Existence and multiplicity of peaked bound states for nonlinear Schrödinger equations on metric graphs

We establish existence and multiplicity of one-peaked and multi-peaked positive bound states for nonlinear Schrödinger equations on general compact and noncompact metric graphs. Precisely, we construct solutions concentrating at every vertex of odd degree greater than or equal to 3. We show that these solutions are not minimizers of the associated action and energy functionals. To the best of our knowledge, this is the first work exhibiting solutions concentrating at vertices with degree different than 1. The proof is based on a suitable Ljapunov–Schmidt reduction.


Introduction
In this paper we are interested in existence and multiplicity of positive solutions of nonlinear Schrödinger equations −u ′′ + λu = u 2µ+1 , (1.1) where µ > 0 and λ ∈ R, on metric graphs.
The class of graphs we will consider is rather general.In what follows, we assume that G = (V, E) is a connected metric graph such that • the set of vertices V and that of edges E are at most countable; • the degree deg(v) of a vertex v, i.e. the number of edges incident at it, is finite for every v ∈ V; • the length |e| of the edge e is bounded away from zero uniformly on e ∈ E, that is inf e∈E |e| > 0.
For the sake of brevity, we will say that G belongs to the class G whenever it satisfies these conditions.As usual, each bounded edge e ∈ E is identified with an interval [0, ℓ e ] with ℓ e := |e|, whereas each unbounded edge (if any) is identified with (a copy of) the positive half-line R + .Graphs in G are noncompact as soon as they have at least one unbounded edge or have infinitely many edges.Figure 1 shows a typical example of metric graph in G.For standard definitions of functional spaces on graphs we redirect e.g. to [13].
The interest in metric graphs has been growing through the decades to become today an active research field with an inter-sectoral popularity within the scientific community.In fact, major contributions to the development of the topic stem from the fact that it attracted attention both in applied sciences and in more theoretical ones.In a wide variety of applications, indeed, metric graphs may serve as simplified models for higher dimensional branched or ramified domains, i.e. structures where the transverse dimensions are negligible with respect to the longitudinal one.At the same time, in many cases the mathematical analysis of problems on graphs proves to be of interest per se, exhibiting elements of novelty compared to standard Euclidean settings.
In the context of differential models on metric graphs, both linear and nonlinear models have been largely investigated (see e.g.[12,29,30,34] and references therein for some recent results in the linear Example of a (noncompact) metric graph in G with 5 unbounded edges and 13 bounded edges, one of which forms a self-loop.
setting, and the reviews [2,32] for comprehensive overviews on nonlinear problems).Within the nonlinear theory, a significant attention has been focused on nonlinear dispersive equations, such as Schrödinger and Dirac equations, also in view of their possible application in the innovative high-tech research field of atomtronics (see [11] for a wide introduction to the subject).In particular, many efforts have been devoted to the search of bound states, i.e. solutions of the associated stationary equation (e.g.(1.1) for the nonlinear Schrödinger equation) coupled with various vertex conditions, unraveling a deep dependence of the problem on both topological and metric properties of graphs (see for instance [1, 3, 5, 6, 8-10, 14-19, 22-28, 31, 36, 38, 40, 41] for Schrödinger equations and [20,21] for Dirac equations).The present paper fits in the investigation of bound states for nonlinear Schrödinger equations on metric graphs focusing specifically on the problem that is (1.1) together with homogeneous Kirchhoff conditions at the vertices.Here, u ′ e (v) denotes the outgoing derivative of u at v along the edge e and e ≻ v means that the sum is extended to all edges incident at v.
The aim of this work is to prove existence and multiplicity of H 1 positive bound states of (1.2) concentrating at given vertices of G as the parameter λ goes to +∞.
