GLOBAL BIFURCATION FOR THE H´ENON PROBLEM

. We prove the existence of nonradial solutions for the H´enon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent α . For sign-changing solutions, the case α = 0 -Lane-Emden equation- is included. The obtained solutions form global continua which branch oﬀ from the curve of radial solutions p (cid:55)→ u p , and the number of branching points increases with both the number of nodal zones and the ex- ponent α . The proof technique relies on the index of ﬁxed points in cones and provides information on the symmetry properties of the bifurcating solutions and the possible intersection and/or overlapping between diﬀerent branches, thus allowing to separate them in some cases.

Some of the results we present here are new also for the latter, and since our techniques allow to deal with both problems simultaneously we shall include the case α = 0 in the reasoning. It is well known that, for α > 0 fixed, the Hénon problem (1.1) admits solutions, and in particular radial solutions, for every p ∈ (1, p α ), being The same holds when α = 0, i.e. for the Lane-Emden equation (1.2), and in this case the threshold exponent for the existence of solutions coincides with the critical the positive solutions, the first eigenvalue alone plays a role and this ensures that the kernel of the linearized operator contains exactly a one-dimensional subspace of the O(N −1)-invariant functions, and this observation was crucial in [3,17]. For nodal solutions, instead, the structure of the kernel is highly nontrivial. We handle this situation by turning to the notion of degree and index of fixed points in cones introduced by Dancer in [11]. It can be applied to the Hénon equation because the exact computations in aforementioned papers rely on a characterization of the Morse index in terms of a singular Sturm-Liouville problem from [7], which allows to describe in full details the kernel of the linearized operator. Furthermore this tool provides a detailed bifurcation analysis also for positive solutions, and in the subcritical case, since it gives informations about the symmetries of the bifurcating solutions and the global properties of the branches. This paper is organized as follows. In Section 2 we outline the positive cones that we will use and the main bifurcation results that we are going to prove. Section 3 deals with the Morse index: after recalling its characterization by means of the singular eigenvalues and the exact computations performed in the aforementioned papers, we check that the Morse changes across the range p ∈ (1, p α ). Next in Section 4 the main results are proved, by taking advantage of the previous discussion on the Morse index and adapting that arguments to compute the index of fixed points in cones.
Notice that radial functions belong to K n for every n. On the other side, only in dimension N = 2 the intersection between two different cones reduces to the radial functions alone. Instead in dimension N ≥ 3 it contains also nonradial functions that do not depend on the angle θ. Throughout the paper we will take the exponent α as fixed and write S m for the curve of radial solutions to (1.1) with m nodal zones, precisely with m nodal zones and u p (0) > 0 .
We will show that a continuum of nonradial solutions in K n detaches from the curve S m , for some integers n depending on the exponent α and the number of nodal zones m. To this aim we introduce the set where the closure is meant according to the natural norm in (1, Remark that the set Σ m n contains also the curves of radial functions S m with m = m, but of course S m and S m are separated. So we say that a couple (p n , u pn ) ∈ S m ∩ Σ m n is a nonradial bifurcation point, meaning that in every neighborhood of (p n , u pn ) in the product space (1, p α ) × C 1,γ 0 (B) there exists a couple (q, v) such that v is a nonradial solution of (1.1) related to the exponent q. In this case we set C m n the closed connected component of Σ m n containing (p n , u pn ) (2.6) and we shall refer it as the "branch"departing from (p n , u pn ), with a little misuse of language. We will also write [t] and t , respectively, for the floor and the ceiling of a real number t, i.e.
Eventually the same reasoning enables us to prove several bifurcation results. First we produce α 2 global branches of positive nonradial solutions, precisely Theorem 2.1 (Bifurcation from positive solutions). In any dimension N ≥ 2 and for every α > 0, there are at least α 2 different points along the curve S 1 where a nonradial bifurcation occurs. More precisely for every n = 1, . . . , α 2 there exists a nonradial bifurcation point (p n , u pn ) ∈ S 1 ∩ Σ 1 n and the respective branch C 1 n has the following global properties i) C 1 n is made up of positive solutions and unbounded, i.e. it contains a sequence (p k , u k ) with u k C 1,γ → ∞ or p k → p α .
ii) In dimension N = 2 the branches are separated, in the sense that their intersection contains at most isolated points along the curve of positive radial solutions S 1 .
iii) In dimension N ≥ 3 two different branches can only have in common couples (p, v), where v are positive solutions to (1.1) which do not depend on the angle θ, and their overlapping can even make up a continuum.
