Energy asymptotics in the Brezis–Nirenberg problem: The higher-dimensional case

. For dimensions N ě 4 , we consider the Brézis-Nirenberg variational problem of ﬁnding


Introduction and main results
1.1.Setting of the problem.Let N ě 4 and let Ω Ă R N be a bounded open set.For ǫ ą 0 and a function V P CpΩq, Brézis and Nirenberg study in their famous paper [3] the quotient functional S ǫV rus :" and the corresponding variational problem of finding SpǫV q :" inf This number is to be compared with S N " πN pN ´2q ˆΓpN {2q ΓpN q ˙2{n , the sharp constant [10,11,1,13] in the Sobolev inequality.Indeed, in [3] it is shown that SpǫV q ă S N as soon as N pV q :" tx P Ω : V pxq ă 0u (1.3) is non-empty.This behavior is in stark contrast to the case N " 3 also treated in [3], where there is an ǫ V ą 0 such that SpǫV q " S N for all ǫ P p0, ǫ V s even if N pV q is non-empty.
The purpose of this paper is, for N ě 4, to describe the asymptotics of S N ´SpǫV q to leading order as ǫ Ñ 0, as well as the asymptotic behavior of corresponding (almost) minimizing sequences and, in particular, their concentration behavior.This is the higher-dimensional complement to our recent paper [6], where analogous results are shown in the more difficult case N " 3.

Notation.
To prepare the statement of our main results, we now introduce some key objects for the following analysis.An important role is played by the Green's function of the Dirichlet Laplacian on Ω, which in the sense of distributions satisfies, in the normalization of [9], Gpx, yq " 0 on BΩ, (1.4) where ω N is the surface of the unit sphere in R N , and δ y denotes the Dirac delta function centered at y.We denote by Hpx, yq " the regular part of G.The function Hpx, ¨q, defined on Ωztxu, extends to a continuous function on Ω and we may define the Robin function φpxq :" Hpx, xq . (1.6) Using this function, we define the numbers σ N pΩ, V q :" sup σ 4 pΩ, V q :" sup xPN pV q ´φpxq ´1|V pxq| ¯, N " 4 , which will turn out to essentially be the coefficients of the leading order term in S N ´SpǫV q.
Another central role is played by the family of functions U x,λ pyq " λ pN ´2q{2 p1 `λ2 |x ´y| 2 q pN ´2q{2 x P R N , λ ą 0. (1.7) It is well-known that the U x,λ are exactly the optimizers of the Sobolev inequality on R N .
Since (1.1) is a perturbation of the Sobolev quotient, it is reasonable to expect the U x,λ to be nearly optimal functions for (1.2).However, since (1.2) is set on H 1 0 pΩq, we consider, as in [2], the functions P U x,λ P H 1 0 pΩq uniquely determined by the properties ∆P U x,λ " ∆U x,λ in Ω, P U x,λ " 0 on BΩ . (1.8) Moreover, let T x,λ :" span P U x,λ , B λ P U x,λ , B x i P U x,λ pi " 1, 2, . . ., N q ( 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.
In what follows we denote by } ¨} the L 2 ´norm on Ω.Finally, given a set X and two functions f 1 , f 2 : X Ñ R, we write f 1 À f 2 if there exists a numerical constant c such that f 1 pxq ď c f 2 pxq for all x P X.
1.2.Main results.Throughout this paper and without further mention we assume that the following properties are satisfied.
Assumption 1.1.The set Ω Ă R N , N ě 4, is open and bounded and has a C 2 boundary.Moreover, V P CpΩq and N pV q ‰ H, with N pV q given by (1.3).
Here is our first main result.It gives the asymptotics of S N ´SpǫV q to leading order in ǫ.
Theorem 1.2.As ǫ Ñ 0`, we have and SpǫV q " S 4 ´exp Here the constants C N are defined in (1.14) below.
