Qualitative analysis on the critical points of the Robin function

Let $\Omega\subset\mathbb{R}^N$ be a smooth bounded domain with $N\ge2$ and $\Omega_\epsilon=\Omega\backslash B(P,\epsilon)$ where $B(P,\epsilon)$ is the ball centered at $P\in\Omega$ and radius $\epsilon$. In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in $\Omega_\epsilon$ for $\epsilon$ small enough. We will show that the location of $P$ plays a crucial role on the existence and multiplicity of the critical points. The proof of our result is a consequence of delicate estimates on the Green function near to $\partial B(P,\epsilon)$. Some applications to compute the exact number of solutions of related well-studied nonlinear elliptic problems will be showed.


Let
where S(x, y) is the fundamental solution given by where C N := 1 N (N −2)ω N , with ω N the volume of the unit ball in R N .H D (x, y) is the regular part of the Green function which is harmonic in both variables x and y.The Robin function is defined as R D (x) := H D (x, x) in D. Note that our definition differs, up to a multiplicative constant, from that of some other authors (see for example [2] and [6] where R D (x) = 2πH D (x, x) for N = 2).However, since we are interested in the critical points of the Robin function, this difference plays no role in our results.
The Robin function plays a fundamental role in a great number of problems (see [2,12] and the references therein).It also plays a role in the theory of conformal mappings and is closely related to the inner radius function (see [19]) and to some geometric quantities such as the capacity and transfinite diameter of sets (see [2,12]).For elliptic problems involving critical Sobolev exponent [16,23], the number of solutions is linked to the the number of non-degenerate critical points of the Robin function.Despite the great interest on the Robin function, many questions are still unanswered and we are far from a complete understanding of its properties.
The only bounded domain where the Robin function is explicitly known is the ball centered at a point Q ∈ R N ; in this case the Robin function is radial and Q is the only critical point (it turns out to be non-degenerate).The computation of the number of critical points as well as other geometric properties (for example the shape of level sets) in general domains of R N interested a great number of experts in PDEs, but anyway very few results are known.One additional difficulty is that it is not known if the Robin function satisfies some differential equation.This is known only for planar simply-connected domains ( [2]) where it solves the Liouville equation.
To our knowledge, one of the first result in general domains is [6], in which Caffarelli and Friedman proved that the Robin function admits only one non-degenerate critical point in convex and bounded domain in R 2 .Note that here a crucial role is played by the Liouville equation.Later, the existence and uniqueness of critical points of Robin function for a convex and bounded domain in higher dimension was proved in [10].However, the non-degeneracy of this critical point is still open.Also some results on the non-degeneracy of the critical points of the Robin function for some symmetric domain can be found in [17].For non-convex domains, for example domains with "rich" topology, we cannot expect the uniqueness of the critical point of the Robin function.But how the topology of the domain impacts the number of critical points of Robin function is still unclear and seems to be a very difficult issue.
In this paper we study what happens when we remove a small hole to a domain Ω ⊂ R N .More precisely, denoting by B(x 0 , r) the ball centered at x 0 of radius r, set Ω ε = Ω\B(P, ε) with P ∈ Ω. (1.2)

