NODAL CLUSTER SOLUTIONS FOR THE BREZIS-NIRENBERG PROBLEM IN DIMENSIONS N ≥ 7

. We show that the classical Brezis-Nirenberg problem ∆ u + | u | 4 N − 2 u + εu = 0 , in Ω , u = 0 , on ∂ Ω admits nodal solutions clustering around a point on the boundary of Ω as ε → 0, for smooth bounded domains Ω ⊂ R N in dimensions N ≥ 7.


Introduction
In this paper we find a new family of sign-changing solutions to the classical Brezis-Nirenberg problem where ε > 0 is a small parameter and Ω is a smooth bounded domain in R N , N ≥ 7.
In their seminal 1983 paper [6], Brezis and Nirenberg initiated the study of positive solutions to (BN) and demonstrated that for dimensions N ≥ 4, the problem admits a solution for ε ∈ (0, λ 1,Ω ), where λ 1,Ω represents the first eigenvalue of −∆ with 0-Dirichlet boundary conditions on ∂Ω.If dimension N is 3, they proved the existence of λ * ,Ω > 0 (whose definition depends on Ω) and of a positive solution to (BN) if λ ∈ (0, λ * ,Ω ).If Ω = B the unit ball, then λ * ,B = λ 1,B 4 ; for general domains see [13].Multiplying the equation in (BN) against the eigen-function associated to λ 1,Ω and integrating by parts on Ω show that no positive solutions exist for ε ≥ λ 1,Ω .Additionally, Pohozaev identity [23] gives that problem (BN) has no non-trivial solutions when Ω is star-shaped and ε = 0. Conversely, Bahri and Coron [2] presented an existence result for a positive solution to problem (1.1) for Ω with a nontrivial topology and ε = 0. Subsequently, considerable attention has been devoted to understanding the possibility of multiple positive solutions to (BN) in the regime ε → 0 [25,3,19,20,11] and also to understanding the limiting behavior of the positive solutions u ε of (BN) as ε → 0 [14,26].
Concerning the existence of sign-changing solutions to (BN), this has been established for all range of ε > 0: it has been proven in [9] for ε ∈ (0, λ 1,Ω ) and N ≥ 6, and in [7] for ε ≥ λ 1,Ω and N ≥ 4. Devillanova and Solimini [12] proved the existence of infinitely many sign-changing solutions to (BN) for any ε > 0 when N ≥ 7. Dimension 7 seems to be a threshold case as for 4 N 6 there are no radial sign-changing solutions for (BN), when Ω is a ball and ε ∈ (0, λ * * ), for some λ * * > 0 [1].This paper wants to give a contribution in the understanding of multiple sign-changing solutions to (BN) in the regime ε → 0. It is well known that in this regime a crucial role is played by the bubbles, namely the positive solutions to (BN) when ε = 0 and Ω = R N .For any δ > 0 and ξ ∈ R N , the bubbles N−2 4 (1.1) are all the solutions of the problem ∆u + u The asymptotic analysis of low-energy sign-changing solutions to (BN) as ε → 0 has been studied in [5] for N ≥ 4: assuming their existence, such solutions u ε have a simple positive and negative blow-up behaviour at two distinct points of Ω as ε → 0, provided the rates of blow-up for the positive and the negative parts are comparable.Roughly speaking, they can be described as follows Construction, asymptotic analysis and multiplicity of sign-changing solutions exhibiting this type of simple blow-up as ε → 0 were obtained in [8,18,4].However, in the case of the unit ball, the low-energy radial sign-changing solutions obtained in [9] do not have a simple blow-up if N ≥ 7. Indeed, both their positive and negative parts blow-up in the form of a positive and a negative bubble both centered at the center of the ball as ε → 0, with non comparable rates of blow-up [24].Roughly speaking, in this case solutions look like u ε (x) ∼ U δ 1ε ,0 (x) − U δ 2ε ,0 (x) with δ iε → 0, i = 1, 2, δ 1ε δ 2ε = o(1), as ε → 0.
This behaviour is known as tower of bubbles (see [10]).In [24] it is proven that signchanging tower of bubbles for (BN) exist as ε → 0 for dimensions N ≥ 7 in a general domain.In contrast, in low dimensions N = 4, 5, 6, sign-changing bubble-towers cannot exist, as shown in [15].
In [30] Vaira constructed a different type of sign-changing solutions to (BN) which blow-up in the form of a concentrated bubble and blow-up occurs at a point of the boundary of Ω. Bubbling at the boundary is not always allowed [26], and some extra requirement on the domain Ω seems to be necessary.In [30] it is assumed that Ω is a smooth bounded domain with non-trivial topology such that the problem has a positive solution u 0 which is non-degenerate, in the sense that the following linear problem 4) admits only the trivial solution v = 0. Existence of solutions to (1.3) for domain with non-trivial topology has been obtained by [2].Besides, for generic Ω these solutions are non-degenerate [28].
Let ν be the unitary outer normal to ∂Ω.Assuming that the function ξ ∈ ∂Ω → ∇u 0 (ξ) • ν(ξ) has a non-degenerate critical point ξ 0 , Vaira proves the existence of a sign-changing solution to problem (BN) of the form Here U δ,ξ is again the bubble defined in (1.1).
The main result of this paper is to prove that a sign-changing cluster solution to (BN) around ξ 0 is possible.Clustering configurations are those where the solutions blow-up as the sum of a finite number of bubbles, of comparable heights, whose centers converge to the same point.Clustering configurations are known to exist in several problems related to semi-linear elliptic equations with critical non-linearity, but none was known for the Brezis-Nirenberg problem (BN).
To state our result, let us denote by P W the projection of a function W onto H 1 0 (Ω), i.e.
Our main result is the following 3) has a solution u 0 , which is non-degenerate in the sense that the linear problem (1.4) has only the trivial solution.Assume there exists a critical point ξ 0 ∈ ∂Ω of the function for ξ ∈ ∂Ω → ∇u 0 (ξ) • ν(ξ), where ν is the unitary outer normal to ∂Ω, such that the second variation D 2 N −1 (∇u 0 (ξ) • ν(ξ)) is positive definite.Let k ∈ N. Then there exist ε > 0 and a constant C > 0 such that, for all ε ∈ (0, ε) there exists a sign-changing solution u ε to (BN) given by where +σ for any σ > 0 arbitrarily small.The solutions described in the theorem are rather delicate to capture, and precise expansions of the parameters δ jǫ and the points ξ jǫ at two consecutive scales are required in the construction.This is described in details in Section 2.
The method we use to prove Theorem 1.1 also applies to the construction of signchanging solutions exhibiting a cluster configuration near the boundary of Ω for the almost critical problem where ε > 0 is a small parameter and Ω is a smooth bounded domain in R N , N ≥ 7.This observation is already present in [30] and we will not elaborate further on this point.
Clustering configurations are known in the literature for perturbation of the Yamabe problem to find metrics on Riemannian manifolds with constant scalar curvature.These have been found in high dimensions N ≥ 7 in [21], in dimensions 4 and 5 in [29], see also [27].We dont't know if clustering sign-changing solutions exist for the Brezis-Nirenberg problem (BN) in low dimensions 4, 5, 6, but if it does the form of the solution should though be different from the one obtained in Theorem 1.1.
Finally, we mention that several interesting results have been obtained on the existence of sign changing solutions to the Brezis-Nirenberg problem in regimes different from the one treated in this paper, namely when ε converges to some fixed ε * > 0. Results in this direction are contained for instance in [16,22].
Acknowledgments.The authors would like to express their gratitude to Angela Pistoia for many interesting discussions around this topic.M. Musso has been supported by EPSRC research Grant EP/T008458/1 while G. Vaira has been supported by Gnampa project "Proprietà qualitative delle soluzioni di equazioni ellittiche".

