Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

For a bounded open set Ω⊂R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^3$$\end{document} we consider the minimization problem S(a+ϵV)=inf0≢u∈H01(Ω)∫Ω(|∇u|2+(a+ϵV)|u|2)dx(∫Ωu6dx)1/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$\end{document}involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of S(a+ϵV)-S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(a+\epsilon V)-S$$\end{document} as ϵ→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow 0+$$\end{document}, where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have S(a+ϵV)<S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(a+\epsilon V)<S$$\end{document} for all sufficiently small ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document}.


Introduction and main results
1.1.Setting of the problem.In their celebrated paper [8] Brézis and Nirenberg considered the problem of minimizing the quotient S a rus :" ş Ω p|∇u| 2 `a|u| 2 q dx p ş Ω u 6 dxq 1{3 over all 0 ı u P H 1 0 pΩq, where Ω Ă R 3 is a bounded open set and a is a continuous function on Ω.We denote the corresponding infimum by Spaq :" inf This number is to be compared with S :" 3 ˆπ 2 ˙4{3 , the sharp constant [24,25,3,30] in the Sobolev inequality One of the findings in [8] is that if a is small (for instance, in L 8 pΩq), then Spaq " S.This is in stark contrast to the case of dimensions N ě 4 where the corresponding analogue of Spaq (with the Date: August 1, 2019.c 2019 by the authors.This paper may be reproduced, in its entirety, for non-commercial purposes.Partial support through US National Science Foundation grant DMS-1363432 (R.L.F.) and Studienstiftung des deutschen Volkes (T.K.) is acknowledged.H. K. has been partially supported by Gruppo Nazionale per Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). 1 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 notion due to Hebey and Vaugon [19].Definition 1.1.Let a be a continuous function on Ω.We say that a is critical in Ω if Spaq " S and if for any continuous function ã on Ω with ã ď a and ã ı a one has Spãq ă Spaq.
Our goal in this paper is to compute the asymptotics of Spa `ǫV q ´S as ǫ Ñ 0 for critical a and to understand the behavior of corresponding minimizers.Here V is a bounded function on Ω, without any restrictions on its sign.
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 [23].If the operator ´∆ `a in Ω with Dirichlet boundary conditions is coercive (which, in particular, is the case if a is critical), then it has a Green's function G a satisfying $ ' & ' % ´∆x G a px, yq `apxq G a px, yq " 4π δ y in Ω , G a px, yq " 0 on BΩ . (1. 2) The regular part of G a is defined by H a px, yq :" 1 |x ´y| ´Ga px, yq . ( It is well-known that for each x P Ω the function H a px, ¨q, which is originally defined in Ωztxu, extends to a continuous function in Ω and we abbreviate φ a pxq :" H a px, xq . It is well-known that the function φ a is relevant for problems involving the critical Sobolev exponent, see, e.g., [26] and [4].For the problem at hand, it was shown in [6,Thm. 7] that if φ a pxq ă 0 for some x P Ω, then Spaq ă S. (In [6] this is attributed to Schoen [26] and a work in preparation by McLeod.)Conversely, it was conjectured in [6] and proved by Druet in [12] that if Spaq ă S, then φ a pxq ă 0 for some x P Ω.An alternative proof, assuming only continuity of a, is given in [15].Thus, the (non-local) condition min Ω φ a ă 0 is necessary and sufficient for Spaq ă S, and replaces the (local) condition min Ω a ă 0 in dimensions N ě 4.
The above results imply that, if a is critical, then min Ω φ a " 0. In particular, the set N a :" tx P Ω : φ a pxq " 0u is non-empty.
1.2.Main results.Let us proceed to a precise statement of our main results.Throughout this paper we work under the following assumption.
Assumption 1.2.The set Ω Ă R 3 is open, bounded and has a C 2 boundary.The function a satisfies a P CpΩq X C 1 pΩq and is critical in Ω.Moreover, apxq ă 0 for all x P N a . (1.4) Finally, V P L 8 pΩq.
We will see in Corollary 2.2 that criticality of a alone implies apxq ď 0 for all x P N a .Therefore assumption (1.4) is not severe.
We set and N a pV q :" tx P N a : Q V pxq ă 0u .
The following is our main result.
Theorem 1.3.Assume that N a pV q ‰ H. Then Spa `ǫV q ă S for all ǫ ą 0 and We supplement this theorem with a result for the opposite case where N a pV q " H.
Theorem 1.4.Assume that N a pV q " H. Then Spa `ǫV q " S `opǫ 2 q as ǫ Ñ 0`.If, in addition, Q V pxq ą 0 for all x P N a , then Spa `ǫV q " S for all sufficiently small ǫ ą 0.
It follows from the above two theorems that the condition N a pV q ‰ H is 'almost' necessary for the inequality Spa `ǫV q ă S for all small ǫ ą 0. Only the case where min Na Q V " 0 is left open.
Example 1.5.When Ω " B is the unit ball in R 3 , then it is well-known that the constant function a " ´π2 {4 is critical and that in this case N a " t0u and G a p0, yq " |y| ´1 cospπ|y|{2q; see, e.g., [6].Thus, with we have and Spa `ǫV q " S for all sufficiently small ǫ ą 0 if q V ą 0.
Remark 1.6.It is instructive to compare our results here with the results for the analogous problem SpǫV q :" inf ş Ω p|∇u| 2 `ǫV u 2 q dx ´şΩ |u| 2N {pN ´2q dx ¯pN´2q{N in dimension N ě 4. Let S N be the sharp constant in the Sobolev inequality in R N .From [8] we know that SpǫV q ă S N if and only if V pxq ă 0 for some x P Ω, and therefore we focus on the case where N pV q :" tx P Ω : V pxq ă 0u ‰ H. Then SpǫV q " S N ´CN sup SpǫV q " S N ´exp with explicit constants C N depending only on N .Note that, as a reflection of the Brézis-Nirenberg phenomenon, V enters pointwisely into the asymptotic coefficient in (1.7) and (1.8), while it enters non-locally through Q V into the asymptotic coefficient in Theorem 1.3.
Asymptotics (1.7) and (1.8) in the case where V is a negative constant are essentially contained in [29]; see also [31] for related results.The case of general V P CpΩq can be treated by similar methods.We emphasize that the proof of Theorem 1.3 is considerably more complicated than that of (1.7) and (1.8), since the expansion in Theorem 1.3 should rather be thought of as a higher order expansion of Spa `ǫV q ´S where the coefficient of the term of order ǫ vanishes due to criticality.In the higher dimensional context, no such cancellation occurs.

