Blow-up of solutions of critical elliptic equations in three dimensions

We describe the asymptotic behavior of positive solutions $u_\epsilon$ of the equation $-\Delta u + au = 3\,u^{5-\epsilon}$ in $\Omega\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_\epsilon$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Br\'ezis and Peletier (1989). Similar results are also obtained for solutions of the equation $-\Delta u + (a+\epsilon V) u = 3\,u^5$ in $\Omega$.


Introduction and main results
We are interested in the behavior of solutions to certain semilinear elliptic equations that are perturbations of the critical equatioń It is well-known that all positive solutions to the latter equation are given by U x,λ pyq :" λ 1{2 p1`λ 2 |y´x| 2 q 1{2 (1.1) with parameters x P R 3 and λ ą 0. This equation arises as the Euler-Lagrange equation of the optimization problem related to the Sobolev inequality ż with sharp constant [33,34,2,36] S :" 3ˆπ 2˙4 {3 .
The perturbed equations that we are interested in are posed in a bounded open set Ω Ă R 3 and involve a function a on Ω such that the operator´∆`a with Dirichlet boundary conditions is coercive. (Later, we will be more precise concerning regularity assumptions on Ω and a.) One of the two families of equations also involves another, rather arbitrary function V on Ω. The case where a and V are constants is also of interest.
We consider solutions u " u ε , parametrized by ε ą 0, to the following two families of equations, While there are certain differences between the problems (1.2) and (1.3), the methods used to study them are similar, and we will treat both in this paper. We are interested in the behavior of the solutions u ε as ε Ñ 0, and we assume that in this limit the solutions form a minimizing sequence for the Sobolev inequality. More precisely, for (1.3) we assume and for (1.2) we assume (1.5) For example, when Ω " B is the unit ball, a "´π 2 {4, and V "´1, then (1.3) has a solution if and only if 0 ă ε ă 3π 2 4 , see [8,Sec. 1.2]. Note that in this case π 2 is the first eigenvalue of the operator´∆ with Dirichlet boundary conditions on Ω.
Returning to the general situation, the existence of solutions to (1.2) and (1.3) satisfying (1.4) and (1.5) can be proved via minimization under certain assumptions on a and V ; see, for instance, [17] for (1.3). Moreover, it is not hard to prove, based on the characterization of optimizers in Sobolev's inequality, that these functions converge weakly to zero in H 1 0 pΩq and that u 6 ε converges weakly in the sense of measures to a multiple of a delta function; see Proposition 2.2. In this sense, the functions u ε blow up.
The problem of interest is to describe this blow-up behavior more precisely. This question was advertised in an influential paper by Brézis and Peletier [9], who presented a detailed study of the case where Ω is a ball and a and V are constants. For earlier results on (1.2) with a " 0, see [1,10]. Concerning the case of general open sets Ω Ă R 3 , the Brézis-Peletier paper contains three conjectures, the first two of which concern the blow-up behavior of solutions to the analogues of (1.2) and (1.3) in dimensions N ě 3 (N ě 4 for (1.3)) with a " 0. These conjectures were proved independently in seminal works of Han [20] and Rey [31,32].
In the present paper, under a natural nondegeneracy condition, we prove the third Brézis-Peletier conjecture, which has remained open so far. It concerns the blow-up behavior of solutions of (1.2) for certain nonzero a in the three-dimensional case. We also prove the corresponding result for (1.3). This latter result is not stated explicitly as a conjecture in [9], but it is contained there in spirit and could have been formulated using the same heuristics. Indeed, it is the version with a ı 0 of the second Brézis-Peletier conjecture, in the same way as, concerning (1.2), the third conjecture is the a ı 0 version of the first one.
A characteristic feature of the three-dimensional case is the notion of criticality for the function a. To motivate this concept, let Spaq :" inf One of the findings of Brézis and Nirenberg [8] is that if a is small (for instance, in L 8 pΩq), but possibly nonzero, then Spaq " S. This is in stark contrast to the case of dimensions N ě 4 where the corresponding analogue of Spaq (with the exponent 6 replaced by 2N {pN´2q) is always strictly below the corresponding Sobolev constant, whenever a is negative somewhere.
This phenomenon leads naturally to the following definition due to Hebey and Vaugon [23]. A continuous function a on Ω is said to be critical in Ω if Spaq " S and if for any continuous functionã on Ω withã ď a andã ı a one has Spãq ă Spaq. Throughout this paper we assume that a is critical in Ω.
A key role in our analysis is played by the regular part of the Green's function and its zero set.
To introduce these, we follow the sign and normalization convention of [32]. Since the operatoŕ ∆`a in Ω with Dirichlet boundary conditions is assumed to be coercive, it has a Green's function G a satisfying, for each fixed y P Ω, #´∆ x G a px, yq`apxq G a px, yq " 4π δ y in Ω , G a p¨, yq " 0 on BΩ . (1.6) The regular part H a of G a is defined by H a px, yq :" 1 |x´y|´G a px, yq . (1.7) It is well-known that for each y P Ω the function H a p¨, yq, which is originally defined in Ωztyu, extends to a continuous function in Ω and we abbreviate φ a pyq :" H a py, yq .
It was proved by Brézis [6] that inf yPΩ φ a pyq ă 0 implies Spaq ă S. The reverse implication, which was stated in [6] as an open problem, was proved by Druet [13]. Hence, as a consequence of criticality we have inf yPΩ φ a pyq " 0 ; (1.8) see also [16] and [17,Proposition 5.1] for alternative proofs. Note that (1.8) implies, in particular, that each point x with φ a pxq " 0 is a critical point of φ a .
Let us summarize the setting in this paper. In the sequel we set N a :" x P Ω : φ a pxq " 0 ( . Assumption 1.1. (a) Ω Ă R 3 is a bounded, open set with C 2 boundary (b) a P C 0,1 pΩq X C 2,σ loc pΩq for some σ ą 0 (c) a is critical in Ω (d) Any point in N a is a nondegenerate critical point of φ a , that is, for any x 0 P N a , the Hessian D 2 φ a px 0 q does not have a zero eigenvalue Let us briefly comment on these items. Assumptions (a) and (b) are modest regularity assumptions, which can probably be further relaxed with more effort. Concerning assumption (d) we first note that φ a P C 2 pΩq by Lemma 4.1, and therefore any point in N a is a critical point of φ a , see (1.8). We believe that assumption (d) is 'generically' true. (For results in this spirit, but in the noncritical case a " 0, see [28].) The corresponding assumption for a " 0 appears frequently in the literature, for instance, in [32,12]. Assumption (d) holds, in particular, if Ω a ball and a is a constant, as can be verified by explicit computation.
To leading order, the blow-up behavior of solutions of (1.3) will be given by the projection of a solution (1.1) of the unperturbed whole space equation to H 1 0 pΩq. For parameters x P R 3 , λ ą 0 we introduce P U x,λ P H 1 0 pΩq as the unique function satisfying ∆P U x,λ " ∆U x,λ in Ω, P U x,λ " 0 on BΩ . (1.9) Moreover, let T x,λ :" span P U x,λ , B λ P U x,λ , B x 1 P U x,λ B x 2 P U x,λ B x 3 P U x,λ ( and let T K x,λ be the orthogonal complement of T x,λ in H 1 0 pΩq with respect to the inner product ş Ω ∇u¨∇v. By Π x,λ and Π K x,λ we denote the orthogonal projections in H 1 0 pΩq onto T x,λ and T K x,λ , respectively. Here are our main results. We begin with those pertaining to equation (1.2) and we first provide an asymptotic expansion of u ε with a remainder in H 1 0 pΩq. Theorem 1.2 (Asymptotic expansion of u ε ). Let pu ε q be a family of solutions to (1.2) satisfying (1.5). Then there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R and pr ε q Ă T K xε,λε such that u ε " α ε´P U xε,λε´λ´1 {2 ε Π K xε,λε pH a px ε ,¨q´H 0 px ε ,¨qq`r ε¯( 1.10) and a point x 0 P Ω with ∇φ a px 0 q " 0 such that, along a subsequence, (1.14) Moreover, if φ a px 0 q " 0, then lim εÑ0 ε λ 2 ε "´32 apx 0 q . (1.15) Our second main result concerns the pointwise blow-up behavior, both at the blow-up point and away from it, and, in the special case of constant a, verifies the conjecture from [9] under the natural nondegeneracy assumption (d).
(1.16) (b) The asymptotics away from the concentration point x 0 are given by for every fixed x P Ωztx 0 u. The convergence is uniform for x away from x 0 .
Strictly speaking, the Brézis-Peletier conjecture in [9] is stated without the criticality assumption (c) on a, but rather under the assumption φ a ě 0 on Ω. (Note that [9] uses the opposite sign convention for the regular part of the Green's function. Also, their Green's function is normalized to be 1 4π times ours.) The remaining case, however, is much simpler and can be proved with existing methods. Indeed, by Druet's theorem [13], the inequality φ a ě 0 on Ω is equivalent to Spaq " S, and the assumption that a is critical is equivalent to min φ a " 0. Thus, the case of the Brézis-Peletier conjecture that is not covered by our Theorem 1.3 is that where min φ a ą 0. This case can be treated in the same way as the case a " 0 in [20,31] (or as we treat the case φ a px 0 q ą 0). Note that in this case the nondegeneracy assumption (d) is not needed. Whether this assumption can be removed in the case where φ a px 0 q " 0 is an open problem.
We note that Theorems 1.2 and 1.3 and, in particular, the asymptotics (1.15) and (1.16), hold independently of whether apx 0 q " 0 or not. We note that apx 0 q ď 0 if φ a px 0 q as shown in [17,Corollary 2.2]. We are grateful to H. Brézis (personal communication) for raising the question of whether apx 0 q " 0 can happen and what the asymptotics of λ ε resp. }u ε } 8 would be in this case, or whether one can show that φ a px 0 q " 0 implies apx 0 q ă 0. Deciding which alternative holds does not appear to be easy, in particular due to the non-local nature of φ a px 0 q. Here is a simple observation that may illustrate the expected level of difficulty: In the spirit of [17, Theorem 2.1 and Corollary 2.2], apx 0 q ă 0 would follow if one could exhibit a family of very refined test functions η x 0 ,λ such that when inf Ω φ a " φ a px 0 q " 0, the Sobolev quotient defining Spaq satisfies S a rη x 0 ,λ s " S´c 1 apx 0 qλ´2´c 2 λ´τ`opλ´τ q for some c 1 , c 2 ą 0 and τ ą 2, say. However, extracting such an explicit term c 2 λ´τ is beyond the precision of both [17] and the present paper.
We also point out that the conjecture in [9] is formulated with assumption (1.4) rather that (1.5). However, the latter assumption is typically used in the posterior literature dealing with problem (1.2), see e.g. [20,19], and we follow this convention.
