Time periodic solutions of completely resonant Klein-Gordon equations on $\mathbb{S}^3$

We prove existence and multiplicity of Cantor families of small amplitude time periodic solutions of completely resonant Klein-Gordon equations on the sphere $\mathbb{S}^3$ with quadratic, cubic and quintic nonlinearity, regarded as toy models in General Relativity. The solutions are obtained by a variational Lyapunov- Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler-Lagrange action functional. Compactness properties of its gradient are obtained by Strichartz-type estimates for the solutions of the linear Klein-Gordon equation on $\mathbb{S}^3$.


Introduction
Motivated by the stability problem of the anti-de Sitter space-time (AdS), the goal of this paper is to prove existence and multiplicity of Cantor families of time-periodic solutions of nonlinear Klein-Gordon equations of the form where φ : R × S 3 → C and ∆ S 3 is the Laplace-Beltrami operator on the 3-dimensional sphere S 3 . For p = 3, time periodic solutions of (1.1) have been very recently constructed by Chatzikaleas and Smulevici in [21]. A mathematical point of interest of Theorems 1.2 and 1.4 below is that, jointly with [21,22], they are the only existence results of time periodic solutions for completely resonant Hamiltonian PDEs on a manifold of dimension higher than one. Their proof is based on a novel combination of variational methods and Strichartz-type estimates for free solutions of the Klein-Gordon equation on S 3 , that we find of theoretical interest in itself and nowhere else available in literature. Let us first shortly outline the physical framework connecting (1.1) with the stability problem of AdS space-time. AdS is the maximally symmetric solution to the vacuum Einstein equations Ric(g) = −Λg with negative cosmological constant Λ. Unlike de Sitter or Minkowski space-times, its stability properties are nowadays still poorly understood. In particular, the stability of AdS depends on the conformal boundary conditions. While, for instance, it is expected that under dissipative boundary conditions AdS is stable, see [30], it has been conjectured, by Dafermos and Holzegel in [25] and by Anderson in [2], that AdS is unstable under fully reflective boundary conditions. The latter instability conjecture is supported by the numerical investigations of Bizon-Rostworowski [16] for the spherically symmetric Einstein-massless-scalar field equations, suggesting that AdS is unstable, against the formation of black holes under arbitrarily small perturbations. Notwithstanding, the work [16] also suggests the existence of small initial data leading to stable solutions, confirmed later by Maliborski-Rostworowski [33] who constructed formal time periodic solutions, supported by numerical evidences. The same existence conjecture of time periodic solutions -called geons-has also been extended to the vacuum Einstein equations in [27,26].
The nonlinear wave equation (1.1) with p = 3 has been introduced in [16,33,15] as a toy model of spherically symmetric Einstein-massless-scalar field equations close to the AdS solution. In [20] Chatzikaleas constructed formal power series expansions of small amplitude time periodic solutions of (1.1) in the spherically symmetric case, which reduces to the 1d wave equation with singular nonlinearity −∂ tt u + ∂ xx u = u 3 sin 2 (x) , u(t, 0) = u(t, π) = 0 , x ∈ (0, π) . (1. 2) The absence of secular terms in the power series expansions is obtained using the method of Maliborski and Rostworowski [33], developed for the Einstein-Klein-Gordon equation. However, the presence of small divisors prevents the convergence of such power series. This difficulty looks analogous to the convergence problem of "Linstedt series" of quasi-periodic solutions in Celestial Mechanics, devised since Poincaré [34], and successfully overcome during the last century by the celebrated KAM theory. The first rigorous existence result of time periodic solutions of (1.1) for p = 3 with strongly Diophantine frequencies ω is given in the very recent paper [21]. Such work constructs solutions of the following form: • spherically symmetric functions, namely φ(t, x) = u(t, cos(x)), x ∈ (0, π), see Definition 1.1 below; • plane waves in Hopf coordinates, namely φ(t, η, ξ 1 , ξ 2 ) = u(t, η)e iµ 1 ξ 1 e iµ 2 ξ 2 , see Definition 1.3, up to restricting to values of the momenta µ 1 = µ 2 ∈ {1, . . . , 5}, or µ 1 = µ 2 large enough.
The results in [21] rely on an abstract theorem by Bambusi and Paleari [3], which uses a Lyapunov-Schmidt approach and whose main assumption is the existence of a non degenerate zero of the "resonant system".
The goal of this paper is to prove existence and multiplicity of periodic solutions of (1.1), for more general values of the nonlinearity degree p and of the momenta µ 1 , µ 2 . More precisely, for • p = 2 and p = 5, we find spherically symmetric solutions, see Theorem 1.2; • p = 3, we find plane waves in Hopf coordinates for any value of the momenta µ 1 , µ 2 ∈ Z, see Theorem 1.4.
These generalizations require new methods, since the verification of the existence of a nondegenerate zero of the associated resonant system seems unapproachable, if ever true. In this work we combine variational methods of mountain pass type, inspired by the works of Berti and Bolle [5,6,7] for 1-d semilinear wave equations, with Strichartz-type estimates for the linear Klein-Gordon equation on S 3 . We now present rigorously our results.