1.3.
Behavior of almost minimizers.We prove Theorems 1.3 and 1.4 by proving upper and lower bounds on Spa `ǫV q.For the upper bound it suffices to evaluate S a`ǫV ru ǫ s for an appropriately chosen family of functions u ǫ .For the lower bound we need to evaluate the same quantity where now u ǫ is an optimizer for Spa `ǫV q.To do so, we will show that u ǫ is essentially of the same form as the family chosen to prove the upper bound.In fact, we will not use the minimality of the u ǫ and show that, more generally, all 'almost minimizers' have essentially the same form as the functions chosen for the upper bound.
Given earlier works and, in particular, those by Druet [12] and Esposito [15] it is not surprising that almost minimizers concentrate at a point in the set N a .One of our new contributions is to show that this concentration happens at a point in the subset N a pV q and, more precisely, at a point in N a pV q where the supremum in (1.6) is attained.
In order to state our theorem about almost minimizers, for x P Ω and λ ą 0, let U x,λ pyq :" λ 1{2 p1 `λ2 |y ´x| 2 q 1{2 .The functions U x,λ and their multiples are precisely the optimizers of the Sobolev inequality (1.1); see the references mentioned above and [21, Cor.I.1].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 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 dy.Finally, 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.
Theorem 1.7.Assume that N a pV q ‰ H. Let pu ǫ q Ă H 1 0 pΩq be a family of functions such that lim ǫÑ0 S a`ǫV ru ǫ s ´Spa `ǫV q S ´Spa `ǫV q " 0 and (1.10) Then there are px ǫ q Ă Ω, pλ ǫ q Ă p0, 8q and pα ǫ q Ă R such that and, along a subsequence, for some s P t˘1u .
There is a huge literature on blow-up results for solutions of equations involving the critical Sobolev exponent.Early contributions related to the problem we are considering are, for instance, [2,10,9,18,22]; see also the book [13] for more recent developments and further references.Here we follow a somewhat different philosophy and focus not on the equation satisfied by the minimizers, but solely on their minimality property.Therefore our proofs also apply to almost minimizers in the sense of (1.10) and we obtain blow-up results for those as well.On the other hand, with our methods we cannot say anything about non-minimizing solutions of the corresponding equation and our blow-up bounds are only obtained in H 1 instead of L 8 norm.Other related works which study Sobolev critical problems from a variational point of view are, for instance, [17,1,16].
As already mentioned before, the works of Druet [12] and Esposito [15], and similarly [17,1] in related problems, show that concentration happens at a point in N a .In terms of Spa `ǫV q, this corresponds essentially to the fact that Spa `ǫV q " S `opǫq.In order to go further than that and to compute the coefficient of ǫ 2 , we need to prove that concentration happens in the subset N a pV q at a point where the supremum in (1.6) is attained.
The strategy of the proof of the lower bound is to expand the quotient S a`ǫV ru ǫ s for an almost minimizer u ǫ as precisely as allowed by the available information on u ǫ , then to use a coercivity bound to deduce that certain terms are small and thereby improving our knowledge about u ǫ .We repeat this procedure three times (namely, in Sections 4, 5 and 6).Therefore, a key tool in our analysis is the coercivity of the quadratic form ż provided that λ distpx, BΩq is sufficiently large; see Lemma 4.3.This coercivity was proved by Esposito [15] and comes ultimately from the non-degeneracy of the Sobolev minimizer U x,λ .Esposito used this bound to obtain an a priori bound on the term α ´1 ǫ u ǫ ´P U xǫ,λǫ in Theorem 1.7.We will use it for the same purpose in Proposition 4.1, but then we will use it two more times in Propositions 5.1 and in Lemma 6.6 in order to get bounds on α ´1 ǫ u ǫ ´P U xǫ,λǫ `λ´1{2 pH a px ǫ , ¨q ´H0 px ǫ , ¨qq and α ´1 ǫ u ǫ ´P U xǫ,λǫ `λ´1{2 Π K x,λ pH a px ǫ , ¨q ´H0 px ǫ , ¨qq, respectively.After the last step we are able to compute the energy to within opǫ 2 q.We emphasize that in principle there is nothing preventing us from continuing this procedure and computing the energy to even higher precision.
Let us briefly comment on a surprising technical subtlety in our proof.While Theorem 1.7 says that almost minimizers are essentially given by P U x,λ ´λ´1{2 Π K x,λ pH a px, ¨q ´H0 px, ¨qq with x P N a pV q a maximum point for the right side in (1.6) and λ proportional to ǫ ´1, to prove the upper bound we use the simpler functions P U x,λ ´λ´1{2 pH a px, ¨q ´H0 px, ¨qq (with the same choices of x and λ).The difference between the two functions, namely ´λ´1{2 Π x,λ pH a px, ¨q ´H0 px, ¨qq , can be shown to be of order ǫ (when λ is proportional to ǫ ´1), but not smaller; see Remark 6.2.Therefore it is not at all obvious that the two families of functions lead to the same (within opǫ 2 q) value of S a`ǫV r¨s.The fact that they do is contained in Lemma 6.3, where the contributions of ´λ´1{2 Π x,λ pH a px, ¨q ´H0 px, ¨qq to the numerator and to the denominator are shown to cancel each other to within opǫ 2 q.
At first sight, the problem considered in this paper resembles the problem of minimizing the quotient ş R N p|∇u| p `ǫV |u| p q dx{ ş R N |u| p dx for p ď N , which is a classical problem for p " 2 [27] motivated by quantum mechanics and which was studied in [14] for general p.The underlying mechanism, however, is rather different.In these works almost minimizers spread out, whereas here they concentrate.The concentration regime is much more sensitive to the local details of the perturbation and necessitates, in particular, the use of orthogonality conditions in T K x,λ and the resulting coercivity.1.4.Notation.Given a set M and two functions f 1 , f 2 : M Ñ R, we write f 1 pmq À f 2 pmq if there is a numerical constant c such that f 1 pmq ď c f 2 pmq for all m P M .The symbol Á is defined analogously.For any p P r1, 8s and u P L p pΩq we denote }u} p " }u} L p pΩq .