We now turn our attention to the results for the second family of equations, namely (1.3). Whenever we deal with that problem, we impose the following additional assumptions; Again, assumption (f) is a modest regularity assumption, which can probably be further relaxed with more effort. Assumption (e) is not severe, as we know from [17, Corollary 2.2] that any critical a satisfies a ď 0 on N a , see also the above discussion of the question by Brezis of whether or not this assumption is automatically satisfied. In particular, it is fulfilled if a is a negative constant. Let Again, we first provide an asymptotic expansion of u ε with a remainder in H 1 0 pΩq. Then there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R and pr ε q Ă T K xε,λε such that and a point x 0 P N a with Q V px 0 q ď 0 such that, along a subsequence, If Q V px 0 q " 0, the right side of (1.21) is to be interpreted as 8.
The following result concerns the pointwise blow-up behavior. (a) The asymptotics close to the concentration point x 0 are given by If Q V px 0 q " 0, the right side is to be interpreted as 8.
(b) The asymptotics away from the concentration point x 0 are given by for every fixed x P Ωztx 0 u. The convergence is uniform for x away from x 0 . Theorems 1.2 and 1.5 state that to leading order the solution is given by a projected bubble P U xε,λε . One of the main points of these theorems, which enters crucially in the proof of Theorems 1.3 and 1.6, is the identification of the localization length λ´1 ε of the projected bubble as an explicit constant times ε (for (1.2) if φ a px 0 q ‰ 0 and for (1.3) if Q V px 0 q ă 0) or ε 1{2 (for (1.2) if φ a px 0 q " 0 and apx 0 q ‰ 0).
The fact that the solutions are given to leading order by a projected bubble is a rather general phenomenon, which is shared, for instance, also by the higher dimensional generalizations of (1.2) and (1.3). In contrast to the higher dimensional case, however, in order to compute the asymptotics of the localization length λ´1 ε , we need to extract the leading order correction to the bubble. Remarkably, this correction is for both problems (1.2) and (1.3) given by λ´1 xε,λε pH a px ε ,¨q´H 0 px ε ,¨qq. In this relation it is natural to wonder whether the above projected bubble P U x,ε can be replaced by a different projected bubble Ą P U x,λ , namely where the projection is defined with respect to the scalar product coming from the operator´∆`a, leading to p´∆`aq Ą P U x,λ " p´∆`aqU x,λ , Ą P U x,λ | BΩ " 0. Such a choice is probably possible and would even simplify some computations, but it would lead to additional difficulties elsewhere (for instance, in the proofs of Propositions 2.2 and 5.1 our choice allows us to apply the classical results by Bahri and Coron).
Moreover, for both problems the concentration point x 0 is shown to satisfy ∇φ a px 0 q " 0. Here, however, we see an interesting difference between the two problems. Namely, for (1.3) we also know that φ a px 0 q " 0, whereas we know from [12,Theorem 2(b)] that there are solutions of (1.2) concentrating at any critical point of φ a , not necessarily in N a . (These solutions also satisfy (1.4).) An asymptotic expansion very similar to that in Theorem 1.5 is proved in [17] for energyminimizing solutions of (1.3); see also [18] for the simpler higher-dimensional case. There, we did not assume the nondegeneracy of D 2 φ a px 0 q, but we did assume that Q V ă 0 in N a . Moreover, in the energy minimizing setting we showed that x 0 satisfies but this cannot be expected in the more general setting of the present paper.
Before describing the technical challenges that we overcome in our proofs, let us put our work into perspective. In the past three decades there has been an enormous literature on blow-up phenomena of solutions to semilinear equations with critical exponent, which is impossible to summarize. We mention here only a few recent works from which, we hope, a more complete bibliography can be reconstructed. In some sense, the situation in the present paper is the simplest blow-up situation, as it concerns single bubble blow-up of positive solutions in the interior. Much more refined blow-up scenarios have been studied, including, for instance, multibubbling, sign-changing solutions or concentration on the boundary under Neumann boundary conditions. For an introduction and references we refer to the books [14,22]. In this paper we are interested in the description of the behavior of a given family of solutions. For the converse problem of constructing blow-up solutions in our setting, see [12] and also [29], and for a survey of related results, see [30] and the references therein. Obstructions to the existence of solutions in three dimensions were studied in [15]. The spectrum near zero of the linearization of solutions was studied in [19,11]. There are also connections to the question of compactness of solutions, see [5,24] and references therein.
What makes the critical case in three dimensions significantly harder than the higher-dimensional analogues solved by Han [20] and Rey [31,32] is a certain cancellation, which is related to the fact that inf φ a " 0. Thus, the term that in higher dimensions completely determines the blow-up vanishes in our case. Our way around this impasse is to iteratively improve our knowledge about the functions u ε . The mechanism behind this iteration is a certain coercivity inequality, due to Esposito [16], which we state in Lemma 2.3, and a crucial feature of our proof is to apply this inequality repeatedly, at different orders of precision. To arrive at the level of precision stated in Theorems 1.2 and 1.5 two iterations are necessary (plus a zeroth one, hidden in the proof of Proposition 2.2).
The first iteration, contained in Sections 2 and 5, is relatively standard and follows Rey's ideas in [32] with some adaptions due to Esposito [16] to the critical case in three dimensions. The main outcome of this first iteration is the fact that concentration occurs in the interior and an order-sharp remainder bound in H 1 0 on the remainder α´1 ε u ε´P U xε,λε .
The second iteration, contained in Sections 3 and 6, is more specific to the problem at hand. Its main outcome is the extraction of the subleading correction λ´1 {2 ε Π K xε,λε pH a px ε ,¨q´H 0 px ε ,¨qq. Using the nondegeneracy of D 2 φ a px 0 q we will be able to show in the proof of Theorems 1.2 and The arguments described so far are, for the most part, carried out in H 1 0 norm. Once one has completed the two iterations, we apply in Subsections 4.3 and 7.2 a Moser iteration argument in order to show that the remainder α´1 ε u ε´P U xε,λε is negligible also in L 8 norm. This will then allow us to deduce Theorems 1.3 and 1.6.
As we mentioned before, Theorem 1.5 is the generalization of the corresponding theorem in [17] for energy-minimizing solutions. In that previous paper, we also used a similar iteration technique. Within each iteration step, however, minimality played an important role in [17] and we used the iterative knowledge to further expand the energy functional evaluated at a minimizer. There is no analogue of this procedure in the current paper. Instead, as in most other works in this area, starting with [9], Pohozaev identities now play an important role. These identities were not used in [17]. In fact, in [17] we did not use equation (1.3) at all and our results there are valid as well for a certain class of 'almost minimizers'.
There are five types of Pohozaev-type identities corresponding, in some sense, to the five linearly independent functions in the kernel of the Hessian at an optimizer of the Sobolev inequality on R 3 (resulting from its invariance under multiplication by constants, by dilations and by translations). All five identities will be used to control the five parameters α ε , λ ε and x ε in (1.10) and (1.18), which precisely correspond to the five asymptotic invariances. In fact, all five of these identities are used in the first iteration and then again in the second iteration. (To be more precise, in the first iteration in the proof of Theorem 1.5 it is more economical to only use four identities, since the information from the fifth identity is not particularly useful at this stage, due to the above mentioned cancellation φ a px 0 q " 0.) Thinking of the five Pohozaev-type identities as coming from the asymptotic invariances is useful, but an oversimplification. Indeed, there are several possible choices for the multipliers in each category, for instance, u, P U x,λ , ψ x,λ corresponding to multiplication by constants, y¨∇u, B λ P U x,λ , B λ ψ x,λ corresponding to dilations and B x j u, ∇ x j P U x,λ , ∇ x j P U x,λ corresponding to translations. (Here ψ x,λ is a modified bubble defined below in (3.1).) The choice of the multiplier is subtle and depends on the available knowledge at the moment of applying the identity and the desired precision of the outcome. In any case, the upshot is that these identities can be brought together in such a way that they give the final result of Theorems 1.2 and 1.5 concerning the expansion in H 1 0 pΩq. As mentioned before, the desired pointwise bounds in Theorems 1.3 and 1.6 then follow in a relatively straightforward way using a Moser iteration.
We believe that our techniques are robust enough to derive blow-up asymptotics for (1.2) and (1.3) in more general situations containing a non-zero weak limit and/or multiple concentration points. Since our main motivation was to solve the Brézis-Peletier conjecture stated for single blow-up in [9], and to limit the amount of calculations needed, we do not attempt to pursue this further here.
Let us also mention that a problem similar to, but different from (1.2) has been studied in the recent article [27] by similar approach. While the analysis in [27], carried out on a Riemannian manifold M of dimension n ě 5, is rather comprehensive and also treats the case of multiple blow-up points, it does not seem to contain an analogue of the vanishing phenomenon for φ a px 0 q nor, as a consequence, of our refined iteration step we described above.
The structure of this paper is as follows. The first part of the paper, consisting of Sections 2, 3 and 4, is devoted to problem (1.3), while the second part, consisting of Sections 5, 6 and 7, is devoted to (1.2). The two parts are presented in a parallel manner, but the emphasis in the second part is on the necessary changes compared to the first part. The preliminary Sections 2 and 5 contain an initial expansion, the subsequent Sections 3 and 6 contain its refinement and, finally, in Sections 4 and 7 the main theorems presented in this introduction are proved. Some technical results are deferred to two appendices.