Before stating our main results, we need to recall what is known about positive solutions of (1.2) on star graphs.Given any integer N ≥ 1, the N -star graph S N is the graph made up of a single vertex, identified with 0, and N half-lines attached to it.Clearly, S 1 = R + and S 2 = R.A function Ψ ∈ H 1 (S N ) can be seen as an N -tuple (ψ 1 , . . ., ψ N ), where ψ i ∈ H 1 (R + ) for every i and ψ i (0) = ψ j (0) for every i, j.Note that, by dilation invariance, for every µ > 0 and λ > 0 any solution of (1.2) on S N is given by λ namely (1.2) with λ = 1.On the real line (i.e.N = 2), it is well-known that the set of L 2 positive solutions of (1.3) is the family of solitons ϕ a (x) := ϕ(x − a), with ϕ(x) := (µ + 1) Similarly, on the half-line R + (that is N = 1), problem (1.3) has a unique L 2 positive solution given by the restriction of ϕ to R + , named the half-soliton.For N ≥ 3, the H 1 solutions to problem (1.3) have been completely classified e.g. in [4].Precisely, if N is odd, (1.3) has a unique positive solution Ψ N = (ψ 1 , . . ., ψ N ) given by where ϕ ∈ H 1 (R + ) is the half-soliton introduced in (1.4).When N is even, on the contrary, (1.3) admits infinitely many positive solutions Ψ a N = (ψ a 1 , . . ., ψ a N ) (with a ∈ R), described (after a possible permutation of indices) through ϕ a in (1.4) as (1.6) Let us note incidentally here that, for general metric graphs, a complete description of the set of solutions of (1.2) is usually out of reach.To the best of our knowledge, besides star graphs the unique case for which this is available is the T -graph (two half-lines and a bounded edge glued together at the same vertex), that has been discussed in the recent paper [10], whereas partial results in this direction have been given e.g. for the tadpole graph (a circle attached to an half-line) and the double-bridge graph (a circle attached to two half-lines at different points) in [35,37].
We are now in position to state the main results of our paper, that are given in the next two theorems.
The above results prove, for sufficiently large λ, existence of one-peaked (Theorem 1.1) and multipeaked (Theorem 1.2) positive bound states concentrating at vertices with odd degree and being negligible (as λ → +∞) on the rest of the graph.The proof of both theorems is based on a Ljapunov-Schmidt procedure using as model function the solution Ψ N of (1.2) on the star graph S N .Note that the notation in Theorem 1.2 is consistent with the fact that one can simultaneously identify all the v i 's with the origin along all edges incident at each of them.This is obvious if such vertices have no common edge, whereas if two of them share one edge this can still be done by considering an additional vertex of degree 2 at the middle point of the shared edge.Also the slightly different definition of ℓ v i with respect to Theorem 1.1 is meant to allow the presence of shared edges between the v i 's.
Note that Theorems 1.1-1.2do not apply when µ < 1 2 or the vertices have even degree.The limitation on µ is technical and unavoidable with our argument, since we need to compute a second order expansion of the function f (s) = s 2µ+1 at s = Ψ N .However, our theorems cover the cubic case µ = 1, which is usually considered the most relevant one in many physical applications.Conversely, it is not clear to us whether one can recover the results of Theorems 1.1-1.2concentrating at vertices with even degree.Heuristically, one may expect that this should be true.
Actually, the approach could be completely different.Let us focus on the simplest case of a graph with one edge and two vertices (i.e. an interval).It is clear that there exists a solution to (1.2) which concentrates at each vertex (i.e.boundary point) whose main profile is the function Ψ 1 .On the other hand it is also well known that there exists a solution to (1.2) which concentrates at a special point inside the edge (i.e. the middle point of the interval) whose main profile is the function Ψ a 1 (1.6) for a special value of the parameter a (see for example [39, Section 2.2] and the references therein).Now, we observe that each point inside the edge can be seen as a vertex of degree 2 and the previous result shows that only one of them is a peak of a concentrating solution.It would be extremely interesting to prove that a similar result holds true for a more general graph with a vertex of even degree.We believe that the difference between the odd and the even case arises in the choice of the ansatz: this is merely a cut-off of Ψ N around the vertex itself in the odd case, while in the even case it should possibly be a global refinement of Ψ a N in order to fit the Kirchhoff conditions on the whole graph (see [39,Remark 2.16]).However, at present the case of even degree is completely open.