Upper bound on Spa `ǫV q
Recall that we always work under Assumption 1.2.In this section (and only in this section), however, we do not assume (1.4).

2.1.
Statement of the bounds and consequences.Our goal in this section is to prove an upper bound on Spa `ǫV q by evaluating the quotient S a`ǫV r¨s on a certain family of trial functions.For x P Ω and λ ą 0, let ψ x,λ pyq :" P U x,λ pyq ´λ´1{2 pH a px, yq ´H0 px, yqq . (2.1) This function belongs to H 1 0 pΩq.We shall prove the following expansions.In particular, In the proof of Theorem 2.1 we do not use the fact that a is critical.We only use the fact that ´∆ `a is coercive.In the following corollary we use criticality.
Corollary 2.2.One has φ a pxq ě 0 for all x P Ω and apxq ď 0 for all x P N a .
The first part of this corollary appears in [6,Thm. 7].Note that the second part is non-trivial since we do not assume (1.4).
Corollary 2.3.Assume that N a pV q ‰ H. Then Spa `ǫV q ă S for all ǫ ą 0 and, as ǫ Ñ 0`, where the right side is to be understood as ´8 if apxq " 0 for some x P N a pV q.
Proof.We fix x P N a and k ą 0 and apply (2.4) with λ " pkǫq ´1.Since Spa `ǫV q ď S a rψ x,λ s, we obtain Thus, which implies the claimed upper bound.
For each u P H 1 0 pΩq, ǫ Þ Ñ S a`ǫV rus is an affine linear function, and therefore its infimum over u, which is ǫ Þ Ñ Spa `ǫV q, is concave.Since Spa `ǫV q ă S for all sufficiently small ǫ ą 0, as we have just shown, we conclude that Spa `ǫV q ă S for all ǫ ą 0.