In the disc solutions enjoying the same symmetry properties have been produced in [16] by the Lyapunov-Schmidt reduction method, and in [1] by minimizing the energy associated to (1.1) in the space H 1 0,n . In this last paper it has been proved that such "least energy n-invariant solutions"are nonradial and different one from another at least for p ∈ (p n , +∞), with p n the same exponent appearing here. On the other hand, they are certainly radial for p close to one, thanks to the uniqueness result in [3]. It is therefore natural to think that the branches of bifurcating solutions shown by Theorem 2.1 are made up by these least energy n-invariant solutions, and so they do exist for every p ∈ (p n , ∞), and are separated.
In higher dimension Theorem 2.1 improves the bifurcation result obtained in [3], which holds for α ∈ (0, 1] and produces only one branch of nonradial solutions. Nonradial solutions with similar symmetries have been produced by the finite-dimensional reduction method: in particular [23] concerns the slightly subcritical case and exhibits solutions which blow up when p approaches the critical Sobolev exponent. Besides nonradial solutions do exist also for p close to p α , as showed in [17]. It is very likely that some of the nonradial solutions found in Theorem 2.1 coincide with the ones in [17], where the specular viewpoint (bifurcation w.r.t. α) is adopted.
ii) Every branch contains a sequence (p k , u k ) with either u k C 1,γ → ∞, or p k → ∞, or possibly p k → 1 and u k converges to an eigenfunction of −∆ω = µ|x| α ω in B, ω = 0 on ∂B, (2.7) which belongs to K n . iii) Two different branches can only have radial solutions in common. Precisely C 2 n ∩ C 2 n ∩ S 2 contains at most isolated points, and if there is some m ≥ 3 such that C 2 n ∩ C 2 n ∩ S m is nonempty, then S m ⊂ C 2 n ∩ C 2 n . The possibility that p k → 1 but u k stays bounded remains open because the uniqueness of nodal solutions does not hold either in a neighborhood of p = 1, see [2,Theorem 1.3]. Concerning property iii), i.e. the possible overlapping of two different branches, we are not aware of any technique which enables to capture the formation of further nodal zones and/or a secondary bifurcation. Consequently a nonradial branch could, in principle, touch another radial curve S m with m ≥ 3, and then incorporate it because of the way in which Σ 2 n and C 2 n have been defined. Theorem 2.2 applies also to α = 0, i.e. to the Lane-Emden equation, giving back [19,Theorem 1.2] since in this particular case 2+α 2 β + 1 = 3 and 2+α 2 κ − 1 = 5. For α > 0 it is worth comparing this existence result with the ones in [2] and in [1], both concerning the least energy n-invariant nodal solutions, that we denote hereafter by U p,n . For n = 1, . . . , 2+α 2 β − 1 , U p,n is nonradial for both p close to 1 and large. It seems that in this case U p,n is nonradial for every p > 1 and the curve p → U p,n does not intersect the curve of radial solutions. This is certainly true for n = 1, i.e. the least energy nodal solution. Conversely for n = 2+α 2 β + 1 , . . . , 2+α 2 κ − 1 , [2, Proposition 4.10] and [1, Theorem 1.6] yield that U p,n are radial for p close to 1, and then nonradial (and different one from another) when p is large. Therefore the curves p → U p,n coincide with the one of radial solutions for p ∈ (1, p n ), and then they give rise to the nonradial bifurcation stated by Theorem 2.2.