The interest in peaked solutions of (1.2) on metric graphs is not new.In [6], positive bound states with a maximum in the interior of a given edge are identified as solutions of a doubly constrained minimization problem, for every µ ∈ (0, 2) and for sufficiently large masses (i.e. the L 2 norm of the function).In [14,33] a Dirichlet-to-Neumann map argument is developed in the cubic case µ = 1 to construct, again for large masses, solutions with maximum points either at vertices of degree 1 or in the interior of any edge.The results of these three papers apply both to compact graphs and to noncompact graphs with finitely many edges.Remarkably, both approaches are able to handle the mass constrained setting.Conversely, in [26] a Ljapunov-Schmidt procedure similar to the one discussed here is used to find solutions concentrating at vertices of degree 1 on compact graphs, for every µ > 0 and λ → +∞.Actually, Theorems 1.2-1.3 of [26] are the analogues of Theorems 1.1-1.2here in the case of vertices of degree 1 and the strategy of the proof is the same.However, we stress that our results here apply to any graph in G and not only to compact ones.Furthermore, from the technical point of view, the Ljapunov-Schmidt argument for vertices with degree greater than 1 is rather different and technically demanding.This is readily seen observing that linearizing (1.2) around Ψ N gives a linear problem that has only the trivial solution when N = 1, whereas it has nontrivial solutions as soon as N ≥ 2 (see Lemma 2.1 below).
With respect to the available literature on bound states of (1.2), the main novelty of Theorems 1.1-1.2 is that these are the first results exhibiting solutions with maximum points at vertices of degree greater than or equal to 3. To better understand why this is worth noting, we recall that almost all the existence results on general metric graphs derived so far are based on minimization arguments.In particular, major attention has been devoted to minimum problems both for the action functional constrained to the associated Nehari manifold and for the energy functional E : constrained to the space of functions with prescribed mass Thorough investigations have been developed for ground states, i.e. global minimizers of both problems (see e.g.[24,38] for the action and [5, 7-9, 27, 28] and references therein for the energy), but local minimizers have been investigated too (see e.g.[6,41]).However, none of these solutions coincides with those in Theorems 1.1-1.2.
(a) (b) Figure 2. Examples of infinite periodic graphs (A) and infinite trees (B) with vertices of odd degree greater than 1.
The proof of Corollary 1.3, which is given for the sake of completeness in Section 7, is actually a straightforward consequence of the fact that, by (1.7), (1.8), one can compute explicitly the action, the energy and the mass of u λ , and a direct comparison with the asymptotic behaviour of the ground state levels shows that u λ is always a solution with action/energy strictly larger than that of ground states.This is no surprise, as ground states are known to concentrate at vertices of degree 1 (when available) or in the interior of the edges (see e.g.[14,26]).
Hence, Theorems 1.1-1.2provide genuinely new existence and multiplicity results.As for the mass of these solutions, observe that, if u λ is as in (1.7), then whereas if u λ is as (1.8) In particular, these are bound states with diverging masses in the L 2 -subcritical regime µ < 2, with masses strictly greater than ∥ϕ∥ 2 L 2 (R) at the L 2 -critical power µ = 2, and with vanishing masses in the L 2 -supercritical regime µ > 2. This is of particular interest both for µ = 2, since in the critical regime it is usually difficult to find solutions with masses larger than ∥ϕ∥ 2 L 2 (R) (see e.g.[7,41]), and for µ > 2, as very few results are available at present in the L 2 -supercritical case.