Additive case: A first expansion
In this and the following section we will prepare for the proof of Theorems 1.5 and 1.6.
The main result from this section is the following preliminary asymptotic expansion of the family of solutions pu ε q. Then, up to extraction of a subsequence, there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R and pw ε q Ă T K xε,λε such that u ε " α ε pP U xε,λε`wε q, (2.1) and a point x 0 P Ω such that This proposition follows to a large extent by an adaptation of existing results in the literature. We include the proof since we have not found the precise statement and since related arguments will appear in the following section in a more complicated setting.
An initial qualitative expansion follows from works of Struwe [35] and Bahri-Coron [3]. In order to obtain the statement of Proposition 2.1, we then need to show two things, namely, the bound on }∇w} and the fact that x 0 P Ω. The proof of the bound on }∇w} that we give is rather close to that of Esposito [16]. The setting in [16] is slightly different (there, V is equal to a negative constant and, more importantly, the solutions are assumed to be energy minimizing), but this part of the proof extends to our setting. On the other hand, the proof in [16] of the fact that x 0 P Ω relies on the energy minimizing property and does not work for us. Instead, we adapt some ideas from Rey in [32]. The proof in [32] is only carried out in dimensions ě 4 and without the background a, but, as we will see, it extends with some effort to our situation.
We subdivide the proof of Proposition 2.1 into a sequence of subsections. The main result of each subsection is stated as a proposition at the beginning and summarizes the content of the corresponding subsection.
2.1. A qualitative initial expansion. As a first important step, we derive the following expansion, which is already of the form of that in Proposition 2.1, except that all remainder bounds are nonquantitative and the limit point x 0 may a priori be on the boundary BΩ. Then, up to extraction of a subsequence, there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R and pw ε q Ă T K xε,λε such that (2.1) holds and a point x 0 P Ω such that |x ε´x0 | " op1q, α ε " 1`op1q, d ε λ ε Ñ 8, }∇w ε } 2 " op1q, where we denote d ε :" dpx ε , BΩq.
Proof. We shall only prove that u ε á 0 in H 1 0 pΩq. Once this is shown, we can use standard arguments, due to Lions [26], Struwe [35] and Bahri-Coron [3], to complete the proof of the proposition; see, for instance, [32, Proof of Proposition 2].
Step 1. We begin by showing that pu ε q is bounded in H 1 0 pΩq and that }u ε } 6 Á 1. Integrating the equation for u ε against u ε , we obtain ż and therefore On the right side, the first quotient converges by (1.4) and the second quotient is bounded by Hölder's inequality. Thus, pu ε q is bounded in L 6 pΩq. By (1.4) we obtain boundedness in H 1 0 pΩq. By coercivity of´∆`a in H 1 0 pΩq and Sobolev's inequality, for all sufficiently small ε ą 0, the left side in (2.4) is bounded from below by a constant times }u ε } 2 6 . This yields the lower bound on }u ε } 6 Á 1.
Step 2. According to Step 1, pu ε q has a weak limit point in H 1 0 pΩq and we denote by u 0 one of those. Our goal is to show that u 0 " 0. Throughout this step, we restrict ourselves to a subsequence of ε's along which u ε á u 0 in H 1 0 pΩq. By Rellich's lemma, after passing to a subsequence, we may also assume that u ε Ñ u 0 almost everywhere. Moreover, passing to a further subsequence, we may also assume that }∇u ε } has a limit. Then, by (1.4), }u ε } 6 has a limit as well and, by Step 1, none of these limits is zero.
We now argue as in the proof of [17, Proposition 3.1] and note that, by weak convergence, Thus, (1.4) gives We bound the left side from above with the help of the elementary inequalitŷ ż and, by the Sobolev inequality for u ε´u0 , we bound the right side from below using Thus, Thus, either u 0 " 0 or u 0 is an optimizer for the Sobolev inequality. Since u 0 has support in Ω Ĺ R 3 , the latter is impossible and we conclude that u 0 " 0, as claimed.
Convention. Throughout the rest of the paper, we assume that the sequence pu ε q satisfies the assumptions and conclusions from Proposition 2.2. We will make no explicit mention of subsequences. Moreover, we typically drop the index ε from u ε , α ε , x ε , λ ε , d ε and w ε .