Only bifurcation from the curve S 2 is taken into account, since the behaviour of nodal solutions as p → ∞ is known only in the case of two nodal zones. When this paper was already finished we came to know that a very recent preprint by Ianni and Saldana [20] describes the asymptotic profile of every radial solutions. Starting from this it is possible, in principle, to compute exactly their Morse index and then the same arguments used here produce bifurcation also in the general case.
nonradial bifurcations take place along the curve S m . More precisely for every n = 2 + α 2 , . . . , n m α there exists a nonradial bifurcation point (p n , u pn ) ∈ S m ∩ Σ m n and the respective branches C m n have the } is made up of nonradial solutions with m nodal zones, one of which contains x = 0 and is homeomorphic to a ball, while the other ones are homeomorphic to spherical shells.
ii) Every branch contains a sequence (p k , u k ) with either u k C 1,γ → ∞, or p k → p α , or possibly p k → 1 and u k converges to an eigenfunction of (2.7) which belongs to K n .
iii) The intersection between two different branches, if non-empty, is made up of nodal solutions which do not depend by the angle θ.
The branches of nodal bifurcating solution in dimension N ≥ 3 can overlap along radial solutions with a different number of nodal zones, but also along nonradial solutions that do not depend by the angle θ.
The statement of Theorem 2.3 is new also in the simpler case α = 0, to the author's knowledge. For the reader's convenience, we state separately the bifurcation result concerning the Lane-Emden equation.

Theorem 2.4 (Bifurcation for the Lane-Emden equation in dimension
For every m ≥ 2 the curve S m bifurcates at 2m − 3 points, at least. More precisely for every n = 2, . . . , n m 0 ≥ 2(m − 1) there exists a nonradial bifurcation point (p n , u pn ) ∈ S m ∩ Σ m n and the continuum detaching at (p n , u pn ), i.e. C m n has the following } is made up of nonradial solutions with m nodal zones, one of which contains x = 0 and is homeomorphic to a ball, while the other ones are homeomorphic to spherical shells, • Global property: every branch contains a sequence (p k , u k ) with either u k C 1,γ → ∞, or p k → p 0 , or possibly p k → 1 and u k converges to an eigenfunction of

8)
which belongs to K n .
• Separation property: the intersection between two different branches, if nonempty, is made up of nodal solutions which do not depend by the angle θ.
There is numerical evidence that n m 0 = 2(m − 1) in any dimension N ≥ 3, so that Theorem 2.4 provides exactly 2m − 3 branches of nonradial solutions. In particular, in the case of 2 nodal zones, there should be only one branch in dimension N ≥ 3, while 3 different branches have been produced in dimension N = 2. The planar case indeed differs from the other ones, as already observed in several occasions.
Let us mention in passing that the number of nonradial branches produced in Theorems 2.1, 2.2 and 2.3 goes to infinity when α → ∞, which is consistent with the specular study (bifurcation w.r.t. α) performed in [21].
3. Preliminaries on the computation of the Morse index. To emphasize the dependence on the exponent p ∈ (1, p α ), we take the exponent α ≥ 0 and the number of nodal zones m as fixed and denote by u p the unique radial solution to (1.1) with m nodal zones which is positive at the origin. We also write for the linearized operator at u p and the related quadratic form, respectively. They will be considered on the space H 1 0 (B), or in one of its subspaces specified case-bycase.
The Morse index, that we denote hereafter by m(u p ), is the maximal dimension of a subspace of H 1 0 (B) in which the quadratic form Q p is negative defined, or equivalently the number of the negative eigenvalues of For radial solutions one can also look at the radial Morse index, denoted by m rad (u p ), i.e. the number of the negative eigenvalues of for (3.3) whose relative eigenfunction is H 1 0,rad (B), the subspace of H 1 0 (B) given by radial functions. As explained in full details in [7], this matter can be regarded through a singular eigenvalue problem associated to the linearized operator L p , which has to be handled in weighted Lebesgue and Sobolev spaces The Morse index (on H 1 0 (B) as well as on some of its subspaces) turns out to be equal to the number of the negative eigenvalues of Concerning radial solutions to the Hénon problem, it turns helpful the transformation t = r which map radial solutions to (1.1) into solutions of one-dimensional problems or, respectively see [7,6]. In both cases M is a real parameter given by In this functional setting it is possible to look at a singular eigenvalue problem associated to (3.7), or equivalently (3.8), that is where and see that, when the infimum stays below the threshold (M − 2)/2, then it is attained by a function φ which solves (3.10) in the weak sense, namely for every test function ϕ ∈ H 0,M . Such function φ can therefore be called an eigenfunction related to the eigenvalue ν 1 , and denoted by φ 1 . Iteratively, if ν i < (M − 2)/2, one can settle the minimization problem Again, as far as ν i+1 < (M − 2)/2, it is attained by an eigenfunction φ i+1 which solves (3.10) in the weak sense. Next, the eigenfunctions related to these singular eigenvalues enjoy the same properties of the standard ones, in particular they are simple, mutually orthogonal, and the i th eigenfunction has exactly i nodal domains.  Then the only nonnegative eigenvalues of (3.10) are ν 1 (p) < ν 2 (p) < · · · < ν m (p) < 0 and satisfy Moreover the Morse index of u p is given by

17)
where s = {min n ∈ Z : n ≥ s} denotes the ceiling function and the multiplicity of the eigenvalue λ j = j(N + j − 2) for the Laplace-Beltrami operator in the sphere S N−1 .