Our first result is the following: Theorem 1.2 (Spherically symmetric solutions). Let p = 2 or p = 5. Fix γ ∈ (0, γ 0 ) and d ∈ (0, 1 4 ). For any n ∈ N, r > 1 2 , and s > 3 2 , there exist ε 0 := ε 0 (n, r, s, d, γ) > 0 and C := C(n, r, s, d) > 0 such that for any ε belonging to with Ω γ defined in (1.4), there exist n different real valued, non zero, T ε -periodic solutions Tε , even in time and with spherical symmetry. They are of the following form: such that, as ε → 0, ε has minimal period T k,ε := Tε m k where {m k } n k=1 is an increasing sequence of positive integers. Correspondingly, the functions {v ε } have minimal periods T k := 2π m k . We point out that Theorem 1.2 holds also in the case p = 3. This is actually the result in [21]. In this case the v (j) ε are close to the "one mode" functions Note that e j are the spherically symmetric functions e j (x) = U j (cos(x)), where U j : R → R are Chebychev polynomials of second kind. The functions v (j) in (1.12) are actually solutions of the "resonant system" where Π V is the L 2 -projector on the infinite dimensional linear space V formed by the solutions of (1.3) (see the definitions (3.7), (3.11) below). On the other hand, for p = 5 the functions v (j) ε in (1.10) are not close to "one modes" as in (1.12). Actually, the v (j) ε are close to functions of the form ε 1 4 v (j) , where v (j) are non zero solutions of the equation (1.14) which does not possess one mode solutions. We actually prove the existence of non trivial solutions of (1.14), exploiting that it is the Euler-Lagrange equation of the action functional 15) which, thanks to the time-space Strichartz-type estimates proved in Section 4, admits mountain pass critical points of class C ∞ . Strichartz estimates are required to imply compactness properties of the action functional, which are not a consequence of Sobolev embeddings (1.6) on S 3 , see Remark 4.6. The case p = 2 is degenerate, since Π V (v 2 ) = 0 (see Lemma 4.4), and the v (j) ε are close to functions of the form ε 1 2 v (j) , where v (j) are non zero solutions of the equation It turns out that equation (1.16) admits mountain pass critical points as well. Further comments are postponed after Theorem 1.4. In the case p = 3 we have new existence results of periodic Hopf plane waves solutions of (1.1) for any value of the momenta (µ 1 , µ 2 ), which we now define: Definition 1.3 (Hopf plane waves). Consider on S 3 Hopf coordinates 0, π 2 × T × T ∋ (η, ξ 1 , ξ 2 ) → (sin(η) cos(ξ 1 ), sin(η) sin(ξ 1 ), cos(η) cos(ξ 2 ), cos(η) sin(ξ 2 )) . (1.17) We say that φ : R × S 3 → C is a Hopf plane wave with momentum (µ 1 , µ 2 ) if φ(t, ·) is a Hopf plane wave with momentum (µ 1 , µ 2 ) for any t ∈ R.
We make the following comments, common to both Theorems 1.2 and 1.4: 1. (Regularity and multiplicity) If r > 5 2 and s > 7 2 the solutions {φ (j) ε } of (1.1) proved in Theorems 1.2 and 1.4 are classical. Actually, the smoother we require the solutions to be in time and in space (i.e., the larger r, s are), the smaller ε 0 (r, s, n) has to be. Analogously, the larger is the number of solutions n, the smaller ε 0 (r, s, n) has to be.
associated to the nonlinearity is finite for any v in the space L ∞ (T t , H 1 (S 3 , dσ)) (which appears in (1.15)) by Sobolev embedding H 1 (S 3 , dσ) ֒→ L 6 (S 3 , dσ). However, it follows to have compact gradient by the Strichartz estimates in Proposition 4.5 (see Remark 4.6). For the supercritical exponents p ≥ 7, the functional G p+1 (v) is not expected to be well defined for any v in L ∞ (T t , H 1 (S 3 , dσ)). If p = 4, then G 5 (v) ≡ 0, as well as for all even values of p.
Then the leading term in the action functional of the corresponding resonant system turns out to have degree 8, which is supercritical.
As already mentioned, Theorems 1.2 and 1.4 are inspired by the variational approach of [5,6,7,4], developed for 1-d semilinear completely resonant wave equations −∂ tt u + ∂ xx u = u p + . . . with Dirichlet boundary conditions. Major difficulties with respect to these works arise because of the 3-dimensional manifold S 3 . This becomes evident for instance in the search of spherically symmetric solutions of (1.1), that reduces to solve the wave equation which has a singular nonlinearity at x = 0, π. Before explaining the main difficulties and ideas of our proof, we present a few related results.
Related literature. The first existence results of 2π-periodic solutions for completely resonant wave equations ∂ tt u − ∂ xx u = |u| p−2 u, p > 2, have been proved by Rabinowitz starting with [37], via global variational methods. These techniques, as well as those in [18,19], enable to find periodic orbits with rational frequency, the reason being that other periods give rise to a small denominator problem. Independently of these global results, the local bifurcation theory of periodic and quasi-periodic solutions was initiated for non resonant 1-d Klein-Gordon equations by Wayne [38], Kuksin [31], Craig and Wayne [24], Poschel [35], Chierchia and You [23], with KAM methods. For semilinear Klein-Gordon equations on T d with convolution potentials, the first result is due to Bourgain in [17], later extended by [9,10] for multiplicative potentials. Bifurcation for periodic and quasiperiodic solutions of non resonant Klein-Gordon equations was obtained in [14,12] for Lie Groups and homogeneous manifolds, in [11] for Zoll manifolds, and in [29] for the sphere S d . These results do not cover the completely resonant case (1.1), where all the linear frequencies of oscillations are integers.
The first existence results of Cantor families of small amplitude time periodic solutions of 1-d completely resonant wave equations −∂ tt u + ∂ xx u = u p , p = 3, was proved in [32] under periodic boundary conditions and in [3] for Dirichlet boundary conditions, for frequencies belonging to the zero measure set (1.4). The latter result was then generalized in [5,6] to arbitrary exponents p, using variational methods. Existence of periodic solutions for a set of frequencies ω ∼ 1 of density one was proved in [7,8] via Nash-Moser implicit function techniques, and in [28] via trees resummation arguments. Existence of time quasi-periodic solutions with two frequencies of completely resonant nonlinear wave equations on the circle were obtained in [36] and [13].
For completely resonant wave equations, or even more general Hamiltonian PDEs in dimension higher than one, not much is known about time periodic solutions besides the aforementioned paper [21] and the present work.