2.2.
Coercivity. The following coercivity inequality from [16, Lemma 2.2] is a crucial tool for us in subsequently refining the expansion of u ε . It states, roughly speaking, that the subleading error terms coming from the expansion of u ε can be absorbed into the leading term, at least under some orthogonality condition.
The proof proceeds by compactness, using the inequality [32, For details of the proof, we refer to [16].
In the following subsection, we use Lemma 2.3 to deduce a refined bound on }∇w} 2 . We will use it again in Section 3.2 below to obtain improved bounds on the refined error term }∇r} 2 , with r P T K x,λ defined in (3.4).
2.3. The bound on }∇w} 2 . The goal of this subsection is to prove Using this bound, we will prove in Subsection 2.4 that d´1 " Op1q and therefore the bound in Proposition 2.4 becomes }∇w} 2 " Opλ´1 {2 q, as claimed in Proposition 2.1.

Proof. The starting point is the equation satisfied by
Integrating this equation against w and using (2.8) We estimate the three terms on the right hand side separately.
The first term on the right side of (2.8) needs a bit more care. We write P U x,λ " U x,λ´ϕx,λ as in Lemma A.2 and expand ż where we again used ş Ω U 5 x,λ w " 0. By Lemmas A.1 and A.2, we have }ϕ x,λ } 2 6 À pdλq´1 and ż Ω U 4 x,λ ϕ x,λ |w| À }w} 6 Putting all the estimates together, we deduce from (2.8) that ż Due to the coercivity inequality from Lemma 2.3, the left side is bounded from below by a positive constant times }∇w} 2 2 . Thus, (2.6) follows.

2.4.
Excluding boundary concentration. The goal of this subsection is to prove By integrating the equation for u against ∇u, one obtains the Pohozaev-type identitý ż where we used (2.6) and Lemmas A.1 and A.2.
The function BP U x,λ {Bn on the boundary is discussed in Lemma A.3. We now control the function Bw{Bn on the boundary.
Proof. The following proof is analogous to [32,Appendix C]. It relies on the inequality This inequality is well-known and contained in [32,Appendix C]. A proof can be found, for instance, in [21].
The function F satisfies the simple pointwise bound which, when combined with inequality (2.12), yields It remains to bound the norms on the right side. The term most difficult to estimate is }ζw 5 } 3{2 , because 5¨3{2 " 15{2 ą 6, and we shall come back to it later. The other terms can all be estimated using bounds on }U } L p pΩzB d{2 pxqq from Lemma A.1, as well as the bound }w} 6 À λ´1 {2`λ´1 d´1 from Proposition 2.4. Indeed, we have In order to estimate the difficult term }ζw 5 } 3{2 , we multiply the equation´∆w " F by ζ 1{2 |w| 1{2 w and integrate over Ω to obtain ż We now note that there are universal constants c ą 0 and C ă 8 such that pointwise a.e.
Indeed, by repeated use of the product rule and chain rule for Sobolev functions, one finds The claimed inequality (2.17) follows by applying Schwarz's inequality v 1¨v2 ě´ε|v 1 | 2´1 4ε |v 2 | 2 to the cross term on the right side with ε ą 0 small enough.
As a consequence of (2.17), we can bound the left side in (2.16) from below by ż Thus, by the Sobolev inequality for the function ζ 1{4 |w| 1{4 w and (2.16), we get For the first term on the right side, we havê ż To control the second term on the right side of (2.18), we use again the pointwise estimate (2.15). The contribution of the |w| 5 term to the second term on the right side of (2.18) iŝ ż which can be absorbed into the left side of (2.18).
It is now easy to complete the proof of the main result of this section.