To conclude, we further observe that the results of this paper apply to graphs with countably many edges.As so, Theorem 1.2 has evident consequences on the set of bound states of (1.2) on graphs of this type, such as infinite periodic graphs (Figure 2(A)) and infinite trees (Figure 2(B)).In particular, graphs like these admit at least countably many H 1 positive solutions of (1.2), each one with an arbitrary number of peaks located at any given subset of vertices with odd degree.Furthermore, in view of (1.10), considering a sufficiently large number of vertices one obtains bound states with arbitrarily large mass, and this is remarkably true independently of µ ≥ 1 2 .The remainder of the paper is organized as follows.Section 2 collects some preliminaries.Sections 3-4-5 provide the proof of Theorem 1.1, with the reduction to a finite dimensional problem (Section 3), its formulation in term of a reduced energy (Section 4) and the analysis of the critical points of such energy (Section 5).Finally, Section 6 discusses the proof of Theorem 1.2 and Section 7 that of Corollary 1.3.
Notation.In the following we will write f ≲ g or f = O(g) in place of |f | ≤ C|g| for some positive constant C independent of λ and f ∼ g in place of f = g + o(g), whenever possible.Furthermore, except for Section 6, we will always write Ψ in place of Ψ N .

Preliminaries
We begin by introducing the basic idea to construct the solutions to (1.2) we are looking for and by collecting some related preliminary results.
Given a vertex v of G with odd degree greater than or equal to 3, our aim is to find, for suitable values of the parameter λ, solutions u λ to (1.2) in the form where W λ concentrates at v and Φ is a smaller order term as λ → +∞.To do this, we first identify a good candidate for the principal part W λ and then derive the correction term Φ with a Ljapunov-Schmidt procedure.
To define W λ , we start with the unique symmetric solution Ψ ∈ H 1 (S N ) to (1.3) on the N -star graph S N , i.e.Ψ = (ϕ, . . ., ϕ), where ϕ ∈ H 1 (R + ) is the half-soliton on R + , and we characterize the set of solutions of the linearization of (1.3) at Ψ. Lemma 2.1.For every N ≥ 2, the set of solutions to is given by the (N − 1)-dimensional space where, for every j = 1, ..., N − 1, and . .
, with e j 1 + ... + e j N = 0 ∀j and e j • e k = 0 if j ̸ = k. (2.4) Proof.Since Ψ = (ϕ, . . ., ϕ), we plainly see that on each half-line of S N , the function ϕ ′ solves the differential equation in (2.2).Hence, by the standard theory of linear ordinary differential equations, the general solution of the first line of (2.2) on the i-th half-line is of the form ) and continuity at the vertex is guaranteed for every values of the c i 's because ϕ ′ (0) = 0. Therefore Z solves problem (2.2) if and only if the homogeneous Kirchhoff condition is satisfied, namely if and only if As ϕ ′′ (0) ̸ = 0, this yields c 1 + ... + c N = 0. Since the equation c 1 + • • • + c N = 0 identifies an (N − 1)dimensional subspace of R N a basis of which is given e.g. by vectors e j as in (2.4), this concludes the proof.□ For the sake of convenience, in the following we will choose the vectors e j , j = 1, ..., N − 1, satisfying (2.4) to be We are now in position to define W λ .Here we denote by B(a, r) the ball of radius r in G centered at the point a.Set N := deg(v), ℓ := min e≻v |e|/2 and let χ : G → [0, 1] be a smooth cut-off function such that χ ≡ 1 on B(v, ℓ) and χ ≡ 0 on G \ B(v, 2ℓ).We then define W λ : G → R as with Z (j) the functions in Lemma 2.1, and the real numbers b j,λ given by for suitably chosen b j ∈ R and α > 0. Here, even though in principle b j may still depend on λ, we decide not to denote explicitly such dependence since, as will be clear in the next sections, the actual value of the b j 's we will consider remains always uniformly bounded in λ.
