Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem

In this paper, we deal with the boundary value problem $-\Delta u= |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon$ in a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n\geq 3$ with homogenous Dirichlet boundary condition. Here $\varepsilon>0$. Clapp et al. in Journal of Diff. Eq. (Vol 275) built a family of solution blowing up if $n\geq 4$ and $\varepsilon$ small enough. They conjectured in their paper the existence of sign changing solutions which blow up and blow down at the same point. Here we give a confirmative answer by proving that our slightly subcritical problem has a solution with the shape of sign changing bubbles concentrating on a stable critical point of the Robin function for $\varepsilon$ sufficiently small.

The critical case corresponds to ε = 0 and we get f0(u) = |u| p−1 u.In such a situation, the Euler functional I0 does not satisfy Palais Smale condition and thus the corresponding variational problem happens to be lacking of compactness.The obtained problem (P0) is especially meaningful in geometry.In fact, the Yamabe problem is a version of this problem on manifolds.
In the sequel we list some known results concerning the critical case ε = 0.When Ω is starshaped, Pohozaev proved in [20] that problem (P0) has no positive solutions.However, when Ω is an annulus, Kazdan and Warner provided in [11] the existence of a positive radial solution.By using critical points at infinity theory, Bahri and Coron [2] proved that such a problem has a positive solution, under the condition that Ω has nontrivial topology.In this paper, we deal with the slightly subcritical case, i.e. ε > 0. In order to state old and new results, it is useful to recall some well known definitions.The space H 1 0 (Ω) is equipped with the norm .and its corresponding inner product ., . is defined by For a ∈ Ω and λ > 0, let (1 + λ 2 |y − a| 2 ) (n−2)/2 , where c0 := (n(n − 2)) (n−2)/4 . (1. 3) The constant c0 is chosen such that δ (a,λ) is the family of solutions of the following problem Notice that the family δ (a,λ) achieves the best Sobolev constant We denote by P δ (a,λ) the projection of δ (a,λ) onto H 1 0 (Ω), defined by −∆P δ (a,λ) = −∆δ (a,λ) in Ω, P δ (a,λ) = 0 on ∂Ω. (1.6) We will denote by G the Green's function and by H its regular part, that is G(x, y) = |x − y| 2−n − H(x, y) for (x, y) ∈ Ω 2 , (1.7) and for x ∈ Ω, H satisfies ∆H(x, .)= 0 in Ω, H(x, y) = |x − y| 2−n , for y ∈ ∂Ω.
We define the Robin function as R(x) = H(x, x), x ∈ Ω.The first paper establishing the existence of blowing up solutions to problem (Pε) is [7] where the authors proved that any x0 non-degenerate critical point of R generates a family of single-peak solutions concentrating around x0 as ε goes to 0. This family of solutions has the following form uε = P δ (aε,λε) + vε, where aε → x0, λε → ∞ and vε → 0 as ε goes to 0. In [17], the asymptotic behavior of radially symmetric solutions of (Pε) was analyzed when Ω is a ball.Very recently, we provided in [5] the existence of positive as well as changing sign solutions that blow up and/or blow down at different points in Ω.Our solutions have the following expansion uε = k i=1 αi,εγiP δ (a i,ε ,λ i,ε ) + v where γi ∈ {−1, 1} for each 1 ≤ i ≤ k, αi,ε → 1 λi,ε → ∞, vε ∈ E (a,λ) and vε → 0 as ε goes to 0.Here where P δi = P δ (a i,ε ,λ i,ε ) and (ai,ε)j is the j th component of ai,ε.Furthermore, the concentration points a1,ε, . . ., a k,ε are far away from each other and converge to distinct points in Ω, which form a non-degenerate critical point of a function defined explicitly in terms of the Green function and its regular part.
In this paper, we focus on a new kind of solutions.In fact, we prove the existence of sign-changing bubble towers solutions.These solutions are constituted by superposition of positive and negative bubbles with different blow up orders and closed concentration points.Musso and Pistoia have proved in [16] the existence of such solutions for the following nonlinear subcritical elliptic problem where ε > 0. As mentioned in [7] and [5], there is an interesting analogy between results obtained for (Pε) and those known for the usual elliptic equation (1.9).In this direction, we prove here that (Pε) shares also with problem (1.9), the existence of tower bubbles solutions of different blow-up orders, as conjectured in [7].