Our second main result shows that the blow-up profile of an arbitrary almost minimizing sequence pu ǫ q is given to leading order by the family of functions P U x,λ .Moreover, we give a precise characterization of the blow-up speed λ " λ ǫ and of the point x 0 around which the u ǫ concentrate.
Here the constants a N , b N and D N are defined in (1.13) and (1.15) below.
The coefficients appearing in Theorems 1.2 and 1.3 are as well as and (1.15) A simple computation using beta functions yields the numerical values ΓpN ´2q , N ě 5. 2).In the special case where V is a negative constant, Brézis and Peletier [4] discussed the concentration behavior of such general solutions and made some conjectures, which were later proved by Han [7] and Rey [8].Probably one can use their precise concentration results to give an alternative proof of our main results in the special case where V is constant and probably one can even extend the analysis of Han and Rey to the case of non-constant V .
Our approach here is different and, we believe, simpler for the problem at hand.We work directly with the variational problem (1.2) and not with the Euler-Lagrange equation.Therefore, our concentration results are not only true for minimizers but even for 'almost minimizers' in the sense of (1.11).We believe that this is interesting in its own right.On the other hand, a disadvantage of our method compared to the Han-Rey method is that it gives concentration results only in H 1 norm and not in L 8 norm and that it is restricted to energy minimizing solutions of the Euler-Lagrange equation.
In the special case where V is a negative constant, our results are very similar to results obtained by Takahashi [12], who combined elements from the Han-Rey analysis (see, e.g., [12,Equation (2.4) and Lemma 2.6]) with variational ideas adapted from Wei's treatment [14] of a closely related problem; see also [5].Takahashi obtains the energy asymptotics in Theorem 1.2 as well as the characterization of the concentration point and the concentration scale in Theorem 1.3 under the assumption that u ǫ is a minimizer for (1.2).Thus, in our paper we generalize Takahashi's results to non-constant V and to almost minimizing sequences and we give an alternative, self-contained proof which does not rely on the works of Han and Rey.
The present work is a companion paper to [6] relying on the techniques developed there in the three dimensional case.In particular, Theorems 1.2 and 1.3 should be compared with [6, Theorems 1.3 and 1.7], respectively.Although the expansions for N ě 4 have the same structure as in the case N " 3, the latter case is more involved.In fact, when N " 3, the coefficient of the leading order term, namely the term of order ǫ, vanishes and one has to expand the energy to the next order, namely ǫ 2 .
Besides the extensions of known results that we achieve here, we also think it is worthwhile from a methodological point of view to present our arguments again in the conceptually easier case N ě 4.
In the three-dimensional case the basic technique is iterated twice, which to some extent obscures the underlying simple idea.Moreover, we hope our work sheds some new light on the similarities and discrepancies between the two cases.
The structure of this paper is as follows.In Section 2 we prove the upper bound from Theorem 1.2 by inserting the P U x,λ as test functions.The proof of the corresponding lower bound is prepared in Sections 3 and 4, where we derive a crude asymptotic expansion for a general almost minimizing sequence pu ǫ q and the corresponding expansion of S ǫV ru ǫ s.Section 5 contains the proof of Theorems 1.2 and 1.3.A crucial ingredient there is the coercivity inequality (5.1) from [9], which allows us to estimate the remainder terms and to refine the aforementioned expansion of u ǫ .Finally, an appendix contains two auxiliary technical results.

Upper bound
The computation of the upper bound to SpǫV q uses the functions P U x,λ , with suitably chosen x and λ, as test functions.The following theorem gives a precise expansion of the value S ǫV rP U x,λ s.
To state it, we introduce the distance to the boundary of Ω as dpxq " distpx, BΩq, x P Ω.
Theorem 2.1.Let x " x λ be a sequence of points such that dpxqλ Ñ 8. Then as λ Ñ 8, we have ż and ż In particular, as λ Ñ 8, (2.4) In view of Proposition 3.1 below, the assumption dpxqλ Ñ 8 in Theorem 2.1 is no restriction, even when dealing with general almost minimizing sequences.