With this definition, to prove Theorem 1.1 we need to find Φ and b j , j = 1, . . ., N − 1, so that u λ as in (2.1) solves (1.2).Note that, setting f (u) := (u + ) 2µ+1 for every u ∈ H 1 (G), it is clear that positive solutions of (1.2) coincide with solutions of Since we will consider the limit λ → +∞, with no loss of generality we can assume from the beginning λ > −λ G , where denotes the bottom of the spectrum of −d 2 /dx 2 (coupled with homogeneous Kirchhoff conditions) on G. Hence, for every such λ we equip the space H 1 (G) with the following equivalent scalar product and denote by ∥ • ∥ λ the corresponding norm.Note that, considering the immersion Therefore, problem (2.7) can be rewritten as and to find u λ as in (2.1) solving (1.2) amounts to find Φ, b j , j = 1, . . ., N − 1, such that u λ satisfies (2.8).Actually, we will further rewrite (2.8) as follows.For every λ, we introduce the linear operator the nonlinear operator N : and the error term Accordingly, (2.8) is equivalent to where of course one still needs to identify both Φ and the coefficients b j 's.
Remark 2.2.Note that, if G is a noncompact graph, i * λ is not compact.However, since by definition W λ is compactly supported in a fixed ball centered at v, the operator from ) is compact for every given λ.As a consequence, the operator L in (2.9) is a compact perturbation of the identity for every λ.
Remark 2.3.Since we will need them in the following, we conclude this section collecting here the following elementary inequalities ) and (2.15)

The finite-dimensional reduction
In this and in the next two sections we solve problem (2.12).Here we start by showing that, once the value of the b i 's is fixed, it is possible to find a unique Φ fulfilling a slightly modified version of (2.12).This will reduce the problem to identify the b j 's to solve (2.12), a task that will be accomplished in the following two sections.
To prove Proposition 3.1, we need the following preliminary lemma.
Lemma 3.2.For every compact subset C of R N −1 , there exists λ 0 > 0 (depending on C) such that, for every (b 1 , . . ., b N −1 ) ∈ C and every λ > λ 0 , the linear operator Proof.Note first that to prove the claim it is enough to show that there exist c > 0 and λ 0 > 0 such that, for every λ > λ 0 and every v ∈ K ⊥ λ , there holds ) which is a compact perturbation of the identity as pointed out in Remark 2.2.
To prove (3.3), we argue similarly to [26,Lemma 3.1].Suppose for contradiction that (3.3) is false, namely that there exist sequences λ n → +∞ and v n ∈ K ⊥ λn such that, as n → ∞, We then write and ∥h n ∥ λn → 0 by assumption.By definition of i * λn , on every edge of G this reads together with homogeneous Kirchhoff conditions at every vertex of G.The rest of the proof is divided in two steps.
Step 1.We claim that ∥k n ∥ λn → 0. Indeed, for some numbers c i,n , For fixed j = 1, . . ., N − 1, we multiply (3.5) by χZ λn and, recalling that v n − h n ∈ K ⊥ λn , we obtain λn χZ As n → ∞, a direct computation shows that for some constant a > 0. Note that in the previous computation we tacitly interpreted the term χ Z (j) ′ as defined both on a compact subset of G and on a compact subset of S N .This is clearly unambiguous, since the support of the cut-off function χ is a finite symmetric star graph with N edges centered at v (recall that here N = deg(v)).The identification of such subsets of G and S N will be frequently used also in the rest of the proof.Arguing as in (3.8) one easily sees that the second term in (3.7) has the same asymptotic behavior, while the last two terms are o λ . Therefore, Let us now focus on A n .Since each Z (j) solves (2.2), the functions Z (j) on every edge of S N , coupled with homogenous Kirchhoff conditions at the vertex.Multiplying by χv n , thinking of the resulting equation as defined on G (due to the cut-off χ) and integrating on G yields λn dx.