Ideas of proof
In order to look for bifurcation of small amplitude time periodic solutions of (1.1) with frequency ω ∼ 1 a natural approach is to implement a Lyapunov-Schmidt decomposition in the spirit of [5,6,7] for 1d semilinear wave equations. Major difficulties arise due to the higher dimension of the space domain, here the sphere S 3 , as we now explain. After a time rescaling, we look for 2π-periodic in time real solutions u(t, z) of −ω 2 ∂ tt u + ∆ S 3 u − u = u p . By splitting where V is the kernel of the operator −∂ tt + ∆ S 3 − 1 (namely the space of solutions of the free Klein-Gordon equation (1.3)) and Π V the corresponding orthogonal projector, it amounts to the system For any ω ∈ Ω γ the operator L ω := −ω 2 ∂ tt + ∆ S 3 − 1 is invertible on the range W := V ⊥ and, for any fixed v ∈ V small enough (in some suitable norm), one may solve first the range equation, , by a contraction argument. Here, in order to control the nonlinearity (v + w) p , it is natural to close the contraction in Sobolev spaces which are an algebra with respect to the product of functions, and where L −1 ω W is bounded. This requires to take v small enough in H r t H s z as well, which amounts, for functions in the kernel V , to require that On the other hand, one needs then to solve the bifurcation equation (Bif. eq) with w = w(v). As observed in [5,6], this turns out to be the Euler-Lagrange equation of the reduced action functional A serious problem which arises is thus the following: Problem: The natural space to find mountain pass critical points for the functional Φ in (1.25) is (a small ball in) the space V 1 t,z (modeled with an H 1 -norm), associated to its quadratic part. This is clearly in contradiction with solving the (Range eq) on the much smaller domain (1.24). How to fill this regularity gap?
We remark that the previous difficulty does not disappear restricting to search solutions which depend on only one space variable, as spherically symmetric functions or Hopf waves. This is evident for instance in the spherically symmetric case, where the reduced equation (1.23) has a singular nonlinearity. If p = 3, this issue is overcome (cfr. [21]) noting that the functional Φ in (1.25) possesses non degenerate critical points of the explicit form v = ε where v is a one mode function as in (1.12), which belong to { v V r+s t,z ≪ 1} for any r + s > 2.
We now describe our strategy. For simplicity, we focus on the case p = 5 and we restrict on spherically symmetric functions. The seminal idea is to note that, neglecting w(v), the functional Φ(v) in (1.25) is a perturbation of the "resonant system" functional The Strichartz estimate (4.11) implies that G 6 is well defined on V 1 t,z and its gradient ∇ V 1 t,z G 6 is a bounded map from V s t,z to V 1 t,z for any s > 5 6 , thus compact on V 1 t,z . Thus Φ 0 possesses a mountain pass critical point v ∈ V 1 t,z (see [1]), which by homogeneity has the form v = ε Such v is not a one mode function, but it is C ∞ by the following bootstrap argument. By the Strichartz estimate (4.11), one has Then, to increase further the regularity of v, we observe that the Strichartz estimate (4.12) implies Iterating this procedure with increasing values of δ ′ , one deduces that v is in C ∞ . In order to adapt the previous arguments to deal with the whole functional Φ in (1.25), we split the bifurcation equation into low and high frequencies. For any N ∈ N (to be determined later large enough) the bifurcation equation is equivalent to the system ω j := j + 1 are the frequencies associated to the eigenfunctions e j defined in (1.12), and Π ≤N , resp. Π >N , is the projector on the time-space Fourier frequencies smaller than N , resp. > N . Then we solve both the high frequency bifurcation equation (1.28) and the range equation (Range eq) arguing by contraction: • In Section 5, we solve first the high frequency bifurcation equation (1.28) for v 2 in a small ball of V 2+ t,z , for any v 1 V 1 t,z ≤ Rε particular values considered in [21]. This is because for µ 1 = µ 2 an explicit formula for the product between the eigenfunctions {e (µ 1 ,µ 2 ) j } j in (1.21) is not available. Then we split our equation (1.1) into the range equation (Range eq) and the high and low bifurcation equations (1.28), (1.27). We solve the low frequency bifurcation equation (1.27) using duality arguments, Hölder inequality and the Sobolev embedding (1.6), without Strichartz-type estimates.
In the degenerate case p = 2, one has Π V (v 2 ) = 0 and the leading nonlinear term in the bifurcation equations (1.27)-(1.28) turns out to be the cubic term Π V vL −1 ω v 2 . The Strichartztype estimates (4.27)-(4.28) are used to solve the high frequency equation (1.28), avoiding to prove if L −1 ω W is bounded on L q (T t , L q (S 3 , dσ)) spaces. Given a ∈ R, we denote a := max{1, |a|}. Given a, b real valued functions, a b means that there exists C > 0 such that a ≤ Cb. If C depends on parameters α 1 , . . . , α r , we write a α 1 ,...,αr b. If a b and b a, we write a ≍ b.