Corollary 2.2.As ǫ Ñ 0`, we have and N´4 can be extended to a continuous function on N pV q which vanishes on BN pV q.Thus there is z 0 P N pV q such that σ N pΩ, V q " φpz 0 q ´2 N´4 |V pz 0 q| N´2 N´4 , N ě 5. (2.8) The corollary for N ě 5 now follows by choosing x " z 0 in (2.4) and optimizing the quantity and (2.5) follows from a straightforward computation.
Similarly, if N " 4, since φpyq |V pyq| is a positive continuous function on N pV q which goes to `8 as y Ñ BN pV q, we find some z 0 P N pV q such that σ 4 pΩ, V q " φpz 0 q |V pz 0 q| .
(2.10) Thus we may choose x " z 0 in (2.4) and optimize the quantity Aλ ´2 ´Bǫλ ´2 log λ in λ ą 0, where A " 8 a 4 φpz 0 q `op1q and B " b 4 |V pz 0 q| `op1q.The optimal choice is λpǫq " ?e exp ˆA Bǫ ˙. (2.11) Inserting this into (2.4),we get where we have used the fact that holds for all a ě 0 and all b ą 0.
This completes the proof of (2.6), and thus of Corollary 2.2.
Proof of (2.1).Since the U x,λ satisfy the equation ´∆y U x,λ pyq " N pN ´2q U x,λ pyq q´1 , y P R N , (2.13) it follows using integration by parts that x,λ P U x,λ dy.
On the other hand, by [9, Prop.1] we know that where (2.15) By putting the above equations together we obtain x,λ Hpx, ¨q dy " λ 1`N 2 `φpxq `Opρ dpxq 1´N q ˘żBρpxq dy p1 `λ2 |x ´y| 2 q pN `2q{2 " λ 1´N 2 a N ´φpxq `Opρ dpxq 1´N q ¯p1 `Oppλ ρq ´2qq and ż ΩzBρpxq U q´1 x,λ Hpx, ¨q dy " Hence for the second term on the right hand side of (2.16) we get (2.20) As for the last term on the right hand side of (2.16), we note that in view of (2.15) The claim thus follows from (2.16) by choosing ρ " dpxq 1{3 λ ´2{3 in (2.20).(Notice that ρ " dpxqpdpxqλq ´2{3 q ď dpxq 2 for λ large enough.) Proof of (2.2).We have Since by [9, Prop.1], together with (2.14), (2.15) and (2.18) we obtain the following upper bound on the last integral in (2.21), To treat the first term on the right hand side of (2.21), first assume N ě 5. Choose a sequence ρ " ρ λ such that ρ ď dpxq, ρ Ñ 0 and ρλ Ñ 8 as λ Ñ 8. (This is always possible, whether or not d Ñ 0.) Then, by continuity of V , Similarly, in the case N " 4 we let B τ pxq and B R pxq be two balls centered at x with radii τ and R chosen such that B τ pxq Ă Ω Ă B R pxq and split the last integration in two parts as follows.
Extending V by zero to B R pxqzΩ we get (2.23) On the other hand, denoting by o τ p1q a quantity that vanishes as τ Ñ 0 and assuming that τ λ Ñ 8 we get By choosing τ " 1 log λ and taking into account (2.23) we arrive at (2.2) in case N " 4.
Recall that q ą 2. Hence from the Taylor expansion of the function t Þ Ñ t q on an interval r0, bs it follows that for any a P r0, bs we have Because of (2.22) and (2.14) we can apply (2.24) with b " U x,λ pyq and a " ϕ x,λ pyq to obtain the following point-wise upper bound: Together with estimate (A.2) this gives ´U q x,λ `q U q´1 x,λ ϕ x,λ ¯dy ˇˇˇ" O ´pdpxq λq ´N ¯. (2.26) On the other hand, the calculations in the proof of (2.1) show that In view of (2.17) and (2.26) this completes the proof.