Afterward, the Morse index has been exactly computed at the ends of the existence range by computing the limits of the eigenvalues ν i (p). The paper [2] dealt with p close to 1, and we need to introduce some more notation to recall the obtained result. For every β ≥ 0 we write J β for the Bessel function of first kind , r ≥ 0, 4806 ANNA LISA AMADORI and z i (β) for the sequence of its zeros on the interval (0, ∞). Since the map β → z i (β) is continuous and increasing, for every fixed integer m there exist a sequence ).
It is clear that After there existsp =p(α) > 1 such that for p ∈ (1,p) the Morse index of u p is given by
It is worth remarking that in dimension N = 2 the parameters β i defined in (3.18) only depend by the total number of nodal zones m, and not by α. In the case of two nodal zones one sees by numerical computation that β := β 1 (α, 2, 2) ≈ 2,305, if α = α n = 2(n + 1)/β − 2, or if α = α n for some for some integer n.
The situation at the supremum of the existence range changes drastically depending if the dimension is N = 2 or greater. In dimension N = 2 only the Morse index of the least energy radial solution (i.e. the positive one) and of the least energy nodal radial solution (i.e. the one with two nodal zones) are known. They have both been computed in the paper [1], where it is shown that  Moreover there exists p > 1 such that for p > p the Morse index of u p is given by when α = α n = 2(n/κ − 1), while when α = α n it holds An analogous result for the radial solution to the Lane-Emden problem with two nodal zones has been obtained in [15]. The number κ appearing in (3.27) is linked, through the formula κ 2 = 2+ 2 2 , to the constant used in that paper. The Morse index in dimension N ≥ 3 is computed in [5] extending some previous results on the Lane-Emden problem in [14]; precisely [5, Propositions 3.3, 3.10 and Theorem 1] state that  After there exists p = p (α) ∈ (1, p α ) such that the Morse index of u p is given by for p ∈ (p , p α ).
Comparing (3.20) with (3.26) or (3.31) one sees that the positive radial solution has Morse index 1 when p is close to 1, and greater than 1 when p is at the opposite end of the existence range (for α > 0).
It is not hard to see that in dimension N = 2 the solution with 2 nodal zones shares the same behaviour, for every α ≥ 0. Indeed in this case formulas (3.23) and (3.24) yield m(u p ) ≤ (2 + α)β + 2 for p ∈ (1,p) for every α ≥ 0, since t < t + 1. Therefore taking q > p we deduce from (3.28), (3.29) that and since clearly t ≥ t we have thanks to the approximated values of β and κ given in (3.22) and (3.27), respectively.
In higher dimensions, the approximation of the parameters β i appearing in (3.20) can be numerically performed after having chosen a specific value for α, which fixes the baseline Bessel function J N −2

2+α
. To have an overall picture it can be useful to establish some estimate. We report here the elementary proof of an estimate of the Bessel zeros that contributes to this aim. Lemma 3.6. For all β ≥ 0 and i, m integers with i < m we have (3.32) Proof. It is known that the m th zero of J β lies in the m th nodal set of J β+1 , i.e.
z m−1 (β + 1) < z m (β) < z m (β + 1), (3.33) which also implies (−1) m J β+1 (z m (β)) < 0. On the other hand Actually the first inequality is obtained by iterating (3.33) and the second one follows since the the map β → z m (β) is increasing. Hence the m th zero of J β can only belong to the (m − 1) th or to the m th nodal set of J β+2 . But by the recurrence relation and therefore also (−1) m J β+2 (z m (β)) < 0, which means that the m th zero of J β lies in the m th nodal set of J β+2 . In particular z m−1 (β + 2) < z m (β). and there isp =p(α) > 1 such that the Morse index of u p is estimated from below by for p ∈ (1,p).