Robin function in Ω Robin function in Ω ε
Note that the Robin function R Ωε blows up at ∂B(P, ε).So, for ε small enough, R Ω and R Ωε look like very differently near P .
Our aim is to study the number of critical points of the Robin function R Ωε as well as their non-degeneracy.Observe that the regular part H Ωε (x, y) satisfies ∆H Ωε (x, y) = 0 in Ω ε , H Ωε (x, y) = S(x, y) on ∂Ω ε .
Hence, by the standard regularity theory, we have that H Ωε (x, y) → H Ω (x, y) (1.3) in any compact set K ⊂ Ω \ B(P, r) as ε → 0 for some small fixed r > 0. Setting x = y in (1. 3) we get that R Ωε (x) → R Ω (x) for any x ∈ K, which is a good information on the behavior of R Ωε far away from P .The behavior of R Ωε close to ∂B ε is much more complicated and is the most delicate problem to be addressed in this paper.Here a careful use of the maximum principle for harmonic functions will be crucial.Actually, sharp estimates up to ∂B(P, ε) will allow us to prove the existence of critical points for R Ωε which converge to P as ε → 0.
Our first result emphasizes the role of the center of the hole B(P, ε).Indeed the scenario is very different depending on whether P is a critical point of R Ω or not.
In all the paper we denote, for x ∈ Ω ε , by O f (ε, x) a quantity such that where C is a constant independent of ε and x.
Remark 1.2.The condition ∇R Ω (P ) = 0 cannot be removed.Indeed, if Ω = B(0, R) we know that 0 is the unique critical point of R Ω and in the annulus Ω ε = B(0, R)\B(0, ε) we have that R Ωε is radial with respect to the origin.Since R Ωε ∂Ωε = +∞ we have that the set of minima of R Ωε is a sphere and then R Ωε admits infinitely many minima.
Remark 1.4.Let us give the idea of the proof of the theorem.Our starting point is the following basic representation formula for the gradient of the Robin function (see [24], and [13] for N = 2, 3 and [4] for any N ≥ 2), where ν ε (y) and ν(y) are the outer unit normal to ∂Ω ε and ∂Ω respectively.By (1.6) we will derive some C 1 -estimates which are crucial to prove our results and, in our opinion, have an independent interest.Let us start to discuss the case N ≥ 3, where we get, uniformly for (1.7) The previous estimate is a second order expansion of the Robin function in Ω ε .It turns out that ∇R Ω (x) and ∇R R N \B(P,ε) (x) are the leading terms of the expansion of ∇R Ωε (x).
Note that from (1.7) we get that ∇R Ωε (x) → ∇R Ω (x) uniformly on the compact sets of Ω not containing P .So, under some non-degeneracy assumptions on the critical points of R Ω (x), we get that the number of critical points of R Ωε (x) far away from P is the same as R Ω (x).Next let us study what happens when x → P .Here the analysis is very delicate but formally, using the uniform convergence in (1.7), we get admits a unique nondegenerate critical point with index (−1) N +1 we get that the same holds for R Ωε , which proves (P 1 ) and (P 2 ) for N ≥ 3.
The case N = 2 is a little bit more complicated because R R 2 \B(P,ε) (x) does not goes to 0 as ε → 0. Actually, an additional term appears in the expansion, (1.8) However, up to some technicalities, the proof follows the same line as in the case N ≥ 3. Finally let us point out that the maximum principle plays a crucial role to get uniform estimates up to ∂Ω in (1.7) and (1.8).
Remark 1.5.An interesting asymptotic formula for the Robin function in Ω ε is the following (see [2], p.198-199): for every x = P and N ≥ 3, (an analogous formula holds for N = 2).(1.9) is a consequence of the Schiffer-Spencer formula (see [25] for N = 2 and [22] for N ≥ 3) but the remainder term O ε N −1 is not uniform with respect to x (as stated at page 771 in [22]).
In Proposition 3.1 we prove the following, where the remainder terms are uniform with respect to x ∈ Ω ε .This can be seen as an extension of (1.9).Theorem 1.1 states the uniqueness and non − degeneracy of the critical points of R Ωε near the hole B(P, ε).Under a non-degeneracy condition on the critical points of R Ω we can compute the exact number of the critical points of R Ωε in Ω ε .
Corollary 1.6.Suppose that Ω and Ω ε are the domains as in Theorem 1.1.If ∇R Ω (P ) = 0 and all critical points of R Ω in Ω are non-degenerate, then for ε small enough, all critical points of R Ωε are non-degenerate and As previously mentioned, the Robin function of the ball B(0, R) has a unique non-degenerate critical point.So Corollary 1.6 applies and then if P = 0 the Robin function R Ωε has two non-degenerate critical points.On the other hand, if P = 0, we are in the situation described in Remark 1.2.Hence a nice consequence of the previous corollary is the following one.
Corollary 1.7.Assume that Ω = B(0, R) ⊂ R N , N ≥ 2 and Ω ε = B(0, R)\B(P, ε).Then, for ε small enough, and if P = 0 the two critical points are non-degenerate.In next theorem we study what happens when ∇R Ω (P ) = 0.This case is more delicate and it seems very hard to give a complete answer.However, in some cases it is possible to compute the number of the critical points, as stated in the following.