The setting of the problem
We consider the Hilbert space H 1 0 (Ω) equipped with the usual inner product (Ω) → H 1 0 (Ω) be the adjoint operator of the immersion i : or equivalently u weakly solves The operator i ⋆ : where S is the best constant for the Sobolev embedding.
In terms of the operator i ⋆ , problem (BN) can be formulated as (2.2) We look for cluster solutions of the problem (BN) which change sign.They have the form Here k is a fixed given integer, u 0 is the positive non-degenerate solution to (1.3), In our construction the scaling parameters δ i will be positive and small, while the points ξ i will collapse into each other, as ε → 0.
In [30] Vaira constructs a solution to problem (BN) of the form (2.3), with k = 1 under the assumption that there exists a non-degenerate critical point ξ 0 of the function ξ ∈ ∂Ω → ∂ ν u 0 (ξ), where ν is the inner unit normal on the boundary.Such solution blows-up, as ε → 0, at ξ 0 , in the sense that the scaling parameter δ and the point ξ in (2.3) can be described at main order as One has δ → 0 and ξ → ξ 0 as ε → 0. The point (d where C and B are the explicit positive constants A direct computation gives that (d 0 , t 0 , ξ 0 ) satisfies the system (2.7) Since by Hopf's Lemma ∂ ν u 0 (ξ 0 ) < 0, the function Ψ has a critical point in the considered region.
The result in [30] indicates that a solution with the form (2.3) and k > 1 would possibly exhibit a cluster behaviour around the point ξ 0 .This suggests the form for the scaling parameters δ i and the points ξ i in (2.3).Let us be more precise.
For the construction of a cluster solution, we assume that ξ 3) have the form ; ) . (2.8) where d 0 , t 0 , ξ 0 , α and β are defined in (2.4), ξi ∈ ∂Ω, τ i ∈ R N −1 , τ i , d i ∈ R are parameters to be found.For the moment, we make the following assumptions on these parameters: we assume there exist a > 0 and ρ > 0 such that τ i , τ i and d i (2.9) Remark 2.1.Without loss of generality, we can choose a coordinate system such that and then (2.10) It is important to observe that α < α and β < β < β.Let us call . (2.12) Now we are able to state our main result.
Theorem 2.1.Assume there exists is a ) is positive definite, and let k ∈ N. Then there exists ε 0 > 0 such that for any ε ∈ (0, ε 0 ) the problem (BN) has a sign-changing cluster solution which blows-up at ξ 0 .More precisely, there exist constants C, a, ρ, a function for some σ > 0 arbitrarily small.Theorem 1.1 is a direct consequence of Theorem 2.1.The rest of the paper is devoted to prove Theorem 2.1.The proof is done via a reduction method.
In Section 3 we prove that, for given ξ )-(2.9), there exists φ ε solution of a projected problem.We then show that a true solution to our problem can be achieved by finding a specific set of parameters This is the reduced finite dimensional problem that we treat in the subsequent sections.