Auxiliary facts.
In this preliminary subsection we collect some expansions that will be useful in the proof of Theorem 2.1 as well as later on.In order to emphasize that criticality is not needed, we state them for a function b P CpΩq X C 1 pΩq such that the operator ´∆ `b in Ω with Dirichlet boundary conditions is coercive.Lemma 2.4.As λ Ñ 8, uniformly in x from compact subsets of Ω, the first bound follows immediately from To prove the second bound, we write The last term on the right side can be bounded as before, using the fact that H b px, ¨q is uniformly bounded in L 8 pΩq for x in compact subsets of Ω, see (2.6) below.The first term on the right side can be bounded using This proves the lemma.
Lemma 2.5.As λ Ñ 8, uniformly for x in compact subsets of Ω, ż (2.9) The asymptotics are uniform for x from compact subsets of Ω.
To prove this, let as well as the fact that ∆|x| " 2|x| ´1 as distributions we see that Ψ x is a distributional solution of where By Step 1 and the assumption b P CpΩq X C 1 pΩq, we have F x P L 8 loc pΩq.In particular, F x P L p loc pΩq for any 3 ă p ă 8 and therefore, by elliptic regularity (see, e.g., [20,Thm. 10.2]), Ψ x P C 1,α loc pΩq for α " 1 ´3{p.Thus, in particular, Ψ x P C 1 pΩq.Inserting the Taylor expansion Ψ x pyq " ∇ y Ψ x pxq ¨py ´xq `op|y ´x|q as y Ñ x into (2.10),we obtain the claim with ξ x " ∇ y Ψ x pxq.The uniformity statement follows from the fact that if x is from a compact set K Ă Ω, then there is an open set ω with K Ă ω Ă ω Ă Ω such that the norm of F x in L p pωq is uniformly bounded for x P K.
The argument in Step 2 is the only place in this paper where we use the C 1 assumption on a.
Clearly the same proof would work if we only assumed a P C 1,α pΩq for some α ą 0.
Lemma 2.6.As λ Ñ 8, uniformly for x in compact subsets of Ω, The proof is similar, but simpler than that of Lemma 2.5 and is omitted.We only note that the constant comes from ż On the complement of B ρ pxq we use the bound (2.5), which gives Choosing ρ " 1{ ln λ we obtain the bound in the lemma.
The same proof shows that if b is merely continuous, but not necessarily C 1 , then the expansion still holds with an error opλ ´2q.This would be sufficient for our analysis.

2.3.
Expansion of the numerator.One easily checks that for all x P R 3 and λ ą 0, This, together with the equation (2.11), the harmonicity of H 0 px, ¨q and (1.9), implies that We now introduce f x,λ by and recall that [23, Prop. 1 (b)], with d :" distpx, BΩq, Hence, by (2.16) and the fact that ψ x,λ vanishes on the boundary, It is easy to see that ż x,λ pyq ´λ´1 2 apyq G a px, yq ˇˇdy " Opλ ´1{2 q and therefore, by (2.18) and the fact that x is in a compact subset of Ω, x,λ pyq ´λ´1 2 apyq G a px, yq ¯fx,λ pyq dy " Opλ ´3q .
A simple computation shows that the first term on the right side of (2.19) is For the second term we use Lemma 2.5 and obtain x,λ pyq H a px, yq dy " 4πφ a pxqλ ´1 ´4πapxqλ ´2 `opλ ´2q .
We will combine the third term with the term coming from ş Ω aψ 2 x,λ dy.
Using again expansion (2.17) of P U x,λ we find Using (2.18) and the fact that x is in a compact subset of Ω it is easy to see that x,λ dy " Opλ ´3p1 `ǫqq .
To summarize, we have shown that Finally, by Lemma 2.4, This proves the first assertion in Theorem 2.1.

2.4.
Expansion of the denominator.By the decomposition (2.17) for P U x,λ we obtain Using (2.6) and (2.18), together with the fact that x is in a compact subset of Ω, we see that the remainder term is Opλ ´3q.Next, we expand `λ´3 }H a px, ¨q} 6 6 q .Using (2.6), together with the fact that x is in a compact subset of Ω, we see that the remainder term is Opλ ´3 ln λq.The first three terms on the right side are evaluated in (2.20) and Lemmas 2.5 and 2.6.This proves the second assertion in Theorem 2.1.

Expansion of the quotient. Expansion (2.3) implies that
Expansion (2.4) now follows by multiplying the previous equation with (2.2).This concludes the proof of Theorem 2.1.