• The Hessian matrix Hess R Ω (P ) has m ≤ N positive eigenvalues Assume that B(P, r) ⊂ Ω admits P as the only critical point of R Ω .If the eigenvalue λ l is simple for some l ∈ {1, • • • , m}, then we have two non-degenerate critical points x ± l,ε of R Ωε (x) in B(P, r) \ B(P, ε) which satisfy where v l is the l-th eigenvector associated to λ l with |v l | = 1, r ε,l is the unique solution of Finally it holds Moreover, if all the positive eigenvalues of R Ω (P ) are simple we have that critical points of R Ωε (x) in B(P, r) \ B(P, ε) = 2m (1.12) and all critical points satisfy (1.10) for l = 1, • • • , m.
Remark 1.9.The condition det HessR Ω (P ) = 0 is widely verified.In [21] it was proved that it holds up small perturbations of the domain Ω.Note that only positive eigenvalue of the Hessian matrix of R Ω (P ) "generates" critical points for R Ω (see Proposition 5.1).Hence saddle points of R Ω give less contribution to the number of critical points of R Ωε .
Corollary 1.10.Let Ω be a convex and symmetric domain (see Gidas, Ni and Nirenberg [14]) with respect to the origin.We have that (i) If P = 0, then R Ωε (x) admits exactly two non-degenerate critical points in Ω ε .
(ii) If P = 0 and all the eigenvalues of Hess R Ω (0) are simple then R Ωε (x) admits exactly 2N non-degenerate critical points in Ω ε .
x + 2,ε → 0 Remark 1.11.An example of a domain Ω which satisfies the conditions in Theorem 1.8 for N ≥ 2 is the following (see Section 6), We will give a precise description of the Robin function R Ω δ for δ small in Theorem 6.1, Section 6.
Remark 1.12.The proof of Theorem 1.8 uses again the estimates (1.7) and (1.8).In this case we get that In other words, after diagonalization we get that admits 2m non-degenerate critical points and the claim follows as in Remark 1.4.On the other hand, if some positive eigenvalue is multiple, say This implies that there is a set of critical points of F given by a sphere S k .Since we have a manifold of critical points, the number of the critical points depends on the approximation function and it leads to a third order expansion of R Ωε (x).Further considerations on multiple eigenvalues deserve to be studied apart.However, when Ω is a symmetric domain, we obtain partial results on the critical points of R Ωε (x) (see Theorem 5.7).
Remark 1.13.Our results can be iterated to handle the case in which k (k ≥ 2) small holes are removed from Ω.Moreover, using similar ideas, our main theorems are true if we replace B(P, ε) by a small convex set.
As in the case ∇R Ω (P ) = 0 we have the following corollary.
Corollary 1.14.Suppose Ω is a bounded smooth domain in R N , N ≥ 2 and P ∈ Ω.If ∇R Ω (P ) = 0, all critical points of R Ω are non-degenerate and Hess R Ω (P ) has m ≤ N positive eigenvalues which are all simple, for small ε it holds Finally all critical points of R Ωε (x) are non-degenerate.
As said before the non-degeneracy of critical points of the Robin function plays an important role in PDEs.Now we would like to give applications of our results to some elliptic problems.For example, let us consider the following, with S the best constant in Sobolev inequality.We have following results.
Proof of Theorem 1.15.Firstly, let us fix ε ∈ (0, ε 0 ] such that Theorems 1.1 and 1.8 apply.Then if ∇R Ω (P ) = 0, from Corollary 1.6 we get that R Ωε admits exact two critical points in Ω ε which are non-degenerate.If ∇R Ω (P ) = 0 and all the eigenvalues of ∇ 2 R Ω (P ) are simple, then Theorem 1.8 gives us that R Ωε admits exact 2N critical points in Ω ε which are non-degenerate.Next it is known by [11,20,23] that the solutions of (1.13) with (1.14) or (1.15) concentrate at critical points of R Ωε when p is large for N = 2 or N +2 N −2 − p > 0 small for N ≥ 4. Moreover from [1,18], we find that, using the non-degeneracy assumption of the critical points of R Ωε , we have the local uniqueness of these solutions.
Remark 1.16.Observe that in above corollary, the assumption N ≥ 4 instead of the natural one N ≥ 3 is due to technical reason in proving the uniqueness result in [1].If the uniqueness result in [1] is extended to N = 3 we will get the claim also in this case.
The paper is organized as follows.In Section 2 we prove some lemmas which estimate the regular part H Ωε in terms of H Ω and H R N \B(P,ε) .Here the maximum principle for harmonic functions allows to get uniform estimates for R Ωε up to ∂B(P, ε).These will be the basic tool to give the expansion of R Ωε and its derivatives which will be proved in Section 3. In Section 4 we consider the case ∇R Ω (P ) = 0 and prove Theorem 1.1 and Corollary 1.6.In Section 5 we consider the case ∇R Ω (P ) = 0 and prove Theorem 1.8 and Corollaries 1.10 and 1.14.In Section 6 we will give an example of domains which satisfy the assumptions of Theorem 1.8.Finally in the Appendix we recall some known properties of the Robin function in the exterior of the ball as well as some useful identities involving the Green function.