Let us review some known facts related to tower bubbles solutions.This phenomenon was firstly discovered by del Pino, Dolbeault and Musso in [9], where the authors dealt with the slightly supercritical Brezis-Nirenberg problem.They proved the existence of positive bubble tower radial solutions obtained by a superposition of several bubbles centered at the origin point but with different scaling parameters.
Their method strongly relies on the symmetry of the problem.Later, the construction described above was extended by Ge, Jing and Pacard in [10] to a more general class of domains, namely domains such that the associated Robin function R admits a non-degenerate critical point in the case of one tower of bubble.They also proved the existence of multiple towers of bubbles under a non-degeneracy condition on a critical point of a certain functional of point of the domain.In [16], the authors studied the same problem and they were able to remove the assumption on the domain.Indeed, they proved that, in any bounded domain Ω, the slightly supercritical Brezis-Nirenberg problem does admit solutions with the shape of a tower of bubbles.Their idea was the following: given any domain Ω, its Robin function is smooth, positive and unbounded as x approaches the boundary, thus it has a minimum in Ω, and hence at least one critical point.
In the same paper [16], as we mentioned before, Musso and Pistoia have constructed sign-changing bubble tower solutions for problem (1.9) which blow up at the minimum of the Robin function R.This work was an extension of that of Pistoia and Weth [18] where they proved that if Ω is symmetric with respect to the x1, . . ., xn axes, problem (1.9) has a sign-changing solution with the shape of a tower of bubbles with alternate signs, centered at the center of symmetry of the domain.Finally, we mention the work in [8], where the authors studied the existence of bubble towers changing sign solutions for the counterpart of (1.9) when the Laplacian operator is replaced by the biharmonic one.
Before introducing our result, we recall the following definition as stated in [14].
We say that ξ0 is a stable critical point of g if ∇g(ξ0) = 0 and there exists a neighborhood U ⊂⊂ Ω of ξ0 such that ∇g(x) = 0, ∀x ∈ ∂U, , where deg denotes the Brouwer degree.
It is easy to see that, if ξ0 is an isolated minimum or maximum point of the function g, then ξ0 is a stable critical point of g.Moreover, a non-degenerate critical point of g is stable according to the previous definition.
As we previously mentioned, the aim of the current paper is, by applying the Liapunov-Schmidt reduction method, proving the existence of bubble tower sign-changing solutions of (Pε) concentrating at a stable critical point of the Robin function R. Precisely speaking, our main result can be stated as follows.
We point out that one can remove the assumption R admits a stable critical point by using the same idea of Musso and Pistoia namely that the Robin function has a minimum in any given domain Ω, and hence it has at least one stable critical point.Arguing as in [15], one can exhibit an example of a contractible domain for which such an assumption holds true.Indeed, following the idea of perturbing domains, Musso and Pistoia constructed a domain for which the function R has a stable critical point.We also mention that the non-degeneracy of the critical point of the Robin function implies the existence of this kind of solutions, since by applying implicit function theorem one may prove that a non-degenerate critical point is a stable one.Thus, our condition is weaker than the non-degeneracy condition.In fact, when k = 1 our result improves the one of Clapp et al. in [7].Note that our family of solutions (uε) converges weakly to zero and its blow up rate satisfies This choice will be justified by exploiting some balancing conditions for the parameters of the concentration given in Lemmas 3.2 and 3.3.Indeed, taking λ k the smallest concentration speed by analyzing these balancing conditions, we obtain that (| ln ε|ε −1 ) 1/(n−2) and λ k needed to be of the same order.This will be the subject of Proposition 3.1 where we investigate the asymptotic profile of a family of sign changing bubble tower solutions uε which blows up in the interior of the domain Ω.Let us point out that, the obtained relations between the parameter λi's and ε goes along the argument developed in [16] concerning (1.9).In fact, Musso and Pistoia have chosen that the smallest concentration speed satisfies in our case.
To prove our main result, we apply the Liapunov-Schmidt Reduction method (see for instance [19] and the references therein).Thanks to the analysis of the gradient of the energy functional performed in [5], we were able to adopt some arguments developed in [16] and that by choosing suitable concentration rates and concentration points.After this choice we have also improved the expansion of the gradient with respect to the new variables.Note that our proof requires also the expansion of the energy functional Iε which is given in Proposition 2.6.In contrast with the nonlinear term |u| p−1−ε u in problem (1.9), one can not write the explicit expression of Fε the antiderivative of the nonlinearity fε of problem (Pε) (see (1.1) and (1.2)).We were able to overcome this technical difficulty and to obtain the asymptotic expansion of Iε, as ε goes to 0. From this expansion we get a new functional defined on finite space.By applying degree theory and taking into account the stability of the critical point of the Robin function we conclude on this new functional.We mention that our new result in Proposition 2.6 is quite involving.We think that this expansion will be useful to study the existence and to describe the blow up profile of some positive and changing sign solutions, as ε goes to zero.