Functions with spherical symmetry
According to Definition 1.1, in spherical coordinates the metric tensor is represented with respect to the basis of the tangent space ∂ ∂x , ∂ ∂θ , ∂ ∂ϕ as Hence the volume form is dσ = sin 2 (x) sin(θ)dxdθdϕ, and the Laplace-Beltrami operator reads For convenience, we introduce the normalized measures chosen in such a way that the measure of the sphere S 3 is 1. We denote L p (S 3 , dσ) := L p (S 3 ). The Laplace-Beltrami operator (2.1) leaves invariant the subspace of spherically symmetric functions (cfr. Definition 1.1), acting as As a consequence, the subspaces of spherically symmetric functions in H s (S 3 , dσ) coincide with We now exhibit a basis of eigenfunctions and eigenvalues for the operator ∆ ss S 3 , see [21]: is an orthonormal basis for H 0 As a consequence, the Sobolev spaces H s x in (2.2) are spectrally characterized as where u j := u, e j H 0 x are the Fourier coefficients with respect to the basis {e j }, with scalar product The eigenfunctions {e n } n∈N satisfy the following product rule: for any integer n ≥ m, (2.7) We will use property (2.7) to prove the Strichartz type Propositions 4.5 and 4.12. It can also be used to prove that the spaces H s x with s > 3 2 enjoy algebra property.