Lower bound. Preliminaries
As a starting point for the proof of the lower bound on SpǫV q, we derive a crude asymptotic form of almost minimizers of S ǫV .The following result is essentially well-known.We have recalled the proof in [6, Appendix B] in the case N " 3, but the same argument carries over to N ě 4.
Convention.From now on we will assume that SpǫV q ă S N for all ǫ ą 0 (3.4) and that pu ǫ q satisfies (1.11).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 .

Lower bound. The main expansion
In this section we expand S ǫV ru ǫ s by using the decomposition (3.2) of u ǫ .We shall show the following result.
In the sequel we denote by c 1 , c 2 , . . .various positive constants which are independent of ǫ.

¯,
where pq ´3q `" maxtq ´3, 0u.Using (2.25), it follows that x,λ `ϕq´pq´3q x,λ In the last inequality we used (2.22) to simplify the form of the remainder terms.Now we use the identity which follows from (1.8), (2.13) and w P T K x,λ , and the fact that ş Ω |w| q Ñ 0, which follows from (3.3) and the Sobolev inequality.Therefore, with the help of the Hölder inequality, we find x,λ ϕ q q´1 x,λ dy x,λ ϕ q q´1 x,λ dy In the last inequality, we used the Sobolev inequality for w and the (3.3) for w, together with It follows from Lemma A.1 and (3.3) that ˜żΩ U qpq´2q q´1 x,λ ϕ q q´1 x,λ dy Thus, we conclude that, as ǫ Ñ 0, Proof of (4.3).We write By the Hölder and Sobolev inequalities we have and Hence (4.3) follows by inserting these estimates into (4.6).

Proof of the main results
We now deduce Theorems 1.2 and 1.3 from Proposition 4.1.To do so, we make crucial use of the following coercivity bound proved in [9, Appendix D].
Proposition 5.1.For all x P Ω, λ ą 0 and v P T K x,λ , one has ż Corollary 5.2.For all ǫ ą 0 small enough, we have, if N ě 5, Proof.Firstly, it follows directly from (5.1) and the definition of Irws in (4.5) that there is a c ą 0 such that for all ǫ ą 0 small enough, we have Using Proposition 4.1 and (5.4) it follows that for ǫ small enough one has x,λ dy , this further implies that for ǫ ą 0 small enough Using (2.2) for the potential term and recalling (3.3), we obtain Now the fact that S N ´SǫV ru ǫ s " p1 `op1qqpS N ´SpǫV qq by (1.11), together with the expansion of S ǫV rP U x,λ s from Theorem 2.1, implies the claimed bounds (5.2) and (5.3).
In the next lemma, we prove that the limit point x 0 lies in the set N pV q.
Lemma 5.3.We have x 0 P N pV q.In particular, d ´1 " Op1q as ǫ Ñ 0 and x P N pV q for ǫ small enough.
Proof.We first treat the case N ě 5.In (5.2), we drop the non-negative gradient term and write the remaining lower order terms as where A " N pN ´2q a N φpxqd N ´2 `op1q, B " ´bN V px 0 qd 2 `op1q. (5.5) Notice that since φpxq Á d 2´N by (2.7), the quantity A is positive and bounded away from zero.Moreover, by (5.2) and the fact that SpǫV q ă S N , which follows from Corollary 2.2, we must have B ą 0. Optimizing in dλ yields the lower bound for some explicit constant c ą 0 independent of ǫ.On the other hand, by Corollary 2.2, there is ρ ą 0 such that the leading term in (5.2) is bounded by p1 `op1qqpS N ´SpǫV qq ě ρ ǫ N´2 N´4 (5.7) for all ǫ ą 0 small enough.Plugging (5.6) and (5.7) into (5.2) and rearranging terms, we thus deduce that B ě ρ (5.8) As observed above, the quantity A is bounded away from zero and therefore (5.8) implies that B is bounded away from zero.Hence, in view of (5.5), d is bounded away from zero and V px 0 q ă 0.