Notice that the first integral in the right hand side vanishes since v n ∈ K ⊥ λn , while , where ϕ is the soliton on R as in (1.4).Since ϕ ′ and ϕ ′′ decay exponentially as x → +∞, there exists Hence, recalling the definition of A n and (3.4), + O e −β √ λn . (3.10) Neglecting for a while the last term and recalling that f (s) = (s + ) 2µ+1 we see that the last inequality using also µ ≥ 1 2 .Coupling with (3.10) and (3.11) entails as n → +∞ and combining with (3.6) and (3.9) we obtain Finally, with the same argument used to compute B n , as n → +∞ since α > 0.
Step 2. We now go back to equation (3.5) and we multiply it by v n , obtaining as n → +∞ λn , and ⟨h n , v n ⟩ λn → 0 by (3.4).In view of this, if we prove that a contradiction arises and the proof is completed.To this aim, we set Note that ṽn , hn , kn are defined on the scaled graph G n := √ λ n G and direct computations show that as n → +∞ by (3.4), (3.12).Since, by construction, one can identify G n ∩ B(v, 2ℓ √ λ n ) with the compact subset of S N given by S N ∩ B(0, 2ℓ √ λ n ), combining (3.5) with the previous formulas shows that, for every compactly supported φ ∈ H 1 (S N ), there exists n large enough so that where with a slight abuse of notation we are thinking of ṽn , hn , kn as functions on S N ∩ B(0, 2ℓ √ λ n ).Hence, ṽn converges weakly in H 1 and strongly in L q , for every q ≥ 2, on compact subsets of S N to a function v 0 .Arguing as in [26,Lemma 3.1], it is easy to see that v 0 ∈ H 1 (S N ) and, letting n → +∞ in the previous formula, that namely that v 0 ∈ K, where K is as in Lemma 2.1.However, since arguing as above and recalling that v n ∈ K ⊥ λn one also has λn ⟩ λn = 0 ∀j = 1, . . ., N − 1 , i.e. v 0 ∈ K ⊥ , it follows that v 0 ≡ 0 on S N .As a consequence, when n → +∞ which together with the second line of (3.14) implies (3.13) and concludes the proof.□ Proof of Proposition 3.1.We prove the claim with a suitable fixed point argument.The proof is divided in two steps.
Step 1: estimates on E .We begin by showing that as λ → +∞ Indeed, by (2.11) and the definition of W λ and i * λ (f ′ (W λ )), it follows where and Multiplying (3.16) by E , integrating over G and using the Hölder inequality we have Now, adapting the argument in the proof of [26, Proposition 3.2] one easily sees that As for E 2 , by (2.14) one has Step 2: the contraction mapping argument.We consider the operator which is well-defined by Lemma 3.2, and the set for a suitable constant c > 0 to be chosen later.Note that, if we prove that T has a unique fixed point Φ in B λ , then Φ satisfies (3.2) and and the proof is completed.
To this aim, we chose c so that, for λ large enough, T is a contraction on B λ .Observe first that, by Lemma 3.2 there exists c 1 > 0 such that for every v, v 1 , v 2 ∈ B λ .Recalling (2.10), we have Now, recalling that f (s) = (s + ) 2µ+1 and µ ≥ 1 2 , the mean value theorem guarantees the existence of functions θ, θ : , whereas by the L ∞ -Gagliardo-Nirenberg inequality of G (see e.g.[9, Section 2]) and plugging into the previous formula yields which coupled with (3.22) gives Combining with the second line of (3.21) shows that there exists a constant c 2 > 0 such that for every λ large enough it holds T is a contraction on B λ since α > 0 by assumption.Furthermore, by (3.21), (3.15) and the previous estimate with v 1 = v, v 2 = 0 we obtain for a suitable constant c 3 .Hence, for sufficiently large λ it is enough to choose e.g.c = 2c 1 c 3 to obtain that T maps B λ into itself, thus concluding the proof.□

The finite-dimensional problem
In view of Proposition 3.1, to complete the proof of Theorem 1.3 it is enough to solve the finitedimensional problem in the coefficients b j , j = 1, . . ., N − 1.This is done by finding b 1 , ..., b N −1 such that the numbers c i 's in (3.1) are zero and so the function W λ + Φ is a genuine solution of problem (2.12).In this section we start doing this by proving a result that relates the b j 's we are looking for with the critical points of a suitable function on R N −1 .To this end, we introduce G : R N −1 → R defined by where the vectors e ℓ are defined in (2.5).This function, which one usually refers to as the reduced energy, plays a crucial role in our discussion, as highlighted by the next result.
