 0 such that inf B xα (R µ α) B α ε R for all α. Since µ α µ α for α 1, there is a whole region in which we see bubbles in C 0, and where bubbles may interact one with another even though they do not interact at the H 1 -level. Any (a priori) C 0 -theory will have to take care of the possible interactions of bubbles at the C 0 -level.

20 Theorem : (C 0 -theory, Druet-H.-Robert, 2004) Let (h α ) α be a sequence in C 0,θ converging in C 0,θ to some h. Let (u α ) α be a bounded sequence in H 1 of solutions of (E α ). Assume that g + h is coercive. There exist k N, a nonnegative solution u of the limit equation (E ), and k bubbles (B i α) α, i = 1,..., k, such that, up to a subsequence, (1 ε α )u (x) + 1 C k Bα(x) i i=1 u α (x) (1 + ε α )u (x) + C k Bα(x) i i=1 (C 0 E) for all x M and all α, where C > 1 is independent of α and x, and (ε α ) is a sequence of positive real numbers, independent of x, such that ε α 0 as α +. Moreover, (H 1 E) holds true with u, k, and these (B i α) α. Rk : The condition that g + h should be coercive is a necessary (and sufficient) condition in order to get (C 0 E).

21 In other words (e.g. when k = 2) :

22 In other words (e.g. when k = 2) : Moreover u α = u + k Bα i + R α, i=1 where R α 0 in H 1. In some sense, though (see Rk1 above) we cannot formally assume that R α L 0 in (H 1 E), the C 0 -theory provides sharp upper and lower bounds where R α 0. The theorem has another variant we get from the Green s representation of the u α s which gives the exact asymptotic formula for the u α s.

23 Let G be the Green s fct of g + h and Φ : M M R + be given by Φ(x, y) = (n 2)ω n 1 d g (x, y) n 2 G (x, y), where ω n 1 is the volume of the (n 1)-sphere. Then Φ is continuous in M M and Φ = 1 on the diagonal. Theorem : (C 0 exact asymptotic formula, Druet-H.-Robert, 2004) For any sequence (x α ) α in M, ( ) u α (x α ) = 1 + o(1) u (x α ) + k ( ) Φ(x i, x α ) + o(1) Bα(x i α ), i=1 where the x i s are the limits of the centers of the B i α s. In particular, the constant C in (C 0 E) can be taken as close as we want to 1 when standing in small balls B xi (δ), 0 < δ 1. Of course, C 0 -theory H 1 -theory L p -theory (when we restrict ourselves to sequences of solutions and not only to PS sequences).

24 Thank you for your attention!