We therefore see that, in dimension N ≥ 3, the Morse index of nodal radial solutions for p close to p α is smaller than the one for p is close to 1.
Corollary 3.8. In dimension N ≥ 3, for every value of α ≥ 0 and m ≥ 2 there exist 1 <p <q < p α such that we have m(u p ) > m(u q ) as 1 < p <p andq < q < p α .
Proof. By (3.31) we know that for every α ≥ 0 as long as q is near p α . So, thanks to the estimate (3.39), the claim follows after checking that (3.40) (3.40) can be proved by induction on the number of nodal zones m ≥ 2, taking advantage from the fact that in dimension N ≥ 3 the multiplicity N j increases with j, i.e. N j+1 > N j as j ≥ 1. (3.41) We first check and this concludes the proof.
In the next section we will see that the changes in the Morse index caused by the first singular eigenvalue ν 1 (p) play a crucial role in establishing bifurcation results. Therefore the parameter β 1 defined in (3.18) deserves a special attention, in particular the integer number n m α := which is characterized by the double inequality Once that the dimension N , the exponent α and the number of nodal zones m have been fixed, the number n m α can be easily computed by using iteratively the function Besselzero in MathLab, for instance. Besides it is already known by (3.36) that For α = 0 (Lane-Emden equation) there is numerical evidence that for every N so that n m 0 = 2(m − 1) indeed.

Global bifurcation.
Here we prove the bifurcation results stated in Section 2. It is well known that if (p, u p ) is a bifurcation point in the curve S m , then the solution u p has to be degenerate, which means that the linearized operator L p defined in (3.1) has nontrivial kernel in H 1 0 (B), or equivalently Λ = 0 is an eigenvalue for (3.3). In Section 3 we have noticed that the Morse index changes within the interval (1, p α ), so that degeneracy values do exist. Besides we can not rely on any variational structure, since we aim to include also supercritical values of p, and bifurcation can be obtained only through an odd change of the Morse index. Hence a better knowledge of the kernel of L p is needed. By [6, Theorem 1.3] the radial solutions are radially nondegenerate, i.e. the kernel of L p does not contain radial functions. Moreover, the degeneracy has been characterized in [7,Proposition 1.5] in terms of the eigenvalues ν i (p) showing that Besides any function in the kernel of L p can be written according to the decomposition formula where ψ i,p is an eigenfunction for (3.10) related to an eigenvalue ν i (p) satisfying (4.1), and Y j stands for an eigenfunction of the eigenvalue j(N − 2 + j) of the Laplace-Beltrami operator.
For a positive solution only the first eigenvalue ν 1 (p) plays a role and one can manage to obtain an odd change in the Morse index by restricting the attention to the subspace of O(N − 1)-invariant functions, as in [3,17]. For a nodal solution, instead, the equality (4.1) can hold for different values of i and j and (4.2) brings out that the kernel of L p has a complex structure. This difficulty can be dealt with by turning to the notion of degree and index of fixed points in the positive cones introduced in Section 2.
Letting T be the operator Denoting by T u (p, ·) the Fréchet derivative of T (p, ·) computed at u, we say that u is an isolated fixed point for T (p, ·) w.r.t. X n when I − T u (p, ·) is invertible in X n , which is assured by the nondegeneracy of u. Starting from the characterization of degeneracy in Proposition 4.1 one can see that radial solutions u p are isolated fixed points, except at most a discrete set of p. It follows from a general regularity result. One can now compute the index of u p relative to the cone K n , see [11], which will be denote by index Kn (p, u p ). It is important to note that, also in the case of nodal solutions, the first singular eigenvalue determines by itself such index.