3.
Lower bound on Spa `ǫV q.Preliminaries 3.1.The asymptotic form of almost minimizers.The remainder of this paper is concerned with proving a lower bound on Spa `ǫV q which matches the upper bound from Corollary 2.3.We will establish this by proving that functions u ǫ for which S a`ǫV ru ǫ s is 'close' to Spa `ǫV q are 'close' to the functions ψ x,λ used in the upper bound for certain x and λ depending on ǫ.We will prove this in several steps.The very first step is the following proposition.Proposition 3.1.Let pu ǫ q Ă H 1 0 pΩq be a sequence of functions satisfying Then, along a subsequence, where α ǫ Ñ s for some s P t´1, `1u , x ǫ Ñ x 0 for some x 0 P Ω , }∇w ǫ } Ñ 0 and w ǫ P T K xǫ,λǫ . (3.3) If the u ǫ are minimizers for Spa `ǫV q, and therefore solutions to the corresponding Euler-Lagrange equation, this proposition is well-known and goes back to work of Struwe [28] and Bahri-Coron [5].
The result for almost minimizers is also well-known to specialists, but since we have not been able to find a proof in the literature, we include one in Appendix B.Here we only emphasize that the fact that u ǫ converges weakly to zero in H 1 0 pΩq is deduced from a theorem of Druet [12] which says that Spaq is not attained for critical a. (Note that this part of the paper [12] is valid for a P L 3{2 pΩq, without any further regularity requirement.) Convention.From now on we will assume that Spa `ǫV q ă S for all ǫ ą 0 (3.4) and that pu ǫ q satisfies (1.10).In particular, assumption (3.1) is satisfied.We will always work with a sequence of ǫ's for which the conclusions of Proposition 3.1 hold.To enhance readability, we will drop the index ǫ from α ǫ , x ǫ , λ ǫ , d ǫ and w ǫ .

A priori bounds
4.1.Statement of the bounds.From Proposition 3.1 we know that }∇w} " op1q and that the limit point x 0 of px ǫ q lies in Ω.The following proposition, which is the main result of this section, improves both these results.
Proof of Lemma 4.2.We will expand separately the numerator and the denominator in S a`ǫV ru ε s.
Expansion of the numerator.Since w is orthogonal to P U , we have The first term on the right side is computed in (A.1).The other terms in the numerator are and, by (A.5), ˇˇˇż Ω pa `ǫV qP U x,λ w dx ˇˇˇď }a `ǫV } 8 }P U x,λ } 6{5 }w} 6 " Opλ ´1{2 }∇w}q .
To summarize, the numerator is α 2 times Expansion of the denominator.We have x,λ w dy `15 x,λ w 2 dy `Op}∇w} 3 q .
The lemma follows immediately from the expansions of the numerator and the denominator.For details of the proof we refer to [15].
By assumption (1.10), this becomes Since all three terms on the right side are non-negative, we obtain (4.1), (4.2) and the first bound in (4.3).The second bound in (4.3) follows from the first one by assumption (1.10).This completes the proof of the proposition.

A priori bounds reloaded
5.1.Statement and heuristics for the improved a priori bound.In order to prove a sufficiently precise lower bound on Spa`ǫV q we need more detailed information on the almost minimizers u ε .Here we extract the leading term from the remainder term w " w ε in (3.2).
In particular, together with Corollary 2.2, we obtain min Ω φ a " 0 for critical a, which is Druet's theorem [12].Our proof, which is closely related to that by Esposito [15], uses another theorem of Druet, which says that Spaq is not attained for critical a [12, Step 1] (see Proposition 3.1), but is otherwise independent of [12].
The proof of Proposition 5.1 is given at the end of this section.Let us explain the heuristics behind the proof.In Lemma 5.2 we will derive the following expansion, x,λ w 2 ¯dy `opλ ´1q . (5.3) Note that this is an improvement over the expansion in Lemma 4.2, which only had a remainder Opλ ´1q.This improvement is possible thanks to the information from Proposition 4.1.
From the expansion (5.3) we want to determine the asymptotic form of w.In order to (almost) minimize the quotient S a`ǫV ru ε s the function w will (almost) minimize the expression This is quadratic and linear in w, so it can be minimized by 'completing a square'.If the term ´15 U 4 x,λ were absent, then the minimum would be ´λ´1 p4πq ´1 ĳ ΩˆΩ G 0 px, yqapyqG a py, y 1 qapy 1 qG 0 py 1 , xq dy dy 1   and the optimal choice for w would be ´λ´1{2 pH a px, ¨q ´H0 px, ¨qq.Using the positive contribution that arises when completing the square, we will be able to show that if u ǫ almost minimizes Spa`ǫV q, then w almost minimizes the above problem and is therefore almost equal to ´λ´1{2 pH a px, ¨q H0 px, ¨qq.Proposition 5.1 provides a quantitative version of these heuristics.
As the above argument shows, the main difficulty will be to show that the term ´15 U 4 x,λ is negligible to within opλ ´1q.This does not follow from a straightforward bound since }∇w} 2 is only Opλ ´1q.The orthogonality conditions satisfied by w will play an important role.