Additive case: Refining the expansion
Our goal in this section is to improve the decomposition given in Proposition 2.1. As in [17], our goal is to discover that a better approximation to u ε is given by the function As in [17], we further decompose with s ε P T xε,λε and r ε P T K xε,λε given by We note that the notation r ε is consistent with the one used in Theorem 1.5 since, writing The following proposition summarizes the results of this section. Then, up to extraction of a subsequence, there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R, ps ε q Ă T xε,λε and pr ε q Ă T K xε,λε such that u ε " α ε pψ xε,λε`sε`rε q (3.6) and a point x 0 P Ω such that, in addition to Proposition 2.1, The expansion of φ a pxq will be of great importance also in the final step of the proof of Theorem 1.5. Indeed, by using the bound on |∇φ a pxq| we will show that in fact φ a pxq " opλ´1q`opεq. This allows us to determine lim εÑ0 ελ ε .
We prove Proposition 3.1 in the following subsections. Again the strategy is to expand suitable energy functionals.
3.1. Bounds on s. In this section we record bounds on the function s introduced in (3.4), and on the coefficients β, γ and δ j defined by the decomposition Since P U x,λ , B λ P U x,λ and B x i P U x,λ , i " 1, 2, 3, are linearly independent for sufficiently small ε, the numbers β, γ and δ i , i " 1, 2, 3, (depending on ε, of course) are uniquely determined. The choice of the different powers of λ multiplying these coefficients is motivated by the following proposition.
Proposition 3.2. The coefficients appearing in (3.8) satisfy Moreover, we have the bounds as well as Proof. Because of (3.5), s ε depends on u ε only through the parameters λ and x. Since these parameters satisfy the same properties λ Ñ 8 and d´1 " Op1q as in [17], the results on s ε there are applicable. In particular, the bound (3.9) follows from [17, Lemma 6.1].
The bounds stated in (3.10) follow readily from (3.8) and (3.9), together with the corresponding bounds on the basis functions P U x,λ , B λ P U x,λ and B x i P U x,λ , i " 1, 2, 3, which come from and similar bounds on B λ U x,λ and B x i U x,λ , compare Lemma A.1, as well as It remains to prove (3.11). Again by (3.8) and (3.9), it suffices to show that (3.12) (In fact, there is a better bound on ∇B x i P U x,λ , but we do not need this.) Since the three bounds in (3.12) are all proved similarly, we only prove the second one.
By integration by parts, we have ż By the bounds from Lemmas A.1 and A.2, the volume integral is estimated by ż By the mean value formula for the harmonic function B λ ϕ x,λ and the bound from Lemma A.2, This implies that |∇pB λ P U x,λ q| À λ´3 {2 on BB d{2 pxq. Thus, the boundary integral is estimated by ż Later we will also need the leading order behavior of the zero mode coefficients β and γ in (3.8).
Similarly, by (3.8), the left side of (3.15) is (The numerical value comes from an explicit evaluation of the integral in terms of beta functions, which we omit.) On the other hand, the right side of (3.15) is by Lemma B.3. Comparing both sides yields the expansion of γ stated in (3.13).
{2 H a px,¨q´f x,λ and bound pointwise When integrated against r, the first term vanishes by orthogonality. Let us bound the contribution coming from the second term, that is, from 5U 4 x,λ s. We write s " λ´1βU x,λ`γ B λ U x,λ`s , sos consists of the zero mode contributions involving the δ i , plus contributions from the difference between P U x,λ and U x,λ in the terms involving β and γ. By orthogonality, we have ż and, by Lemmas A.1 and A.2, as well as Proposition 3.2, It remains to bound the remainder terms in (3.21). We write H a px, yq " φ a pxq`Op|x´y|q and bound ż Hencěˇˇˇż Ω U 4 x,λ´λ´1 À´λ´1φ a pxq`λ´2¯}r} 6 . (3.23) Since α 4 Ñ 1 and r P T K x,λ , the coercivity inequality (2.5) implies that for all sufficiently small ε ą 0 the left side is bounded from below by c}∇r} 2 2 with a universal constant c ą 0. Thus, For all sufficiently small ε ą 0, the last two terms on the right side can be absorbed into the left side and we obtain the claimed inequality (3.18).
Proposition 3.4 is a first step to prove the bound (3.7) in Proposition 3.1. In Section 3.4 we will show that φ a pxq " Opλ´1`εq and λ´1 " Opεq. Combining these bounds with Proposition 3.4 we will obtain (3.7).
The following lemma provides the expansions of the terms in (3.25).
which immediately imply the bound in (a).
This completes the proof.
Proof of Proposition 3.6. The claim follows from (3.25) and Lemma 3.7.

3.4.
Expanding φ a pxq. In this subsection we prove the following important expansion.
Before proving it, let us note the following consequence.
Proof. The fact that φ a px 0 q " 0 follows immediately from (3.28). Since φ a pxq ě 0 by criticality and since apx 0 q ă 0 by assumption, we deduce from (3.28) that Q V px 0 q ď 0 and that Reinserting this into (3.28) we find φ a pxq " Opεq. Inserting this into Proposition 3.4, we obtain the claimed bound on }∇r} 2 , and inserting it into (3.24) and (3.13), we obtain the claimed expansion of α 4 .
The proof of (3.28) is based on the Pohozaev identity obtained by integrating the equation for u against B λ ψ x,λ . We write the resulting equality in the form ż The involved terms can be expanded as follows.
We emphasize that the proof of Lemma 3.10 is independent of the expansion of α 4 in (3.24). We only use the fact that α " 1`op1q.
Proof of Lemma 3.10. (a) Because of (3.26), the quantity of interest can be written as ż We discuss the three integrals on the right side separately. As a general rule, terms involving f x,λ will be negligible as a consequence of the bounds }f x,λ } 8 " Opλ´5 {2 q and }B λ f x,λ } 8 " Opλ´7 {2 q in Lemma A.2. This will not always be carried out in detail.
We have ż Ω´U 5 x,λ´α x,λ´ψ 5 x,λ¯B λ ψ x,λ . (3.32) The first integral is, since ψ p1`r 2 q 4ˇd r " Opλ´4q. Next, by Lemma B.3, This completes our discussion of the first term on the right side of (3.32). For the second term we have similarly, ż Ω´U 5 x,λ´ψ 5 x,λ¯Bλ ψ x,λ " ż Ω U 5´k x,λ H a px,¨q k`1`o pλ´3q . (3.36) Finally, the two sums are bounded, in absolute value, by This completes our discussion of the second term on the right side of (3.32) and therefore of the first term on the right side of (3.31).
For the second term on the right side of (3.31) we get, using ψ The second integral is negligible since, by Lemma A.4,ˇˇˇ1 Since a is differentiable, we can expand the first integral as Using similar bounds one verifies that This completes our discussion of the second term on the right side of (3.31).
For the third term on the right side of (3.31), we write ψ x,λ " λ´1 {2 G a px,¨q´f x,λ´gx,λ and get ż In the last equality we used the bounds from Lemma A.4 and the fact that G a px,¨q P L 2 pΩq. This completes our discussion of the third term on the right side of (3.31) and concludes the proof of (a).