The paper is organized as follows: In Section 2, we collect some basic tools which includes the asymptotic expansions of the energy functional and its gradient.We would like to warn the reader that the expansion of the gradient is quoted from [5] and we did not repeat the proofs.However, we provide the proof of the expansion of the energy functional in case of close concentration points.The description of the appropriate solutions is obtained in Section 3 and that by analyzing the asymptotic profile of sign changing bubble tower solutions.In Section 4 we introduce the precise profile of our searched solutions and we study its remainder term.Then we derive an asymptotic expansion for the reduced energy functional in Section 5. Finally, we complete the proof of our main result in Section 6. Section 7 is an appendix where we collect some technical Lemmas used in this work.Throughout this paper, we use the same c to denote various generic positive constants independent of ε.

The Technical Framework
Recall that each critical point of the energy functional Iε (defined by (1.2)) is a solution of (Pε).To construct solutions of the form (1.10), our argument require the expansion of Iε and its gradient for u = k i=1 αiγiP δ (ξ i ,λ i ) + v i.e. u belongs to a neighborhood of potential concentration sets.Here γi ∈ {−1, 1}.We mention that the expansions introduced in this section will be given in general setting i.e. we will not use, at this stage, the parameters information presented in Theorem 1.2.For η > 0, k ∈ N and (γ1, . . ., γ k ) ∈ {−1, 1} k , let us define where Note that, the variable εij comes from the scalar product (see [1] page 4).For simplicity, we denote P δ (ξ i ,λ i ) by P δi.Recall that simple computations show that As in [5], we are looking for a solution of (Pε) in a small neighbourhood of k i=1 αiγiP δi.The authors in [5] investigated the gradient of the functional Iε in V (k, η).In the sequel, we recall some expansions extracted from [5].
For each i ∈ {1, ..., k} and j ∈ {1, ..., n}, we have the following expansion where ∂H ∂(a) j denotes the partial derivative of H with respect to the j−th component of the first variable.Now, our aim is to obtain the expansion of the energy functional Iε around u ∈ V (k, η) with close concentration points.We start by the following useful lemmas.
Furthermore, the following integral is needed in the proof of Proposition 2.2 and it was computed in [5].We have (for more details see the proof of Lemma 2.9 in [5]).

Asymptotic behavior
In this section, we investigate the asymptotic profile of a family of sign changing bubble tower solutions uε blowing up in the interior of the domain Ω.Our aim is to look for suitable conditions on the parameters λi and ξi to construct such solution in the next sections.
For simplicity, we will assume that uε is a solution of (Pε) having the following form where and vε satisfies vε = min Therefore we get vε, ∂P δ i (3.5) The proof of (3.5) is similar to that of Eq. (3.7) in [5].
Our main result in this section is stated as follows.
Proposition 3.1 Let n ≥ 6 and let (uε) be a family of sign-changing solutions of (Pε) having the expansion (3.1) and satisfying (3.2)- (3.4).Assume that Then the concentration points ξ1,ε, ξ2,ε and the concentration speeds λ1,ε, λ2,ε satisfy λ2,ε|ξ1,ε − ξ2,ε| → 0 as ε → 0, and where Γ2 is a positive constant defined in (3.14).Furthermore, ξ0 is a critical point of the Robin function R and Λ satisfies We point out that the condition ln λ1,ε ≤ c ′ ln λ2,ε is very helpful in our argument.In fact, thanks to this condition and λ1,ε/λ2,ε → ∞ which implies ln λ2,ε ≤ ln λ1,ε, we derive that the quantities ln λi,ε's are of the same order, and this will help us to analyze the balancing conditions introduced in Lemma 3.2.
Next, we provide two balancing conditions, that need to be satisfied by the parameters of concentration, required in the proof of Proposition 3.1.The first one concerns the concentration speeds.Multiplying the equation −∆uε = fε(uε) by λi ∂P δ i ∂λ i and integrating by parts with taking into account (3.5) and (3.3), we obtain Lemma 3.2 Let n ≥ 6 and uε = γ1P δ1 + γ2P δ2 + vε a solution of (Pε).For each i, l ∈ {1, 2} such that l = i, we have the following expansion where Γ1 and c1 are the same constant defined in Proposition 2.2 and τ is a positive constant small enough.