Hopf symmetry
According to Definition 1.3, in Hopf coordinates the metric tensor is represented with respect to the basis of the tangent space ∂ ∂η , ∂ ∂ξ 1 , ∂ ∂ξ 2 as Hence the volume form is dσ = 1 2 sin(2η)dηdξ 1 dξ 2 and the Laplace-Beltrami operator reads We introduce the normalized measure so that the measure of the sphere S 3 is 1.
Representing a function φ in Hopf coordinates (see Definition 1.3) and expanding in Fourier series with respect to the variables ξ 1 , ξ 2 , we have (2.10) In these coordinates, the Laplace-Beltrami operator (2.8) reads As a consequence, the space of Hopf plane waves is left invariant by ∆ S 3 and recalling (2.8), (2.9), the subspaces of Hopf plane waves in H s (S 3 , dσ) coincide with (2.14) We now exhibit a basis of eigenfunctions and eigenvalues of −∆ µ 1 ,µ 2 , see [21, Section 3.2].
By Lemma 2.2, the Sobolev spaces H s η in (2.13) are spectrally characterized as η are the Fourier coefficients with respect to the basis {e

Sobolev spaces in time-space
Since equation (1.1) is time reversible, we look for functions which are even in time. For this reason, we consider the Sobolev spaces of time periodic even real functions and u ℓ,j are the time-space Fourier coefficients of u, u ℓ,j := For any r > 1 2 and for any s ∈ R the space for some C r > 0. Moreover, since the spaces H s z for s > 3 2 are an algebra, also the spaces H r t H s z are an algebra for any r > 1 2 and s > 3 2 : there exists a constant C s,r > 0 such that . (2.25) and recalling the definition of dσ, dξ 1 , dξ 2 in (2.9), one has Then applying at any time t the generalized Hölder inequality with p 1 = p 2 = p 3 = p 4 = 4 for functions on S 3 , and the Sobolev embedding H dt .