Because of this equation, the quantity of interest can be written as
We discuss the two integrals on the right side separately.
We have ż The first integral is, by the orthogonality condition 0 " ş Ω ∇w¨∇B λ P U x,λ " 15 For the second integral on the right side of (3.38) we have ż ż Ω qU 4 x,λ |x´y|˙.
Using the bound (3.27) on q and Lemma A.1 we geťˇˇˇż The remainder term is better because of the additional factor of |x´y|. We gain a factor of λ´1 since Another typical term, ż can be treated in the same way, since the bounds for B λ U x,λ are the same as for λ´1U x,λ ; see Lemma A.1. The remaining terms are easier. This completes our discussion of the first term on the right side of (3.37).
The second term on the right side of (3.37) is negligible. Indeed, ż where we used Lemma A.4 and the same bound on q as before. This completes our discussion of the second term on the right side of (3.37) and concludes the proof of (b).
The proof of (d) uses similar bounds as in the rest of the proof and is omitted.
On the other hand, the next lemma allows to control the error terms involving q.
Before proving these two lemmas, let us use them to give the proof of Proposition 3.11. In that proof, and later in this subsection, we will use the inequality This follows from the bound (3.10) on s and the bounds in Corollary 3.9 on λ´1 and r. Note that (3.48) is better than the bound (3.27) in the L 6 norm.
Proof of Proposition 3.11. We shall make use of the bounds The first bound follows by writing ψ x,λ " U x,λ´λ´1 {2 H a px,¨q`f x,λ and using the bounds in Lemmas A.1 and A.2 and in (B.1). For the second bound we write ψ x,λ " P U x,λ´λ´1 {2 pH a px,¨qH 0 px,¨qq and use the bounds in Lemmas A.3 and B.1.
It remains to prove Lemmas 3.12 and 3.13.
This shows that the first term on the right side of (3.50) gives the claimed contribution.
On the other hand, for the second term on the right side of (3.50) we have ż Ω apf x,λ`gx,λ q∇ψ x,λ " ż Ω apf x,λ`gx,λ q∇pU x,λ´λ´1 {2 H a px,¨qq Here we used bounds from Lemmas A.2 and A.4 and from the proof of the latter. Finally, we write apyq " apxq`Op|x´y|q and using oddness of g x,λ ∇U x,λ to obtain ż This proves the claimed bound on the second term on the right side of (3.50).
Proof of Lemma 3.13. The proof is analogous to that of Lemma 2.6. By combining equation (2.7) for w with ∆pH a px,¨q´H 0 px,¨qq "´aG a px,¨q, we obtain´∆q " F with F :"´3U 5 x,λ`3 α 4 pψ x,λ`q q 5´a q`apf x,λ`gx,λ q´εV pψ x,λ`q q .
(We use the same notation as in the proof of Lemma 2.6 for analogous, but different objects.) We define the cut-off function ζ as before, but now in our bounds we do not make the dependence on d explicit, since we know already d´1 " Op1q by Proposition 2.5. Then ζq P H 2 pΩq X H 1 0 pΩq and´∆ pζqq " ζF´2∇ζ¨∇q´p∆ζqq .
(Notice that for the estimate on s it is crucial that the integral avoids B d{2 pxq.) Moreover, by To bound the remaining term }ζq 5 } 3{2 we argue as in Lemma 2.6 above and get We use the pointwise estimate (3.51) on ζF , which is equally valid for ζ 1{2 F . The term coming from |q| 5 is bounded bŷ ż which can be absorbed into the left side. The contributions from the remaining terms in the pointwise bound on ζ 1{2 |F | can by easily controlled and we obtain Collecting all the estimates, we obtain the claimed bound.  Once this is shown, we will be able to find a relation between λ and ε. The proof of (4.1) (and only this proof) relies on the nondegeneracy of critical points of φ a .
We already know that φ a px 0 q " 0 and that φ a pyq ě 0 for all y P Ω, hence x 0 is a critical point of φ a . In this subsection we collect the necessary ingredients which exploit this fact.
Lemma 4.1. The function φ a is of class C 2 on Ω.
Since we were unable to find a proof for this fact in the literature, we provide one in Appendix B.2.
Thus, the following general lemma applies to φ a .

(4.2)
Suppose additionally that Hess up0q ě c for some c ą 0 in the sense of quadratic forms, i.e. the origin is a nondegenerate minimum of u. Then, as x Ñ 0, Proof. We abbreviate Hpxq " Hess upxq and make a Taylor expansion around x to get and 0 " ∇up0q " ∇upxq´Hpxqx`op|x| 2 q . (4.5) We infer from (4.5) and the invertibility of Hp0q that x " Hpxq´1∇upxq`op|x| 2 q .
To prove ( (3.5). The facts that x 0 P N a and that Q V px 0 q ď 0 follow from Corollary 3.9.
By Lemma 4.1 and the assumption that x 0 is a nondegenerate minimum of φ a , we can apply Lemma 4.2 to the function upxq :" φ a px`x 0 q to get φ a pxq À |∇φ a pxq| 2 .
The remaining claims in Theorem 1.5 follow from Proposition 3.1. 8 . In this subsection, we prove a crude bound on the L 8 norm of the first-order remainder w appearing in the decomposition u " αpP U x,λ`w q, and also on some of its L p norms which cannot be controlled through Sobolev, i.e. p ą 6. This bound was not needed in the proof of Theorem 1.5, but will be in that of Theorem 1.6. Proposition 4.3. As ε Ñ 0, }w} p À λ´3 p for all p P p6, 8q.

A bound on }w}
(4.8) Moreover, for every µ ą 0, Our proof follows [31, proof of (25)], which concerns the case N ě 4 and a " 0. Since some of the required modifications are rather complicated to state, we give details for the convenience of the reader.
Proof. We begin by proving the first bound in the proposition, which we write as }w} r`1 3pr`1q À λ´1 for all r P p1, 8q .
Putting these estimates together, we obtain by the relationship between ε and λ proved in Theorem 1.5. Moreover, U x,λ pxq " λ 1{2 " }U x,λ } 8 . This finishes the proof of part (a) in Theorem 1.6.
The proof of part (b) necessitates much fewer prerequisites. It only relies on the crude expansion of u given in Proposition 2.1 and the rough bounds on w from Proposition 4.3.
The second term on the right side of (4.14) is easily bounded by εˇˇˇˇż Ω G a pz, yqV pyqupyqˇˇˇˇÀ ε}G a pz,¨q} 2 p}U } 2`} w} 2 q À ελ´1 {2 by the bounds from Proposition 2.1 and from Lemma A.1. Collecting the above estimates, part (b) of Theorem 1.6 follows.

Subcritical case: A first expansion
In the remaining part of the paper we will deal with the proof of Theorems 1.2 and 1.3. The structure of our argument is very similar to that leading to Theorems 1.5 and 1.6. Namely, in the present section we derive a preliminary asymptotic expansion of u ε and the involved parameters, which is refined subsequently in Section 6 below. Because of the similarities to the above argument, we will not always give full details.
The following proposition summarizes the results of this section.
Proposition 5.1. Let pu ε q be a family of solutions to (1.2) satisfying (1.5). Then, up to the extraction of a subsequence, there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R and pw ε q Ă T K xε,λε such that u ε " α ε pP U xε,λε`wε q (5.1) and a point x 0 P Ω such that 5.1. A qualitative initial expansion. As a first step towards Proposition 5.1, we observe that the qualitative expansion from Proposition 2.2 still holds true, that is, there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R and pw ε q Ă T K xε,λε such that (5.1) holds and a point x 0 P Ω such that, along a subsequence, where, as before, d ε :" dpx ε , BΩq.
On the other hand, the right side is Á 1 by coercivity of´∆`a, which is a consequence of criticality, and by Hölder. This gives }u ε } 6´ε Á 1, and hence }∇u ε } 2 Á 1 by Sobolev and Hölder. This completes the analogue of Step 1 in the proof of Proposition 2.2.
Let us now turn to Step 2 in that proof. We denote by u 0 a weak limit point of u ε in H 1 0 pΩq, which exists by Step 1. Still by Step 1, we may assume that the quantities }u ε } 6´ε and }∇u ε } 2 have non-zero limits. The only difference to Proposition 2.2 is now that we modify the definition of M to where the exponent is 6´ε instead of 6. Thanks to the uniform bound }u ε } 6´ε À 1 by Step 1, it can be easily checked that the proof of the Brézis-Lieb lemma (see e.g. [25]) still yields Then the modified assumption (1.5) can be used to conclude The rest of the proof is identical to Proposition 2.2.
We again adopt the convention that in the remainder of the proof we only consider the above subsequence and we will drop the subscript ε.