A second expansion.
In this subsection, we shall prove the following lemma.
Note that here we have kept the term ş Ω apP U 2 x,λ `2P U x,λ wq dy instead of estimating it.We now treat this contribution more carefully.We expand P U x,λ as in (2.17), which leads to x,λ dy .
Expansion of the denominator.Combining the bound from the proof of Lemma 4.2 with the bounds on d and }∇w} from Proposition 4.1, we obtain x,λ w 2 dy `opλ ´1q . (5.6) Expansion of the quotient.Multiplying (5.5) and (5.6) gives x,λ w 2 ¯dy `opλ ´1q .
The resolvent identity together with the symmetry G 0 px, yq " G 0 py, xq implies This completes the proof of the lemma.

Regularization and coercivity.
In this subsection we will show that the coercivity bound from Lemma 4.3 remains essentially true after regularization.A convenient regularization procedure for us is a spectral cut-off.Namely, we denote by 1p´∆ `a ď µ q the spectral projection for the interval p´8, µs of the self-adjoint operator ´∆ `a in L 2 pΩq with Dirichlet boundary condition.The parameter µ here will be later chosen large depending on ǫ.
The integral is easily seen to be bounded by a universal constant times This proves the claimed bound.
From [23, Appendix B] we know that, as λ Ñ 8, uniformly in x with λd ě T ˚, where T ˚is any fixed constant.Here " means that the quotient of both quantities is bounded from above and away from zero.Let φj :" and By [23, Appendix B] and (5.10), G j,k :" Opλ ´1q for all j ‰ k and G j,j " 1 for all j . (5.12) Hence, if λ is large enough, which follows from dλ ě T ˚with sufficiently large T ˚since Ω is bounded, then G is invertible and pG ´1{2 q j,k " δ j,k `Opλ ´1q . (5.13) Hence, by the Gram-Schmidt procedure, is an H 1 0 pΩq-orthonormal basis of T x,λ .

5.4.
Completing the square.The following lemma gives a lower bound on the term in (5.4) which involves w.As explained above, this is the crucial step in the proof of Proposition 5.1.
We now choose c " mint1, ρ{p2C 1 qu.Moreover, by (5.20) we can choose a δ ą 0, independent of ǫ and µ such that From now on, we fix this value of δ.
It remains to show that R 2 pδq is Opλ ´3{2 q for an appropriate choice of µ.By (4.1) and (5.25) and by the orthogonality (5.22) we have (5.26) Thus, since a P L 8 pΩq and since G 0 px, ¨q is uniformly bounded in L 2 pΩq, we have Moreover, by Lemma 5.3, ż Thus, With the choice µ " λ the right side becomes Opλ ´3{2 q, as claimed.
Now we prove the main result of this section.
Since ´∆ `a is coercive, the last bound implies By the resolvent identity, p´∆ `aq ´1aG 0 px, ¨q " G 0 px, ¨q ´Ga px, ¨q " H a px, ¨q ´H0 px, ¨q , and therefore, setting q :" w `λ´1{2 pH a px, ¨q ´H0 px, ¨qq, the previous bound can be rewritten as }∇q} 2 " opλ ´1q.This completes the proof of the proposition.

A refined decomposition of almost minimizers
From Proposition 5.1 we infer that any sequence pu ε q satisfying (1.10) can be decomposed as where ψ x,λ " P U x,λ ´λ´1{2 pH a px, ¨q ´H0 px, ¨qq is as in the proof of the upper bound, see (2.1), and where }∇q} " opλ ´1{2 q .
Thus, expanding S a`ǫV ru ǫ s leads to an expression that coincides with the upper bound in Corollary 2.2 up to additional terms involving q.Using coercivity we will be able to show that the contribution from r :" Π K x,λ q , the orthogonal projection of q onto T K x,λ in H 1 0 pΩq, is negligible; see Lemma 6.6 below.The main focus in this section is on Π x,λ q " Π x,λ ´w `λ´1{2 pH a px, ¨q ´H0 px, ¨qq ¯" λ ´1{2 Π x,λ pH a px, ¨q ´H0 px, ¨qq, where the last identity follows from w P T K x,λ .In Lemma 6.3 we will prove that the contribution from Π x,λ q is negligible.This is not obvious and, in fact, somewhat surprising since Π x,λ q is of order λ ´1 and not smaller.

Preliminary estimates. Let us write
Since P U x,λ , B λ P U x,λ and B x j P U x,λ , j " 1, 2, 3, are linearly independent for sufficiently large λ, the numbers β, γ and δ j , j " 1, 2, 3, (depending on ǫ, of course) are uniquely determined.The choice of the different powers of λ multiplying these coefficients is motivated by the following lemma.
Formulas for the Laplacians ∆ φj are given in (5.19) and the quantities }∇φ j } appearing there were estimated in (5.10).For a 1 , the integral ş Ω U 5 x,λ pH a px, yq ´H0 px, yqq dy is Opλ ´1{2 q according to Lemma 2.5, which proves the claim in (6.1).To bound a j for j " 2, . . ., 5 we compute This expression and straightforward bounds lead to the claim for a 2 in (6.1).