Reduction to a finite dimensional problem
The purpose of this section is to find the term ).The term φ ε will be small in ε and satisfy a set of orthogonal conditions.To introduce these orthogonality conditions, consider the linear problem −∆ψ = pU p−1 1,0 ψ in R N , whose set of solution is spanned by the functions and define where Let us denote by G(x, y) the Green's function of the Laplace operator with Dirichlet boundary condition and let H(x, y) be its regular part, namely We remark that in [25] it was shown that for every ξ ∈ Ω By using the previous observations we get that where and for all j = 1, Proof.See Proposition 1 in [25].
In particular Lemma 3.2 implies that P U δ,ξ ≥ 0 and that there exists a positive constant C such that In [25] it is also shown that More generally we can estimate the L q (Ω) norm of ϕ δ,ξ and the Lemma 3.3.Let N > 6, q ∈ p+1 2 , p + 1 and δ, ξ satisfy (2.8) and (2.9) as in Section 2. Then there exists a constant C = C(a) > 0 and ε 0 > 0 such that for any ε ∈ (0, ε 0 ) for some σ > 0 arbitrarily small.
Proof.The proof easily follows from Lemma 2.2 in [30].Now let us define the operators Moreover the problem (2.2) is equivalent to solve the system Let us also define the linear operator the error term and the nonlinear terms where f (u) = |u| p−1 u.
Two important inequalities we will use throughout all our work are the following.
Proof.We will argue as in the proof of Lemma 1.7 in [19].The main difference is that our points are converging to the same point ξ 0 ∈ ∂Ω while in [19] they are converging to different points in Ω.In view of a contradiction, assume that there exist where w n ∈ K n = K δn,ξn , i.e.
Observe that where η i is defined in 3.4.Let us also observe that w n ∈ K n is orthogonal to φ n , h n ∈ K ⊥ n .
• Step 1.It holds that Observe that Using (3.14) and using Lemma 3.4, we have ,Ω go to zero as ε n → 0 by (3.3), (3.15) and Lemma 3. Moreover, Putting together (3.16) and (3.17), we have that c j in are bounded and consequently Then φin is bounded and there exists φi∞ such that φin → φi∞ weakly in D 1,2 (R N ).We observe that where i ⋆ in is the adjoint operator of the immersion i ⋆ in : where the first term goes to zero by the Dominated Convergence Theorem.Hence at the limit We want to prove that the limit function φi∞ is null.It is sufficient to show that φi∞ ∈ ker(−∆ − pU p−1 ) ⊥ , i.e.
Indeed, if j = 0 we have that , by the non degeneracy of u 0 , we have that φ ∞ = 0.
• Step 5. Now we have to prove that L δ,ξ is invertible and its inverse is continuous.