5.2.
The bound on }∇w} 2 . The goal of this subsection is to prove Proposition 5.2. As ε Ñ 0, Note that, in contrast to Proposition 2.4, there appears an additional error Opεq. We will prove in an extra step (Proposition 5.5) that ε " Oppλdq´1q, so this extra term will disappear later.
The proof of Proposition 5.2 is somewhat lengthy and we precede it by an auxiliary result, which is a simple consequence of the fact that α Ñ 1.
A useful consequence of this lemma is that Indeed, this follows from the lemma together with the fact that U x,λ Á λ´1 {2 in Ω.
Hence equation (5.5) combined with the fact that α Ñ 1 implies ż we have λ´ε 2 Ñ 1 and hence the claim.
We are now in position to give the Proof of Proposition 5.2. From equation (1.2) for u we obtain the following equation for w, ∆w`aw "´3U 5 x,λ´a P U x,λ`3 α 4´ε pP U x,λ`w q 5´ε . (5.10) Integrating this equation against w gives ż Ω p|∇w| 2`a w 2 q "´ż As before, the first term on the right hand side is controlled easily by Hölder,ˇˇˇż In order to control the second term we use the fact that P U x,λ " U x,λ´ϕx,λ . Moreover, by Taylor and (5.4), pP U x,λ`w q 5´ε " pU x,λ´ϕx,λ`w q 5´ε " U 5´ε x,λ`p 5´εqU 4´ε x,λ ẁ O´U 4 x,λ ϕ x,λ`U 3 x,λ w 2`| w| 5´ε`ϕ5´ε x,λ¯. (5.12) We estimate the first term on the right side using Lemma 5.4. For the second term on the right side we argue as in the proof of Proposition 2.4 and obtain ż Ω U 4 x,λ ϕ x,λ |w| " O´pλdq´1 }∇w} 2¯.

5.3.
The bound on ε. The goal of this subsection is to prove Proposition 5.5. As ε Ñ 0, ε " Oppλdq´1q . (5.14) We note that the analogue of this proposition is not needed in Section 2 when studying (1.3).
The proof of Proposition 5.5 is based on the Pohozaev-type identity ż which arises from integrating equation (4.4) against B λ P U x,λ and inserting the following bounds.
Before proving Lemma 5.6, let us use it to deduce the main result of this subsection.
In the proof of Lemma 5.6 we need the following auxiliary bound.
The proof of this lemma is analogous to that of Lemma 5.4 and is omitted.
Next, we prove (5.17). Using (5.12) and (5.4) we bound pointwise The integral over Ω of the two remainder terms is bounded by a constant times 2 6 , where in the last inequality we used the bounds from Lemmas A.1 and A.2.
By Lemma 5.7, the integral over Ω of the second term on the right side of (5.19) is bounded by a constant times ελ´1}∇w} 2 " opελ´1q.
Finally, by an explicit calculation, ż where, in the last step, we used Lemma 5.3. This completes the proof of (5.17).

5.4.
Excluding boundary concentration. The goal of this subsection is to prove The proof is very similar to that of Proposition 2.5 and we will be brief. Integrating the first equation in (1.2) against ∇u implies the Pohozaev-type identitý ż The volume integral on the left side can be estimated as before, since by Propositions 5.2 and 5.5 we have the same bound }∇w} 2 2 À λ´1`pλdq´2 as before. To bound the surface integral, we use the fact that ż This is the analogue of Lemma 2.6. We only note that by (5.10) we have and that this function satisfies (2.15). Therefore, using the above bound on }∇w} 2 we can proceed exactly in the same was as in the proof of Lemma 2.6.

Subcritical case: Refining the expansion
As in the additive case, we refine the analysis of the remainder term w ε in Proposition 5.1, which we write as w ε " λ´1 {2 ε pH 0 px ε ,¨q´H a px ε ,¨qq`s ε`rε with s ε and r ε as in (3.4).
The following proposition summarizes the main results of this section. Proposition 6.1. Let pu ε q be a family of solutions to (1.2) satisfying (1.5). Then, up to the extraction of a subsequence, there are sequences px ε q Ă Ω, pλ ε q Ă p0, 8q, pα ε q Ă R, ps ε q Ă T xε,λε and pr ε q Ă T K xε,λε such that u ε " α ε pψ xε,λε`sε`rε q (6.1) and a point x 0 P Ω such that, in addition to Proposition 5.1, We will prove Proposition 6.1 through a series of propositions in the following subsections.
The prefactor λ´ε {2 on the right side tends to 1 by Lemma 5.3. The first integral in the parentheses is bounded in (3.22). For the second integral we proceed again as in (5.13) and obtaiňˇˇˇˇż Ω U 4 x,λˆe ε log where we used (3.10) in the last inequality. Thus, recalling the bound on ε in (5.2),ˇż The fourth term on the right side of (6.8) is bounded, in absolute value, by a constant times ż Ω U 4 x,λ´λ´1 {2 |H a px,¨q|`|f x,λ |¯|r| À´λ´1φ a pxq`λ´2¯}r} 6 , where we used (3.23).
Using Lemma 5.4 to control the first term on the right hand side of (6.8) and putting all the estimates into (6.7) we finally get ż This, in combination with the coercivity inequality of Lemma 2.3, implies the claim.
Proof. As in the proof of Lemma 5.3 we integrate equation (1.2) against u. However, this time we write u " αpψ x,λ`q q and obtain ż Ω |∇pψ x,λ`q q| 2`ż Ω apψ x,λ`q q 2 " 3α 4´ε ż Ω pψ x,λ`q q 6´ε , which we write as ż We discuss separately the three terms that are involved in the identity (6.10).
The first term on the right side was already computed in the proof of Lemma 3.7 (b) and the last term on the right side can be bounded in the same way as there, except that now, instead of (3.27), we use the bound }∇q} 2 À λ´1 , (6.11) which follows from the bounds on s and r in Propositions 3.2 and 6.6. For the second term on the right side we proceed as in the proof of Lemma 5.4 and obtaiňˇˇˇż Ω q´U 5´ε x,λ´U 5 x,λ¯ˇÀ ελ 1´ε{2 ż Ω |q|U 5 x,λ |x´y| ď ελ 1´ε{2 }q} 6 À ε}q} 6 À ελ´1 .
Finally, we bound R 0 , the term on the right side of (6.10). Because of (6.11), the first integral in the definition of R 0 is Opλ´2q. The second integral is bounded, in absolute value, by a constant times ż Inserting all the bounds in (6.10), we obtain the claimed bound.
6.3. Expanding φ a pxq. In this subsection we prove the following important expansion. Proposition 6.4. As ε Ñ 0, φ a pxq " π apxq λ´1`π 32 ελ`1`op1q˘`opλ´1q . (6.12) The proof of this proposition, which is the analogue of Proposition 3.8, is a refined version of the proof of Proposition 5.5. We integrate equation (1.2) for u against B λ ψ x,λ and we write the resulting equality in the form ż Lemma 6.5. As ε Ñ 0, the following holds. (a) The proof of Lemma 6.5 is independent of the expansion of α 4´ε in Proposition 6.3. We only use the fact that α " 1`op1q.
We write the first integral on the right side as ż Ω´U 5 x,λ´α x,λ¯B λ ψ x,λ . (6.14) As shown in the proof of Lemma 3.10 (a), ż x,λ´U 5 x,λ˘H a px,¨q`opλ´3q .
(6.15) This will complete our discussion of the right hand side of (6.14) and hence the proof of (a).
The proof of (6.15) is similar to the corresponding argument in the proof of Lemma 3.10 (a), but we include some details. We bound pointwise Using the bounds from Lemmas A.1 and A.2, we easily find that the remainder term, when integrated against |B λ ψ x,λ | is opλ´3q. Using expansion (B.5) we obtain, by an explicit calculation similar to (B.11) and (B.13), ż where we used Lemma 5.3. In the same way, we get ż This proves (6.15).
Consequently, using (6.11) and (6.16),ˇˇˇż Collecting all the bounds, we arrive at the claimed expansion in (b).
The proof of (d) uses similar bounds as in the rest of the proof and is omitted.
Using the expansions (3.13) of β and γ, this can be simplified to which is the assertion.
6.4. Bounding ∇φ a . In this subsection we prove the bound on ∇φ a pxq in Proposition 6.1.
Proposition 6.6. For every µ ă 1, as ε Ñ 0, Note that together with (5.2) it follows from Proposition 6.6 that x 0 is a critical point of φ a .
The proof of Proposition 6.6 is a refined version of the proof of Proposition 5.8 and is again based on the Pohozaev identity (5.21). The latter reads, in the notation of (3.46), 0 " Irψ x,λ s`2 Irψ x,λ , qs`Irqs . (6.18) To control the boundary integrals involving q in this identity, we need the following lemma, which is the analogue of Lemma 3.13.
Before proving this lemma, let us use it to complete the proof of Proposition 6.6. In that proof, and later in this subsection, we will use the inequality This follows from the bound (3.10) on s and the bound in Proposition 6.2 on r.
Equation (1.13) follows from (6.5). In case φ a px 0 q ‰ 0 this is immediate, and in case φ a px 0 q " 0 we use, in addition, the expansion of β from Proposition 3.3 and the fact that ε " opλ´1q by (7.1).
Appendix B. Properties of the functions H a px, yq In this appendix, we prove some properties of H a px, yq needed in the proofs of the main results. Since these properties hold independently of the criticality of a, we state them for a generic function b which satisfies the same regularity conditions as a, namely, b P CpΩq X C 2,σ loc pΩq for some 0 ă σ ă 1 .
(In fact, in Subsection B.1 we only use b P CpΩq X C 1,σ loc pΩq for some 0 ă σ ă 1.) In addition, we assume that´∆`b is coercive in Ω with Dirichlet boundary conditions. Note that the choice b " 0 is allowed. with C uniform for x in compact subsets of Ω. Proof.
Step 1. We first prove the bounds for the special case b " 0, which we shall need as an ingredient for the general proof. Since H 0 px,¨q is harmonic, we have ∆ y ∇ y H 0 px, yq " 0. Moreover, we have the bound ∇ y G 0 px, yq À |x´y|´2 uniformly for x, y P Ω [37, Theorem 2.3]. This implies that for x in a compact subset of Ω and for y P BΩ, |∇ y H 0 px, yq| " |∇ y p|x´y|´1q´∇ y G 0 px, yq| ď C .
We now conclude by the maximum principle.
The proof for the bound on ∇ x H 0 px, yq is analogous, but simpler, because ∇ x G 0 px, yq " 0 for y P BΩ.
Step 2. For general b, we first prove the bounds for both x and y lying in a compact subset of Ω. By [17, proof of Lemma 2.5] we have with }Ψ x } C 1,µ pKq ď C for every 0 ă µ ă 1 and every compact subset K of Ω, and with C uniform for x in compact subsets. This shows that |∇ y H b px, yq| ď C uniformly for x, y in compact subsets of Ω. By symmetry of H b , this also implies |∇ x H b px, yq| ď C uniformly for x, y in compact subsets of Ω.
Step 3. We complete the proof of the lemma by treating the case when x remains in a compact subset, but y is close to the boundary. In particular, we may assume for what follows |x´y|´1 À 1.