The proof of Lemma (3.2) that we omit here looks like the expansion of ∇Iε(u), λi ∂P δ i ∂λ i given in Proposition 2.2.Indeed, being a solution of (Pε), uε is a critical point of Iε and therefore we have ∇Iε(uε), λi ∂P δ i ∂λ i = 0. Our second balancing condition focus on the concentration points and it is obtained in similar way.Namely, we have Lemma 3.3 Let n ≥ 6 and uε = γ1P δ1 + γ2P δ2 + vε a solution of (Pε).For each i, l ∈ {1, 2} such that l = i and j ∈ {1, ..., n}, we have the following expansion γi 2 where ∂H ∂(a) j denotes the partial derivative of H with respect to the j−th component of the first variable.
Proof of Proposition 3.1 : We recall that we have λ1/λ2 → +∞ as ε → 0. From Proposition 3.2, it is easy to obtain the following estimate (3.9) Let K be a compact set in Ω.It is easy to see that Note that the concentration points ξ1 and ξ2 satisfy (3.3).Thus, there exists a compact set K in Ω such that ξ1, ξ2 ∈ K for any ε and therefore We start by the following lemma.for some real number κ ≥ 1.In the sequel, we distinguish two cases: κ > 1 and κ = 1.
Using again (7.3) and (3.43) and Lemma 3.4, we obtain where we have used (3.9), (3.31) and the fact that Observe that, using (3.43), we get Thus, using (3.51),Lemma 3.4 and the antisymmetry of the function δ p ∂(ξ 1 ) 1 with respect to (x − ξ1)1 in B(ξ1, d1/2), we obtain where we have used, in the last equality, (3.25) i.e. ε12 and (λ2/λ1) are of the same order and the fact that In view of (3.47), (3.49)-(3.52)and (3.54), we get (3.55) To estimate B ′ 3 , we split the integral as follows: where Ω1 := {x : Here M is a positive constant greater than 4c where c is the constant introduced in (3.43).Using this fact, it is easy to see that in Ω c 1 we have Thus in view of (3.43), (7.5), Lemma 3.4 and the fact that p − 2 ≤ 0 for n ≥ 6, we get where we have used (3.53) and the fact that ε12 and ( λ 2 λ 1 ) n 2 are of the same order.Observe that, in Ω1, it holds |γ1δ1 + γ2 δ2| ≤ cλ2|ξ1 − ξ2| δ2, by using (3.43).Therefore, the expansion (3.42), (7.2), (3.43) and (3.53) imply Hence, (3.56), (3.57) and (3.58) assert Lastly, we compute < u, P ψ 1 1 >.In view of Proposition 7.2, (2.3), (3.31), (3.51) and Holder's inequality, we obtain Recall that we have d(ξ2, ∂Ω) ≥ c.Thus, the functions H and its derivatives are bounded.Furthermore, the function H satisfies (3.10).Therefore, from (3.63), it is easy to see that Λ is bounded from above and it is also bounded from below by some positive constant.Hence, Λ converges to Λ > 0 (up to a subsequence).Let ξ0 be the limit of ξ2 when ε goes to 0. Passing to the limit in (3.63) and (3.64), we get which imply that ξ0 is a critical point of the Robin function R and Λ satisfies (3.8).Therefore, in view of Lemma 3.4 and (3.31), the conclusion in (3.7) holds.The proof of Proposition 3.1 is thereby completed.✷

Description of the solution and its lower order term
We denote by i * : L 2n n+2 (Ω) → H 1 0 (Ω) the adjoint operator of the embedding i : is the unique solution of the equation −∆u = w in Ω and u = 0 on ∂Ω.By the definition of the operator i * , it is clear that (Pε) can be written as follows: where fε is introduced in (1.1).Next, we describe the shape of the solutions we are looking for.Let ξ be a point in Ω and, given an integer number k, let λj for j = 1, ..., k, be positive parameters defined as multiple of proper power of ε | ln ε| , namely Let ξj, for j = 1, ..., k, also be k points in Ω given by ξj = ξ + λ −1 j σj, with σ2, ..., σ k ∈ R n and σ1 = 0. ( Let αj , for j = 1, ..., k, be positive parameters.Fix a small η > 0 and assume that It is an immediate observation that We look for a tower of sign-changing bubbles solution to (Pε), of the form: where we set α = (α1, ..., n .The term v has to be thought as a remainder term of lower order.Let where P δi = P δ (ξ i ,λ i ) and ξ j i is the j th component of ξi.Let Π : H 1 0 (Ω) → E ⊥ and Π ⊥ : H 1 0 (Ω) → E be the orthogonal projections.In order to solve (4.1) we will solve the couple of equations: Given ξ, α, ρ and σ satisfying conditions (4.4), one can solve uniquely (4.8) in v ∈ E. This solution, which will be denoted by v is the lower order term in the description of the ansatz (4.6).This is the content of: Proposition 4.1 There exists ε0 > 0 such that for any 2)-(4.4) and for any ε ∈ (0, ε0) there exists a unique function v = v(ε, α, ρ, σ, ξ) ∈ E such (4.10) In addition, there exists The proof of such a result is contained in [5] (see the proof of [5,Proposition 3.2]).In the following, v is the solution of (4.8).