The fact that x P N pV q for ǫ small enough is a consequence of the continuity of V .This completes the proof in case N ě 5.
Now we consider the case N " 4 in a similar way.In (5.3), we drop the non-negative gradient term and write the remaining lower order terms as where A " 8a 4 φpxqd 2 `op1q, B " ´b4 pV px 0 q `op1qqd 2 p1 ´log d log dλ q. (5.10) Since φpxq Á dpxq ´2 by (2.7), the quantity A is positive and bounded away from zero.
Moreover, by (5.3) and the fact that SpǫV q ă S 4 , we must have B ą 0. Optimizing (5.9) in dλ yields the lower bound On the other hand, by Corollary 2.2, there is ρ ą 0 such that the leading term in (5.3) is bounded by p1 `op1qqpS 4 ´SpǫV qq ě expp´ρ ǫ q. (5.12) Plugging (5.11) and (5.12) into (5.3),we thus deduce that which leads to ´2A B `ǫ logp Bǫ 2e q ě ´ρ. (5.13) Since φpxq Á d ´2 by (2.7), the quantity A is bounded away from zero and moreover B is bounded.Using this fact, the left hand side of (5.13) can be written as Together with (5.13), this easily implies, if ǫ ą 0 is small enough, that As before, in view of (5.10), we deduce that d is bounded away from zero and that V px 0 q ă 0. The fact that x P N pV q for ǫ small enough is again a consequence of the continuity of V .
Proof of Theorem 1.2.We first treat the case N ě 5.In view of Lemma 5.3, the lower bound (5.2) can be written as (upon dropping the non-negative gradient term) by optimization in λ.Therefore where the last inequality uses the fact that x 0 P N pV q by Lemma 5.3.
Since the matching upper bound has already been proved in Theorem 2.1, the proof in case N ě 5 is complete.
Since the matching upper bound has already been proved in Theorem 2.1, the proof in case N " 4 is complete.
Proof of Theorem 1.3.We start again with the bounds from Corollary 5.2, but this time we need to take into account the various nonnegative remainder terms more carefully.
Proof for N ě 5. We rewrite (5.2), using Lemma 5.3, as where we have set Notice that both summands of R are separately nonnegative.Inserting the upper bound from Corollary 2.2 into (5.14),we get Since each one of the first two summands on the right hand side is nonnegative, we deduce that (5.15) In particular, (5.15) implies that }∇w} 2 2 " opǫ N´2 N´4 q.
(5.16) Denote by the unique value of λ for which the first summand of R vanishes.Using Lemma A.2, the bound (5.15) implies that ǫpλ ´1 ´λ0 pǫq ´1q 2 " opǫ N´2 N´4 q, which is equivalent to with D N given in (1.15).This completes the proof of Theorem 1.3 in the case N ě 5.
Lemma A.2. Let f ǫ : p0, 8q Ñ R be given by f ǫ pλq " A ǫ λ N ´2 ´Bǫ ǫ λ 2 with A ǫ , B ǫ ą 0 uniformly bounded away from 0 and 8. Denote by the unique global minimum of f ǫ .Then there is a c 0 ą 0 such that for all ǫ ą 0 we have .6) Note that this bound uses the C 2 assumption on Ω.) Since, moreover, V " 0 on BN pV qzBΩ, the function φ ´2 N´4 |V | .17) Finally, to obtain the asymptotics of α, by (4.2), (1.11), (2.3) and (5.16), we have that This gives the bound claimed in (A.1) in each case, provided we can bound the integral on the complement ΩzB d pxq.On this region, we have by Hölder˜żΩzB d pxq U Prop. 1(c)].Combining all the estimates, we deduce (A.1).Proof of (A.2).We split the domain of integration Ω again into B d pxq and ΩzB d pxq.On B d pxq, by (2.14),