for every λ large enough and for a suitable constant a > 0, since by the same computations in (3.7), (3.8) and χZ as λ → +∞.Hence, system (4.2) is diagonal in the c j 's and, to prove that it admits only the trivial solution c j = 0 for every j, it is enough to find suitable values of λ and of the b j 's for which the left hand side of (4.2) is equal to zero.
To this end, we show that, for sufficiently large λ, there exist b λ 1 , . . ., b λ N −1 as in the statement of the proposition and making the left hand side of (4.2) vanish by proving that where Observe that this is enough to conclude, since for large λ we can interpret the right hand side of (4.3) as −Aλ where E 1 , E 2 as in (3.17),(3.18)(the previous identity follows by (3.16)).We have where ϕ is the soliton in (1.4), and since both ϕ, ϕ ′ decay exponentially as x → +∞ this implies that for some σ > 0. As for the second integral on the right hand side of (4.4), by (3.18) we write Since f (s) = (s + ) 2µ+1 , by (2.15) and the definition of b i we obtain where the finiteness of the integrals appearing in last line is guaranteed by the properties of ϕ and the last equality makes use of 2µ + 1 ≥ 2. Combining with (4.6) then yields which coupled with (4.4), (4.5) entails ⟨E , χZ for every λ large enough.

5.
The reduced energy and the end of the proof of Theorem 1.1 This section characterizes the critical points of the reduced energy G introduced in (4.1).We observe that the whole analysis developed so far is insensitive of the degree N of the vertex v, which for the results of Sections 3-4 to hold needs to be just greater than or equal to 2. Conversely, the result of this section is the only point in our work where we need to impose N odd, as the next lemma clearly shows. .
Proof.We first note, for future reference, that from the definition (2.5) of the vectors e i we have (5.1) In view of this, we can write G as Since the right hand sides do not depend on k, this entails that x 2 i = x 2 k for every i, k, so that all critical points have the form (x 1 , . . ., x N −1 ) = (σ 1 t, . . ., σ N −1 t), with σ j ∈ {+1, −1} for every j = 1, . . ., N − 1, for some t ∈ R. Now, if n denotes the number of negative σ j 's in (σ 1 t, . . ., σ N −1 t), then by (5.2) As N is odd, the coefficient of t 2 never vanishes, so that ∇G(x) = 0 if and only if x = 0, and the same holds of course for ∇G.However, x = 0 is degenerate, since the Hessian matrix of G at 0 is the null matrix.To compute the local degree of 0 we therefore perturb G by defining, for fixed small ε > 0, G ε (x 1 , . . ., x N −1 ) = G(x 1 , . . ., x N −1 ) − ε 2 N −1 j=1 x j .Now, as above, we see that if x is a critical point for G ε , then + ε 2 ∀k = 1, . . ., N − 1. (5.4) Since, again, the right hand sides do not depend on k, we find once more that if x is critical for G ε , then (x 1 , . . ., x N −1 ) = (σ 1 t, . . ., σ N −1 t), with σ j ∈ {+1, −1} for every j = 1, . . ., N − 1, for some t ∈ R. Denoting again by n the number of negative σ j 's, we see from (5.4) that Now it is immediate to check that the coefficient of t 2 is negative if and only if n = N −1 2 , in which case the preceding equation reduces to t 2 = ε 2 , yielding t = ±ε.