Indeed, we know that Π
(Ω) is a compact operator, then L δ,ξ = Id − K where K is a compact operator.By (3.13) we also know that L δ,ξ is injective.Thus, by the Fredholm's alternative theorem is also surjective.Now we want to estimate the error term defined in (3.10).Proposition 3.6.Let N > 6 and δ, ξ satisfy (2.8) and (2.9) as in Section 2. Then there exists C = C(a) > 0 and ε 0 > 0 such that for any ε ∈ (0, ε 0 ) the error term satisfies for some σ > 0 arbitrarily small, where θ is given by (2.12).

Proof. Observing that
and by (2.1) we get .
and that by (3.18) and (3.5).Now and analogously Then, by (3.19) and (3.20) In a similar way, by (3.5) and (3.18) Now we can solve (3.8) using a fixed point argument.

The reduced energy
In this section we want to reduce the original problem to a finite dimensional one and solve (3.7).Define J ε : Then the critical points of J ε are solutions to (BN).Let us also define the reduced functional Jε (t, d, τ for some σ > 0 and θ = Proposition 4.1.Let N > 6 and δ, ξ satisfy (2.8) and (2.9) as in Section 2. There exists ε 0 > 0 such that for any ε ∈ (0, ε 0 ) it holds where J ε is defined in (4.1).
Proof.It's easy to see that Cε θ+σ by (3.2).Now by (3.1) and (3.22) as in the proof of Proposition 3.6.Now we want to evaluate and find an expansion of Proposition 4.2.Let N > 6 and δ, ξ satisfy (2.8) and (2.9) as in Section 2. Then it holds and where For the definition of Jε we refer to (4.2).
Proof.We can write J ε (u 0 − k i=1 P U i ) as where f (u) = |u| p−1 u and F (s) = s 0 f (t)dt.Now we evaluate each term.Recalling that The other important term is (III).Indeed We have to show that the other terms are of higher order.Reasoning as the third term of (III) we have Now we can split the term (V III) + (IX) into the sum of integrals on the balls B η h (ξ h ) and the integral on Ω\ ∪ B η h (ξ h ) that we call A h and B respectively, where we can evaluate the integral B on Ω \ ∪B η h (ξ h ) as and in a similar way for the integral A h on a ball B η h (ξ h ) we get that At the end we get We put Now by (2.10) we have that At the end, observing that and We need this equal to zero as ξi ∈ Ω and ξ 0 is a critical point of ∂ ν u 0 (ξ), we have where the lower order terms are O ε θ+σ .Now the zero order terms are The first order terms are The quantities in the previous box are zero as d 0 and t 0 satisfy the system in (2.7).Then the order zero terms are a function of (d 0 , t 0 ), namely Now the second order terms are We have the following estimates on the integrals where A 1 depends only on (d 0 , t 0 , ξ 0 ) and Since the constants A, C are strictly positive, d = t = (0, • • • , 0) ∈ R 2k is the unique critical point for Φ in d and t.Since the matrix D 2 N −1 ∂ ν u 0 (ξ 0 ) is positive definite there also exists a critical point τ 0 in τ .The point (0, 0, τ ) is a critical point for Φ, which is stable under small perturbation of the function.Hence for all ε sufficiently small there exists (δ ε , t ε , τ ε ), satisfying (2.9), critical point for Jε , which is close to (0, 0, τ ).Now observe that for all i = 1, • • • , k and r = 1, • • • , N − 1 Hence the assumptions of Proposition 4.3 are satisfied and we conclude that u ε = W δε,ξε + φ δε,ξε is a solution of (BN).

2 φ2in
where the last term converges to zero by the Dominated Convergence Theorem.Moreover φ 2 n and φ2 hn are uniformly bounded respectively in L N N−2 (Ω) and L N N−2 (R N ) and they converge to zero almost everywhere in Ω and R N .Hence φ 2 n and φ2 hn converge weakly to zero in L N N−2 (Ω) and