Uniform estimates on the regular part of the Green function
Set Observe that B ε ⊂ Ω for ε sufficiently small and we will always assume this is the case.We denote by B c ε := R N \B ε and without loss of generality, we take P = 0 ∈ Ω.
In this section we will prove two crucial lemmas that will be repeatedly used in the proof of our expansion of the Robin function and its derivatives.In order to clarify their role, let us write down the following representation formula for the gradient of the Robin function proved in [4], p.170: for x ∈ Ω ε and letting ν y be the outer unit normal to the boundary of the domain, = using the identities Hence in order to estimate the integrals in the boxes we need the behavior of H Ω (x, y) − H Ωε (x, y) on ∂Ω and H B c ε (x, y) − H Ωε (x, y) on ∂B ε respectively.It will be done in the next lemmas.
For any y ∈ Ω with |y| ≥ C 0 > 0, we have that By the representation formula for harmonic function, we obtain ∂G Ω (0,y) ∂G Ω (0,y) So, by the maximum principle, we get that for any x ∈ Ω ε which implies (2.4) for N ≥ 3.
and exactly as for Next lemma concerns the estimate for H B c ε (x, y) − H Ωε (x, y) as y ∈ ∂B ε .Since the corresponding integrals of (2.1) are harder to estimate, we will need to write the leading term of the expansion as ε → 0. Lemma 2.2.For any x ∈ Ω ε , y ∈ ∂B ε we have where the function φ ε (x, y) is given by Remark 2.3.It is possible to improve the estimate (2.5) for N ≥ 3 in order to have a lower order term like o(1).This can be achieved adding other suitable terms to φ ε like in the 2-dimensional case.However the remainder term O(1) will be enough for our aims.
Proof.As in the proof of Lemma 2.1, we can prove that In this case we have, for x ∈ ∂B ε (recall that y ∈ ∂B ε ), On the other hand for x ∈ ∂Ω, using (A.1), we have So by the maximum principle we get In this case we have, for x ∈ ∂B ε , As in the previous case the maximum principle gives the claim.Lemma 2.1 and Lemma 2.2 will allow to give sharp estimates for R Ωε and ∇R Ωε in Ω ε .For what concerns H B c ε (x, y) − H Ωε (x, y) we need additional information on its second derivative.For this it will be useful to use the following known result for harmonic function.Proof.See [15], page 22.
By the previous lemma we deduce the following corollary.
Corollary 2.5.For any x ∈ Ω ε we have that, where φ ε is the function introduced in Lemma 2.2.
Proof.Let N = 2 and |y| ≥ C 0 .From (2.4) and by Lemma 2.4 with r = dist(x, ∂Ω ε ) we get which gives the claim.In the same way we get, for N ≥ 3, which proves (2.6) for N ≥ 3.In the same way applying Lemma 2.4 to the function ∇ y H B c ε (x, y) − H Ωε (x, y) − φ ε (x, y) and using (2.5), we have (2.7).