From
Step 1 in the proof of Lemma 2.5, recalling (4.2), we infer that there are ρ ą 0 and C ą 0, both independent of ǫ, such that ˇˇH a px, yq ´H0 px, yq ´Ha px, xq `H0 px, xq ˇˇÀ |y ´x| for all y P B ρ pxq .
Since the function U 4 x,λ B x j U x,λ is odd, we have ż Bρpxq pH a px, xq ´H0 px, xqqU 4 x,λ B x j U x,λ dy " 0 .
On the other hand, using the above expression for B x j U x,λ we find This proves (6.1) for j " 3, 4, 5.
Step 2. Let us deduce the statement of the lemma.We have In view of (5.10), the assertion of the lemma is equivalent to With respect to the orthonormal system ψ j , j " 1, . . ., 5, from (5.14) we have p∇ψ j , ∇qqψ j .
Using (5.14) twice to express ψ j in terms of φk 's we obtain Similarly as in (5.13) one finds pG ´1q j,k " δ j,k `Opλ ´1q , and then (6.2) follows from (6.1).This completes the proof of the lemma.
6.2.A third expansion.In this subsection, we shall prove the following lemma. with x,λ dy ( and x,λ H a px, yqr dy `15 x,λ r 2 dy `20 x,λ r 3 dy .( We emphasize that the coefficients β, γ and δ j enter only into the remainders opλ ´2q `opǫλ ´1q.This is somewhat surprising since β enters to orders λ ´1 and λ ´2 and γ enters to order λ ´2 in the expansion of the numerator and the denominator.
We will also use the fact that }∆h} 1 " Opλ ´5{2 q .(6.7) This follows from Lemma 6.1 together with (5.19) and the same bounds that led to (5.18).
We will obtain Lemma 6.3 from separate expansions of the numerator and the denominator, which we state in the following two lemmas.
Expanding the numerator.We abbreviate and write E ǫ rv 1 , v 2 s for the associated bilinear form.Recall that N 0 was defined in (6.4).We shall show Lemma 6.4.As ǫ Ñ 0, where Proof.
Step 2. We now extract the relevant contribution from g and show Indeed, E ǫ rψ x,λ `g `rs " E ǫ rψ x,λ `rs `2 E ǫ rψ x,λ `r, gs `Eǫ rgs .
We have, since r P T K x,λ and g P T x,λ , ż Ω ∇r ¨∇g dy " 0 .
The lemma follows by collecting the estimates from the three steps.
Step 2. We now extract the relevant contribution from g and show ż Ω pψ x,λ `g `rq 6 dy "  (6.6), the last term is opλ ´2q.Finally, by (5.19) and the fact that r P T K x,λ , ż This proves (6.12).
Step 3. We finally extract the relevant contribution from r and show ż Ω pψ x,λ `rq ānd by (6.6) the last term is opλ ´2q.We need to extract Irrs from the three terms on the right side involving r.We begin with the term which is linear in r, ż By (A.7), (6.6) and }U x,λ } 3 18{5 " Opλ ´1q, the last term is opλ ´2q.Since r P T K x,λ , the first term is ż Ω U 5 x,λ r dy " 3 ´1 ż Ω ∇P U x,λ ¨∇r dy " 0 .
The lemma follows by collecting the estimates from the three steps.
Proof of Lemma 6.3.Note that, by (6.6), D 1 " Opλ ´1q and Irrs " opλ ´1q.Moreover, by (2.3), D 0 stays away from zero.Therefore, the expansion from Lemma 6.5 implies that Combining this with the expansion from Lemma 6.4 and using N 1 " Opλ ´1q (again from (6.6)), we obtain S a`ǫV ru ǫ s " S a`ǫV rψ x,λ s `A `D´1{3 Thus, the assertion of the lemma is equivalent to A " opλ ´2q `opǫλ ´1q.We write It follows from (2.2) and (2.3) that This, together with D 1 " Opλ ´1q, yields We shall show in Appendix A that (6.17) 6.3.Coercivity.To complete the proof of our main results, it remains to prove that the terms involving r in the expansion (6.3) give a non-negative contribution.Recall that Irrs was defined in (6.5) and N 0 and D 0 in Lemmas 6.4 and 6.5, respectively.Lemma 6.6.There is a ρ ą 0 such that for all sufficiently small ǫ ą 0, Proof.We bound, using (4.2),Lemma 2.6 and (5.1), for any δ ą 0, x,λ r 2 dy `δ´1 opλ ´2q .
Since r P T K x,λ , Lemma 4.3 implies that for all sufficiently small ǫ ą 0, the first term on the right side is bounded from below by ρ ş Ω |∇r| 2 dy for some ρ ą 0 independent of ǫ.On the other hand, by (5.20), choosing δ ą 0 small, but independent of ǫ, and then ǫ small, we can make sure that ´´2δ `Opλ ´2q `Opǫλ ´1q This completes the proof of the lemma.