Variational Lyapunov-Schmidt decomposition
We look for time periodic solutions of (1.1) with time frequency ω close to 1, via a Lyapunov-Schmidt decomposition. More specifically we look for a 2π ω -time periodic real valued spherically symmetric solution u(t, x) of (1.1) which solves We consider the cases p = 2, 5 only, because the case p = 3 is covered in [21]. If p = 3 we look for a 2π ω -time periodic Hopf plane wave solution φ(t, η, ξ 1 , ξ 2 ) = u(t, η)e iµ 1 ξ 1 e iµ 2 ξ 2 of (1.1), with u(t, η) real. The function u(t, η) solves with ∆ µ 1 ,µ 2 defined in (2.12). Both the equations in (3.1) and (3.2) are of the form where A denotes the unbounded, self-adjoint, positive operator  We shall exploit the variational structure of (3.3) in Section 7, after a suitable finite dimensional reduction. We perform a Lyapunov-Schmidt decomposition of equation (3.3). We define

Equation (3.3) admits a variational formulation. It is the formal Euler Lagrange equation of the action functional
We decompose the space V into low and high frequencies: given N ∈ N, we define respectively, so that any u can be decomposed as (3.11) We then observe that a function u satisfies (3.3) if and only if it is a solution of the system We shall solve the equation (3.13) for v 2 by a contraction argument in Section 5. Then in Section 6 we shall solve the range equation (3.14), arguing again by a contraction argument and using the following lemma.

Properties of functions in V
In this section we prove some properties of functions in the kernel V which will be used to solve the system (3.12)-(3.14). Given s ∈ R, we shall denote V s t,z : Proof. In order to prove (4.2) it is sufficient to observe that v 2 with · H s z defined according to (2.18) and since | cos(·)| ≤ 1. By (4.3) and algebra property of the spaces H s z , for any v (1) , v (2) ∈ V s t,z and s > 3 For any s < s ′ the following smoothing properties hold (cfr. (3.9), (3.10)): for any v ∈ V Since Ae j (z) = ω 2 j e j (z) for any j (see Lemmas 2.1 and 2.2) and recalling (4.1) it results We will also use that by Lemma 4.1 and the Sobolev embedding Lemma 4.3. Let s > 3 2 , r > 1 2 and q ∈ N, then there exists a positive constant C = C(s, r, q) such that for any j < q and any v (1) In particular, if n is even then Proof. Performing in the integral in (4.9) the change of variables (t, x) and, thus, since q is odd, namely I = 0.

Strichartz-type estimates for p = 5
The aim of this section is to prove a set of Strichartz-type estimates for solutions of (1.3) in the case of spherical symmetry. We shall use the following duality property: for any Proposition 4.5. (Generalized Strichartz-type estimates) The following estimates hold: Remark 4.6. From (4.11) the functional G 6 (v) : with compact gradient.
The rest of this section is devoted to the proof of Proposition 4.5. We use the following definition.
We now prove Proposition 4.5.
Note that, due to Lemma 4.4, one has (4) ) is well defined. The rest of this section is devoted to the proof of Proposition 4.12.
Proof. With no loss of generality, we suppose that j 1 ≤ j 2 ≤ j 3 ≤ j 4 . By the product rule (2.7) one has e j 1 e j 2 e j 3 e j 4 = Now for each fixed j 1 , j 2 , j 3 , j 4 , k there is at most one value of h such that j 2 −j 1 +2k = j 4 −j 3 = 2h. Moreover the sum over k runs over j 1 + 1 = ω 1 elements. This proves I j 1 j 2 j 3 j 4 ≤ ω j 1 . The lower bound I j 1 j 2 j 3 j 4 ≥ 0 directly follows because (4.30) is the sum of non-negative integers.
Proof of Proposiition 4.12, Item 2. First we prove that, defining v (k) with the factor γ −1 in (4.35) replaced by 1 if ω = 1. Once that (4.35) has been proved, Item 2 follows by the following claim: for any v (1) , v (2) , v (3) , t,x using also Lemma 3.1. Moreover (4.28) for a general index l follows by self-adjointness of L −1 ω . The first step in the proof of (4.35) is the following: Lemma 4.14.
The sum J in (4.37), using its symmetry in the indexes j 1 , j 2 , j 3 , is bounded by , from which (4.35) follows.

Solution of the v 2 equation
In this section we solve the equation (3.13) for the high frequency components v 2 in the kernel. We argue separately for the cases p = 5, p = 3, and for the degenerate case p = 2. Given ρ 1 ∈ (0, 1), ρ 2 ∈ (0, 1), ρ 3 ∈ (0, 1), we define and for some δ > 0 In the sequel δ will always denote a positive small constant.
We now prove Proposition 5.4. We define the map and show that it is a contraction.