Estimates on the Robin function R Ωε and its derivatives
In this section we prove an asymptotic estimate for R Ωε and its derivatives in Ω ε = Ω\B ε .It is worth to remark that we get uniform estimates up to ∂B ε for ε small.These allow us to find the additional critical point for R Ωε (which will be actually close to ∂B ε ), but we believe that these estimates are interesting themselves.There is a common strategy in the proof of the estimates both for R Ωε and its derivatives.We start using some representation formula and after some manipulations (as in (2.1)) we reduce our estimate to some boundary integrals.Lastly we use the Lemmas of Section 2 to conclude.
The main result of this section is the following.Proposition 3.1.We have that, for any )), we have that (3.1) can be written in this way, Proof.Starting by (1.1) and the representation formula for harmonic function we have and so By (2.4) we get that

Computation of K 2,ε
First we observe that and so the claim follows for N ≥ 3.
Case N = 2.In the same way we get by (3.3),The main result of this section is the following.
Proposition 3.3.We have that for any x ∈ Ω ε , Proof.Recalling (2.1) we have that We will show that, for N ≥ 3, the integrals I 1,ε , • • • , I 4,ε are lower order terms with respect to ∇R Ω and ∇R B c ε .If N = 2 the situation is more complicated, because both integrals I 3,ε and I 4,ε give a contribution.

Computation of I 1,ε
By (2.4) we get that

Computation of I 2,ε
By (2.4) we get that

Computation of I 3,ε
Then we look at the cases N ≥ 3 and N = 2 separately.
Case N ≥ 3 By (3.3) we get Case N = 2 Again by (3.3) we have that

Computation of I 4,ε
As in the previous step let us consider the case N ≥ 3 firstly.
These estimates will be crucial to prove the uniqueness of the critical point of R Ωε close to ∂B(0, ε).The basic result of this section is the following.Proposition 3.4.For any i, j = 1, • • • , N and for x ∈ Ω ε such that dist(x, ∂Ω) ≥ C > 0, we have .
We have to estimate the integrals J 1,ε , • • • , J 6,ε .The computations are very similar to those of Proposition 3.3.

Computation of J 1,ε
By Lemma 2.1 and (2.6) we get

Computation of J 2,ε
Here we use the assumption that x satisfies dist(x, ∂Ω) ≥ C > 0. We need it to have Using Lemma 2.1 we immedialtely have

Computation of J 6,ε
We will show that We have, by (3.10) and N ≥ 3, and for N = 2, which proves the claim.
Now we look at the cases N = 2 and N ≥ 3 separately.If N ≥ 3, collecting the previous estimates we have that If N = 2, collecting the previous estimates we have that which ends the proof.
We will apply the C 2 estimates of R Ωε at the critical points of R Ωε .This leads to the following corollary.Corollary 3.5.Set D ε,q,c = x ∈ Ω ε such that dist(x, ∂Ω) ≥ C > 0 and for some c > 0, 0 Then for any i, j = 1, • • • , N and x ∈ D ε,q,c we have that for N = 2.
Proof.First of all we have that, for N ≥ 2 and since |x| ≥ cε q , Next we observe that dist(x, ∂B ε ) = |x| − ε ∼ |x| since x ∈ D ε,q,c and by Remark 4.2.So we have that (3.8) becomes for N = 2, which gives the claim.
We start this section with a necessary condition on the location of the critical points of R Ωε .Basically it is a consequence of Proposition 3.3.
If x ε is a critical point of R Ωε (x), then for ε → 0 we have that either where r ε is defined in (1.5).
Proof.Let x ε be a critical point of R Ωε and first consider N ≥ 3.By (3.4) we get If x ε → x 0 = 0 we have that (4.2) holds.So let us suppose that x ε → 0.

By (3.4) we have
and, since which gives the claim.