By
Step 1, the derivatives of H 0 px, yq are uniformly bounded.
We thus only need to consider the integral term. Its B x i -derivative equals ż where we again used the fact that (B.2) holds for b " 0, together with (B.4). This completes the proof of (B.2).
The proof of (B.3) can be completed analogously. It suffices to write the resolvent formula as This gives (B.5) provided we can show that for each fixed x P Ω, Indeed, by using (B.7) twice with the roles of x and y exchanged, subtracting and recalling H b px, yq " H b py, xq, we get φ b pyq´φ b pxq " p∇Ψ y pyq`∇Ψ x pxqqpy´xq`b pyq´bpxq 2 |x´y|`Op|x´y| 1`µ q " p∇Ψ y pyq`∇Ψ x pxqqpy´xq`Op|x´y| 1`µ q , (B.9) because b P C 0,µ loc pΩq. We now argue that Ψ y Ñ Ψ x in C 1 loc pΩq, which implies ∇Ψ y pyq Ñ ∇Ψ x pxq. Together with this, (B.8) follows from (B.9).
To justify the convergence of Ψ y we argue similarly as in [17,Lemma 2.5]. We note that´∆ z Ψ y " F y pzq with F y pzq :" bpzq´bpyq |z´y|´b pzqH b py, zq .
We claim that F y Ñ F x in L p loc pΩq for any p ă 8. Indeed, the first term in the definition of F y converges pointwise to F x in Ωztxu and is locally bounded, independently of y, since b P C 0,1 loc pΩq. Thus, by dominated convergence it converges in L p loc pΩq for any p ă 8. Convergence in L 8 loc pΩq of the second term in the definition of F y follows from the bound on the gradient of H b in Lemma B.1 . This proves the claim.
We omit the details.
For the proof of (B.12) we use the explicit formula for B x i U x,λ in Lemma A.1. We split the integral into B d pxq and ΩzB d pxq. In the first one, we used the bound (B.1) and the expansion (B.5). By oddness, the contribution coming from φ a pxq cancels, as does the contribution from ř k‰i B k φ b pxqpy k´xk q. For the remaining term we use ż As similar computation shows that the contribution from the error |x´y| 1`µ on B d pxq is Opλ´1 {2´µ q. Finally, the bounds from Lemma A.1, show that the contribution from ΩzB d pxq is Opλ´5 {2 q. This completes the proof.
B.2. C 2 differentiability of φ a . In this subsection, we prove Lemma 4.1. The argument is independent of criticality of a and we give the proof for a general function b P C 0,1 pΩq X C 2,σ loc pΩq for some 0 ă σ ă 1. The following argument is similar to [17,Lemma 2.5], where a first-order differentiability result is proved, and to [12,Lemma A.1], where it is shown that φ b P C 8 pΩq for constant b.
Since b P C 2,σ loc pΩq and since H b is Lipschitz by Lemma B.1, the right side is in C 0,σ loc pΩq as a function of y. By elliptic regularity, Ψpx, yq is in C 2,σ loc pΩq as a function of y. Since Ψpx, yq is symmetric in x and y, we infer that Ψpx, yq is in C 2,σ loc pΩq as a function of x.
It remains to justify the existence of mixed derivatives B y j B x i Ψpx, yq. For this, we carry out a similar elliptic regularity argument for the function B x i Ψpx, yq. We havé Since b P C 1,1 loc pΩq, and since B x i H b is bounded by Lemma B.1, the right side is in L 8 loc pΩq as a function of y. By elliptic regularity, B x i Ψpx, yq P C 1,µ pΩq for every µ ă 1, as a function of y. In particular, the mixed derivative B y j B x i Ψpx, yq is in C 0,µ loc pΩq as a function of y. By symmetry, the same argument shows that the mixed derivative B x j B y i Ψpx, yq is in C 0,µ loc pΩq as a function of x.
The proof of Lemma 4.1 is therefore complete.