As in [5], we estimate the numbers A, B and C. Note that the functions ∂P δ i ∂λ i and ∂P δ i ∂ξ j i , which appear in (4.11), are multiplied respectively by λi and 1 λ i in [5].Taking into account this change and the ε-orders of the λ ′ i s given in (4.2), we get the following.Proposition 4.2 For i = 1, ..., k and j = 1, ..., n, we have As in [21] and [16], since |fε(u)| ≤ |u| p , we have the following estimate.where Since ζv is a solution of(4.12), the following inequality holds: Hence, we need to estimate |g| 2 2n/(n+1) .We observe that Let us denote q := 2n/(n + 1).We have to estimate each term of (4.15) in the L q (Ω)-norm.It holds and by using Proposition 4.2, we get Furthermore, since q < 2 < p + 1, by Holder's inequalities, we have The only remaining term is Ω (ζ|v| p ) q 2/q which is the most difficult to estimate because pq > p + 1.We v, and we integrate on Ω. Arguing as in [21] (see also [16]),integration by parts and the Sobolev embedding theorem lead to the inequality On the other hand, we write Arguing exactly as in [21, Appendix C], we get: . By using Holder's inequality, the definition of the function ζ and Proposition (4.2), we also have the validity of the following estimates: Taking account of (4.2) and (4.10), the desired result follows.✷

The reduced energy
We are left now to solve (4.9), more precisely to find the point ξ, the points σ2, ..., σ k ∈ R n and the parameters ρ1, ..., ρ k and α1, ..., α k so that (4.9) is satisfied.It happens that this problem has a variational structure, in the sense that solving (4.9) is reduced to find critical points to some given explicit finite dimensional functional. Let As in Part 1 of [16, Proposition 2.2], we have the following: if (α, ρ, σ, ξ) is a critical point of Iε, then u := k i=1 (−1) i αiP δ (ξ i ,λ i ) + v is a critical point of Iε.This result reduces the existence of solutions for (Pε) to the problem of finding critical points of the reduced energy functional Iε.Now, we introduce the asymptotic expansion for the reduced energy functional Iε and its partial derivative in terms of the parameters (α, ρ, σ, ξ).Using Propositions 2.1 and 4.1, we have the following.Proposition 5.1 For h = 1, ..., k, we have C 0 -uniformly with respect to ξ in compact sets of Ω, σ in compact sets of R (k−1)n , α in compact sets of R k + and ρ in compact sets of R k + .
The expansion of Iε can be stated as follows.
For h = 1, ..., k, in view of (4.2) and (4.3), we have (5.12) In similar way, from Proposition 2.3 we derive, for r = 2, ..., k and j = 1, ..., n that  In fact, by integrating by parts and using (5.17We recall that, in the previous section, we have mentioned that our aim is prove that the reduced energy has a critical point.This fact is equivalent to the existence of the requested solution. Let us perform the change of variables: si = ρi+1/ρi, for 1 ≤ i ≤ k − 1 and s k = ρ k .where |.|q denotes the usual norm in L q (Ω) for each 1 ≤ q ≤ ∞.
We also need the following lemma.(a) To prove (a) we distinguish two cases.

2 and 2 and
ε12 are of the same order, found in the first case.This will help us to conclude in the first step.Concerning the second step, namely excluding the case c ′ 1 < λ2|ξ1 − ξ2| < c ′ 2 , we subtract the equations (3.20) and (3.21) we get, as in the case κ > 1, 1/λ n−2 ε12 are of the same order and we conclude similarly.✷ Notice that Lemma 3.4 implies, as ε → 0