Remark 4.2. Let us point out that
In fact, taking g(r) = r − ln r ln ε , then Remark 4.3.Proposition 4.1 implies that the critical points of R Ωε that converges to 0 belong to the ball where δ > 0 is a small fixed constant.So, if x ε is a critical point of R Ωε then either In the next lemma we introduce a function F which plays a crucial role in the proof of the main results.
Proof.By a straightforward computation we have that y 0 is the unique critical point of F (y).
Next we have that, for N ≥ 2, the Hessian matrix of F (y) computed at y 0 = (y 1,0 , .., y N,0 ) is given by Note that λ 1 = 3 − 2N is the first eigenvalue of the matrix A ij with associated eigenvector v 1 = y 0 .Next, we observe that any vector of the space X := {x ∈ R N , x⊥y 0 } is an eigenvector of the matrix A ij corresponding to the eigenvalue λ = 1, so that det(A ij ) = 3 − 2N .Hence we have that which proves (4.12).Now we are in position to prove Theorem 1.1.
Proof of Theorem 1.1.Recall that we assume that P = 0 ∈ Ω.If x ε ∈ B(0, r) \ B(0, ε) is a critical point of R Ωε , then, since B(0, r) ⊂ Ω is chosen not containing any critical point of R Ω , by Proposition 4.1, we necessarly have that x ε → 0 and the expansion in (4.3) holds.Then by Proposition 4.1 and Remark 4.3 it is enough to prove the existence and the uniqueness of the critical point x ε in the ball C ε .

Let us introduce the function
where y 0 is defined in (4.1) and δ is chosen as in Remark 4.5.Let us show that where F is the function defined in Lemma 4.4.Indeed, using Proposition 3.3 we have that which gives the uniform convergence of ∇F ε to ∇F in B(y 0 , δ).
Let us show the C 1 convergence.By Remark 4.5 we can apply Corollary 3.5 with q = N −2 2N −3 and a suitable c > 0 such that, for N ≥ 3 which gives the claim.In the same way, if N = 2, using (1.5) and again the Corollary 3.5 with any q < 1 and a suitable c > 0, which gives the claim.Finally, the C 1 convergence of ∇F ε to ∇F and (4.12) gives that which, jointly with the non-degeneracy of y 0 , implies the existence and uniqueness of a critical point y ε ∈ B(y 0 , δ) of F ε .By the definition of F ε this implies the existence of a unique critical point x ε for R Ωε in C ε .Finally, by the definition of C ε , x ε → 0 and by (4.3) of Proposition 4.1 we get (1.4).Moreover We end the proof showing that R Ωε (x ε ) → R Ω (0).By Proposition 3.1 we have that, for N ≥ 3, which gives the claim.For N = 2 we have, by Remark 3.2, =R Ω (0)+o( 1) where r is such that det Hess R Ω (x) = 0 in C 1 .By Proposition 3.3 and Corollary 3.5 we have that R Ωε → R Ω in C 1 and so by the choice of r, the non-degeneracy of the critical points of R Ω and the Finally from Theorem 1.1, we get that critical points of R Ωε (x) in B(P, r)\B(P, ε) = 1, which proves the claim.
As in the previous sections we assume that P = 0. We will follow the line of the proof of Theorem 1.1.So we start with a necessary condition satisfied by the critical points of R Ωε (x).Proposition 5.1.If 0 is a non-degenerate critical point for R Ω (x) and x ε is a critical point of R Ωε (x), then for ε → 0 we have that either where λ is a positive eigenvalue of the Hessian matrix Hess R Ω (0) , v is an associated eigenvector with |v| = 1 and r ε is the unique solution of .This can be seen observing that r ε,1 = 1 (λπ| ln ε|) Furthermore from Proposition 3.3 and ∇R Ω (0) = 0, we get for N ≥ 3, for N = 2.
Analogously to the previous section let us introduce "the limit function" of a suitable rescaling of R Ωε .
for N ≥ 3, where D N (N ≥ 2) is the same as in (4.11).Suppose that Hess R Ω (0 where v (l) is an eigenvector of the matrix ∂ 2 R Ω (0) ± are non-degenerate critical points and then it is possible to select δ (l) > 0 such that in B ȳ(l) ± , δ (l) we have det Hess F y = 0.
The claim (5.6) will be proved by diagonalizing the matrix Hess R Ω (0) .Here we consider the case N ≥ 3 (N = 2 can be handled in the same way).Let P the orthogonal matrix such that Taking Z = P T (y) we get that the system ∇ F (y) = 0 becomes Note that these zeros are critical point of the function (5.8) Next step is the computation of the determinant of the hessian matrix of F .We know det Hess F y (l) = det P T Hess T y (l) P = det Hess T y (l) .
By the basic theory of linear algebra, we can find that the eigenvalues of M 0 are λ s − λ l for s Assume that λ l is a simple eigenvalue.Then, following the notations of Remark 5.5, analogously to the previous section we define and ε,± .
Proposition 5.1 implies that, under the assumption that all the eigenvalues of ∂ 2 R Ω (0) ∂x i ∂x j i,j=1,..N are positive, the critical points x ε verify either Proof of Theorem 1.8.Recall that we assume that P = 0 ∈ Ω.As in the proof of Theorem

Let us introduce the function F
Furthermore, using Corollary 3.5 with q = N −2 2N −2 for N ≥ 3 and q < 1 for N = 2 and arguing as in the proof of Theorem 1.1 we get that l) .