By (4.5) and Lemma 4.3, one has
We now prove that it is a contraction. One has Applying Lemma 4.3, (4.5), using (5.20) and the smallness condition (5.21), one obtains which is (5.27).
We now prove Proposition 5.6. For any (v 1 ,w) ∈ D ρ 1 × D W ρ 3 we look for a solution of (5.30) as a fixed point of the map which to v 2 associates with A = −∆ ss S 3 + 1 as in (3.4). We shall use the following technical lemma: Lemma 5.7. Let ρ 1 and ρ 2 as in (5.28). There exists ǫ R,c 2 > 0 such that, if N −1−2δ γ −1 < ǫ R,c 2 , then for any s ∈ [0, 2 + 2δ], any v 1 ∈ D ρ 1 and v 2 ∈ D V 2 ρ 2 one has Proof. The estimate on v 1 V s t,x follows from (4.5). For any s ∈ [0, 2 + 2δ] and v 2 ∈ D V 2 ρ 2 , by (4.5) and (5.28) The next Lemma is based on the Strichartz-type estimates of Proposition 4.12: As a consequence, for any and we estimate each term separately. T 1 is estimated using Item 2 of Proposition 4.12, which gives 3 .

(5.39)
T 2 is estimated using (4.5) and Lemma 4.3: one has and Finally, T 4 is estimated using algebra property (2.24): provided (5.32) holds for some b ∈ (0, 1) and ǫ R,δ small enough and c 2 ≥ 2C δ . We now prove that T v 1 ,w is a contraction. We actually prove that . We proceed estimating separately all terms. By Item 2 of Proposition 4.12, by Lemma 5.7 and using the definitions of the parameters ρ 1 , ρ 2 , ρ 3 , one has The estimate of D 3 [h 2 ] is the same, and gives The estimate of D 2 [h 2 ] is analogous to the estimate of T 2 , and yields We finally estimate D 4 [h 2 ] using algebra property (2.24) and Lemma 3.1. One gets Then using Lemma 5.7 and the definitions of parameters ρ 1 , ρ 2 , ρ 3 as in (5.28) one has Thus, combining (5.44), (5.46), (5.45), (5.47) and assuming (5.32), one gets , which implies (5.38) and that T v 1 ,w is a contraction. Finally, since Finally, with analogous arguments to the ones in the proof of Lemma 5.8, one obtains differentiability of v 2 (v 1 ,w) with respect to v 1 andw with estimates (5.34), (5.35).

Solution of the range equation
In this section we solve the range equation (3.14) in the algebra spaces H w) is the solution of (3.13), namely we find w such that (6.1)
We are going to prove that the map is a contraction, with v 2 (v 1 , w) as in Proposition 5.1.

Restricted Euler-Lagrange functional
We start observing that (7.1) has a variational structure.

Cases p = 5 and p = 3
We first consider the case p = 5. Since the functional G 6 in (3.6) is positive, we have m(G 6 ) = m + (G 6 ).
We now prove properties of the functional R 6 defined in Lemma 7.3.
Proof of Theorem 7.2 for p = 3. By (7.22) and Lemma 7.9, one has with C the constant whose existence is stated in Theorem 7.2, provided (5.21) holds with ǫ R,δ small enough. By (7.22) one observes that Then the existence of a critical point v as ε → 0 follows.

Multiplicity of solutions
In this section we prove multiplicity of solutions.
The following estimates hold:

2)
and v The remaining part of this section is devoted to prove Theorem 8.1. Since the dependence of the spaces V 1 , V 2 on the parameter N plays a significant role, in this section we denote them respectively by V ≤N , V >N . We regard equations (3.1), (3.2) on the space of 2π n time periodic functions X n := u(t, z) = ℓ∈N j∈N u ℓ,j cos(nℓt)e j (z) .
We define the restrictions to X n of the kernel and range subspaces V, W, V ≤N , V >N defined in (3.7), (3.8), (3.9), (3.10): V n := V ∩ X n , W n := W ∩ X n , V ≤N,n := V ≤N ∩ X n , V >N,n := V >N ∩ X n . (8.4) We note that for any n ∈ N * the space X n is left invariant both by the spatial operator A defined in (3.4) as well as by L ω defined in (3.3).
Proof. The functions v 2 and w are respectively obtained as the fixed point of the contractions T v 1 ,w(v 1 ) and T v 1 , defined in (5.10) and (6.4) in the case p = 5 (the case p = 3, p = 2 are analogous). Then the lemma follows observing that, for any v 1 ∈ V ≤N,n ∩ D ρ 1 and w ∈ W n ∩ D W ρ 3 the operator T v 1 ,w maps V >N,n into itself, and that for any v 1 ∈ V ≤N,n ∩ D ρ 1 , the operator T v 1 maps W n into itself.
In order to find 2π n periodic solutions of (7.1), we look for critical points of Ψ n :=Ψ| V ≤N,n ∩Dρ 1 .
We remind thatΨ has the expansion (7.5) in the cases p = 5, p = 3 and (7.9) in the case p = 2.
Proof. By Theorem 7.5 and (8.11),Ψ n admits a critical point v (n) 1 ∈ V ≤N,n which is proportional to a point y (n) satisfying We note that, since r (n) = O(α n (R)), provided αn(R) mn(G) < C with C = C(β) small enough, one has (1 − β)m n (G) > |r (n) |, which by (8.13) gives Combining the latter inequality with hypothesis (8.10), one gets , thus for any m > m 0 one has that y (n) belongs to V ≤N,n ⊂ V but y (n) does not belong to V ≤N,mn , namely y (n) has minimal period ≥ 2π m 0 n . Since v (n) 1 and y (n) are proportional, the same holds for v (n) 1 .
It remains to prove that v (n) 1 is also a critical point for the functionalΨ. To fix ideas, we prove the result for p = 5. The cases p = 3, p = 2 follow analogously. By Lemma 7.1, a point v 1 ∈ D ρ 1 is critical forΨ if and only if (8.14) Since v (n) 1 is critical forΨ n , one already has that (8.14) holds for h ∈ V ≤N,n , thus it remains to prove it for h ∈ V ≤N ∩ V ⊥ ≤N,n . Then it is sufficient to observe that, by Lemma 8.
1 )) 5 belongs to V ≤N,n , namely it is orthogonal to any h ∈ V ≤N ∩ V ⊥ ≤N,n , which gives the thesis.
Theorem 8.1 follows from an iterative application of Lemma 8.5 and Proposition 8.6. In the next sections we verify the assumptions (8.10) arguing separately for the cases p = 5, p = 3, p = 2.

Cases p = 5 and p = 3
We start with p = 5 and we prove lower and upper bounds for m n (G 6 ) defined in (8.8).
Lemma 8.7 (Estimate of m n (G 6 )). For any δ > 0 there exists C δ > 0 and for any n ∈ N * there exists κ n > 0 such that Proof. We take v n = cos(nt)e n−1 (x), then v n V 1 t,x = n. One has G 6 (v n ) = 1 6 T cos 6 (nt) dt 2 . This proves that for any n, m ∈ N there exists m 0 = m 0 (n) ∈ N such that, if m > m 0 (n) and N ≥ n, one has , namely (8.10) is satisfied. We then define n 0 := 1, n k+1 := m 0 (n k )n k + 1 and we apply Proposition 8.6 with n = n k for any k = 1, . . . , k * . In particular, assumptions (8.11) hold for any n k , observing that α n k (R 6 ) ≤ α(R 6 ) δ N − 7 6 +9δ R 4 by Lemma 7.7 and m n k (G 6 ) ≥ κ n by Lemma 8.7, and assuming R ≥ R k * = max k (κ n k C) −1 and N δ inf k (R 4 κ n k ) 1 7 6 −9δ , which is ensured by (5.5). Thus by Proposition 8.6 the functionalΨ admits a critical point v has the same minimal period T n k . Remark 8.8. With careful estimates on m(G 6 ) one can obtain n k+1 = 3n k + 1 for any k.

Strong solutions
In this section we prove higher regularity of the solutions found in Theorem 8.1.