Lemma 4.4. Let p be such that u p is nondegenerate. Then Here the symbol deg Xn (I − T (p, )) stands for the Leray-Schauder degree of the operator I − T (p, ) restricted at X n , computed in a neighborhood of (p, u p ) which does not contain nonradial solutions (this choice is possible since u p is nondegenerate by assumption).
In this way, the prove reduces to show that the so-called property α holds if and only ν 1 (p) + ( 2 2+α ) 2 n(N − 2 + n) < 0. Several characterizations of the property α are provided in Lemma 3 and the following Remark in [11]. To state the one which will be used here we need the sets W + := {v ∈ X n u p + γv ∈ K n for some γ > 0}, W 0 := {v ∈ W + up : −v ∈ W + up }, V the orthogonal (in the H 1 0 sense) complement to W 0 in X n . Notice that the functions in W 0 do not depend by the angle θ. Next, T has the property α if there exists t ∈ (0, 1) such that the problem has a solution. We follow the proof of [12,Theorem 1] and look at the family of eigenvalue problems and let Λ t be its first eigenvalue. When t = 0 (4.4) reduces to an eigenvalue problem for the Laplacian and certainly Λ 0 > 0. When t = 1, instead, (4.4) gives back the eigenvalue problem (3.3), but only eigenfunctions in V matter. Furthermore the variational characterization yields that the first eigenvalue Λ t is strictly decreasing w.r.t. t. If T has the property α, then Λ t ≤ 0 for some t < 1 and therefore Λ 1 < 0. This in turn means that the eigenvalue problem (3.3) has a negative eigenvalue with related eigenfunction in V and then ν i (p) + 2 2+α 2 j(N − 2 + j) < 0 for some i = 1, . . . , m and j such that the related spherical harmonic belongs to V , by the characterization in [7,Theorem 1.4]. Taking advantage from the description of the spherical harmonics given in the proof of Theorem 1.1 in [4], one sees that j must be a multiple of n and so, in particular, On the other hand if ν 1 (p) + ( 2 2+α ) 2 n(N − 2 + n) < 0 we let ψ be the first radial eigenfunction for (3.10) and Y n the spherical harmonic related to n(N−2+n) belonging V (which does exist for what we have said before). Now v(r,θ,φ) = ψ(r 2+α 2 )Y n (θ,φ) is in V and an easy computation shows that using the change of variable t = r 2+α 2 and the notation in (3.11) and as ψ solves (3.10) and Y n is a spherical harmonics we get Hence the first eigenvalue Λ 1 is negative, and since Λ 0 > 0 there exists t ∈ (0, 1) such that Λ t = 0, which means that T has the property α.
Relying on Lemma 4.4 one can see that a sufficient condition for bifurcation is lim p→1 ν 1 (p)+ 2 2+α for some integer n.
Proposition 4.5. If n is an integer which fulfills (4.5), then there exists at least one p n ∈ (1, p α ) such that (p n , u pn ) is a nonradial bifurcation point and the branch C n defined according to (2.6) is global, in the sense that it contains a sequence (p k , u k ) with i) either u k C 1,γ (B) → +∞, ii) or p k → p α , iii) or p k → 1.
So Proposition 4.5 yields that, for any of such values of n, there exist a nonradial bifurcating point (p n , u 2 pn ) and a bifurcating branch in K n . The local property mentioned i) is a plain consequence of the continuity of the branch. As for property ii), Proposition 4.5 states that the branch C n contains a sequence (p k , u k ) such that either u k C 1,γ → ∞, or p k → ∞, or p k → 1. If p k → 1 but u k C 1,γ stays bounded, then [2, Lemma 2.1] ensures that u k converges to an eigenfunction of (2.7). Coming to property iii), since K n ∩ K n is the set of radial functions, then the intersection point between two branches C 2 n and C 2 n should be another nonradial bifurcation point (p, u m p ). Here possibly m = 2, because the number of nodal zones of the bifurcating solutions could become larger that 2, far from the bifurcation point. If this happens, then the very definition of C 2 n yields that S m ⊂ C 2 n and this concludes the proof.
The proof of the bifurcation in higher dimensions is quite similar.