Since ȳ(l)
+ is a non-degenerate critical point of F then ∇ F ε (y) admits a unique critical point y l) and also y Next let us assume that all the eigenvalues of the Hessian matrix Hess R Ω (0) are simple.Again by Proposition 5.1, Remark 5.5 and the discussion above, we have that if x ε is a critical point belonging to B(0, r) \ B(0, ε) then x ε ∈ C ε .The simplicity of the eigenvalue and the previous claim prove (1.12).
Proof of Corollary 1.10.In this case the Robin function R Ω (x) has only one critical point P = 0 which is a non-degenerate minimum point (see [10,17]).This means that Hess R Ω (0) has N positive eigenvalues and, if they are simple, the Robin function R Ω\B(0,ε) (x) has exactly 2N critical points for ε small enough.So the claim follows by Theorems 1.1 and 1.8.In this case we are not able to give a complete description of the critical points of R Ωε .Suppose that we have a multiple eigenvalue satisfying In this case, the function T (y) defined in (5.8) admits a manifold of critical points given by By (5.6) we have that the Hessian matrix is non degenerate in the directions different from x j , • • • , x j+k and so it is a non − degenerate manifold of critical points for N in the sense of Morse-Bott theory.However, even in this explicit case, it seems hard to get existence results of critical points for T (y) under non radial small perturbations.Of course no possible information can be deduced about the non-degeneracy.
Without additional assumptions, as pointed out in the introduction, it is even possible to have infinitely many solutions (the radial case).For these reasons the case of multiple eigenvalues is unclear.
Before the close of this section, we give a partial result when Ω is a symmetric domain.
Theorem 5.7.Let Ω ⊂ R N be convex and symmetric with respect Then we have that R Ωε (x) has at least 2N critical points which are located on the coordinate axis, i.e.
Moreover we have Proof.We prove the claim by constructing 2N zeros for ∇R Ωε (x).For the case N ≥ 3, as in the previous theorem we study the equation Here we introduce the points and similarly, This implies that there exists Z Taking a small, we can write b = α 1 + o(1).Hence there exists Z 1,ε = then repeating the argument above for any positive eigenvalue, we have that R Ωε (x) has at least 2N critical points x ± i,ε (i = 1, • • • , N ) in B(0, r)\B(0, ε), where B(0, r)\{0} ⊂ Ω does not contain any critical point of R Ω (x).Moreover we can write Also using (3.13) for N ≥ 3, we compute For the case N = 2, as in the previous theorem we study the equation Then similar the idea in proving the case N ≥ 3, we can write x ± 1,ε = 1 + o(1) r ε,1 , 0 and x ± 2,ε = 0, 1 + o(1) r ε,2 .
matrix with i-th eigenvalue λ i = For future aims it will be useful to remark that δ , a straightforward (and tedious) computation gives that which ends the proof.
The final lemma computes some useful integrals.
D ⊂ R N , N ≥ 2 be a smooth domain.For (x, y) ∈ D × D, x = y, denote by G D (x, y) the Green function in D. It verifies    −∆ x G D (x, y) = δ x (y) in D, G D (x, y) = 0 on ∂D, in the sense of distribution.We have the classical representation formula G D (x, y) = S(x, y) − H D (x, y), (1.1)

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Picture of two solutions concentrating at critical points of R Ωε where the solution u ε,p satisfies either lim p→+∞ p Ω |∇u ε,p | 2 = 8πe, for N = 2,

Lemma 4 . 4 .
Let us consider the function F : R N \ {0} → R as

Lemma 5 . 4 .
Let us consider the function F : R N \ {0} → R as