( This is the only place in the proof of Theorem 1.3 where we need assumption (1.4).
Proof.We recall the upper bound from Corollary 2.3, Combining this with (6.18) and using R ě 0, we find By the assumptions N a pV q ‰ H and (1.4), both C 1 and C 2 tend to some positive quantities as We now assume N a pV q ‰ H and complete the proof of Theorems 1.3 and 1.7.We can write Inserting this into (6.18)we obtain pS{3q ´1{2 `QV px 0 q `op1q ˘2 4 `2π 2 |apx 0 q| `op1q ˘ǫ2 ě p1 `op1qq `S ´Spa `ǫV q ˘`R 1 (6.20) with Since R 1 ě 0 we obtain, in particular, S ´Spa `ǫV q ď p1 `op1qqpS{3q ´1{2 `QV px 0 q `op1q In the last inequality we used x 0 P N a pV q.This proves the claimed lower bound on Spa `ǫV q and completes the proof of Theorem 1.3.
We now proceed to the proof of Theorem 1.7, still under the assumption N a pV q ‰ H. Combining the lower bound on S ´Spa`ǫV q from Corollary 2.3 with the upper bound in (6.22) we obtain Moreover, inserting the lower bound on S ´Spa `ǫV q into (6.20)we infer that R 1 " opǫ 2 q.Thus, by (6.19) and (6.21) and, reinserting the last expression into R " opǫ 2 q, also φ a pxq " opǫq .
Since both C 1 and C 2 are positive for all sufficiently small ǫ ą 0, we arrive at a contradiction.Thus, assumption (3.4), under which we have worked so far, is not satisfied.By the concavity argument in the proof of Corollary 2.3 this means that Spa `ǫV q " S for all sufficiently small ǫ ą 0. This concludes the proof of Theorem Putting everything together and using the resolvent identity (2.8) as in the proof of (A.14), we obtain (A.15).
This completes the proof of (6.15).
A.3.Proof of (6.16).We have For the first term we use (A.3) and for the second term we use Lemma 2.5.
For the first and the second term we argue as in the proof of (A.13) and for the third one we use Lemma 2.5.
The bounds (A.18), (A.19) and (A.20) follow from the corresponding relations where ψ x,λ and P U x,λ are replaced by U x,λ and where B λ P U x,λ is replaced by B λ U x,λ .
This completes the proof of (6.16).
Indeed, by Step 1 and Rellich's compactness theorem we have u ǫ Ñ 0 in L 2 pΩq and therefore Thus, the u ǫ , extended by zero to functions in 9 H 1 pR 3 q, form a minimizing sequence for the Sobolev quotient.By a theorem of Lions [21] there exist pz ǫ q Ă R 3 and pµ ǫ q Ă R `such that, along a subsequence, µ ´1{2 ǫ u ǫ pµ ´1 ǫ ¨`z ǫ q converges in 9 H 1 pR 3 q to a function, which is an optimizer for the Sobolev inequality.By the classification of these optimizers (which appears, for instance, in [21, Cor.I.1]) and taking the normalization of the u ǫ into account, we can assume, after modifying the µ ǫ and z ǫ , that µ ´1{2 ǫ u ǫ pµ ´1 ǫ ¨`z ǫ q Ñ s U 0,1 in 9 H 1 pR 3 q for some s P t˘1u.By a change of variables (which preserves the 9 H 1 pR 3 q norm) this is the same as (B.2).Thus, µ ǫ Ñ 8 and distpz ǫ , Ωq Ñ 0. Using, in addition, that the boundary of Ω is C 1 , we conclude that µ ǫ distpz ǫ , R 3 zΩq Ñ 8.In particular, after passing to a subsequence, z ǫ Ñ x 0 P Ω.

Note that ż
Step 3. We now conclude the proof of the proposition.
Theorem 2.1.As λ Ñ 8, uniformly for x in compact subsets of Ω and for ǫ ě 0, We claim that for any x P Ω there is a ξ x P R 3 such that [15]nd (4.2) were shown in [15, Lem.2.2 and Thm.1.1]inthecasewhere u ǫ is a minimizer for Spa `ǫV q.Since the proof in[15]uses the Euler-Lagrange equation satisfied by minimizers, this proof is not applicable in our case.We will replace the use of the Euler-Lagrange equation by a suitable expansion of S a`ǫV ru ǫ s, which is carried out in Subsection 4.2.The other ingredient in the proof of [15, Lem.2.2] and in our proof is the coercivity of a certain quadratic form, see Lemma 4.3 in Subsection 4.3.Finally, in Subsection 4.4 we will prove Proposition 4.1.