William Borrelli
email: borrelli@ceremade.dauphine.fr

WEAKLY LOCALIZED STATES FOR NONLINEAR DIRAC EQUATIONS

 to more general nonlinearities. Contents 1. Introduction 1 2. Existence and asymptotics 5 3. Variational characterization 10 4. The massive case 18 References 19

Introduction

The Dirac equation has been widely employed to build relativistic models of particles (see, e.g., the survey paper [START_REF] Esteban | Variational methods in relativistic quantum mechanics[END_REF]). Recently, it made its appearance in Condensed Matter Physics in connection with two-dimensional honeycomb structures. They appear, for instance, in the study of new twodimensional materials possessing Dirac fermions low-energy excitations, the most famous being graphene (see, e.g. [START_REF] Neto | The electronic properties of graphene[END_REF], [START_REF] Cayssol | Introduction to dirac materials and topological insulators[END_REF]). In those materials electrons at the Fermi level have zero apparent mass and can be described using the massless Dirac equation. The laser beam propagation in certain photonic crystals can also be described by honeycomb structures [START_REF] Peleg | Conical diffraction and gap solitons in honeycomb photonic lattices[END_REF].

More generally, Schrödinger operators of the form

H = -∆ + V per (x), x ∈ R 2 ,
where V per ∈ C ∞ (R 2 , R) is a honeycomb potential (in the sense of [START_REF] Fefferman | Honeycomb lattice potentials and dirac points[END_REF]), that is possessing the symmetries to a honeycomb lattice Λ, generically exhibit conical intersections (the so-called Dirac points) in their dispersion Date: July 19, 2018. bands, as proved in [START_REF] Fefferman | Honeycomb lattice potentials and dirac points[END_REF]. This leads to the appearance of Dirac as the effective operator, describing the electron dynamics for wave packets spectrally concentrated around such conical degeneracies.

More precisely, by the Bloch-Floquet theory [START_REF] Reed | Methods of modern mathematical physics. IV. Analysis of operators[END_REF], the spectrum Spec(H) ⊆ R is the union of spectral bands, obtained through the following k-pseudoperiodic eigenvalue problem:

HΦ(y, k) = µ(k)Φ(y, k), y ∈ Y Φ(y + v, k) = e ik•v Φ(y, k), v ∈ Λ ( 1
)
where k ∈ Y * varies in the Brillouin zone, that is the fundamental zone of the dual lattice Λ * , and Y ⊆ Λ is the fundamental zone of the physical lattice.

The eigenvalues given by (1) form a sequence µ 0 (k) µ 1 (k) µ 2 (k) ... and the corresponding pseudo-periodic eigenfunctions Φ n (•, k), called Bloch functions. In [START_REF] Fefferman | Honeycomb lattice potentials and dirac points[END_REF] Fefferman and Weinstein proved that if k = K * is a vertex of the Brillouin zone, then there exists m ∈ N such that the bands µ n (•), µ n+1 (•) intersect conically at µ * := µ n (K * ) = µ n+1 (K * ) and ker(H -µ * ) = span{Φ 1 (x), Φ 2 (x)}.

(

More details and precise definitions can be found in [START_REF] Fefferman | Honeycomb lattice potentials and dirac points[END_REF].

Remark 1. Physically, the energy µ * of the Dirac point is the Fermi level, in the case of graphene. In turns, this corresponds to the zero-energy for the Dirac operator. However, there is no interpretation of the Dirac spectrum in terms of particles/antiparticles. Rather, the positive part of the spectrum corresponds to massive conduction electrons, while the negative one to valence electrons.

The following nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation.

i∂ t u = -∆u + V (x)u + κ|u| 2 u, x ∈ R 2 , κ ∈ R, (3)
describes, in the quantum setting, the dynamics of Bose-Enstein condensates, and u is the wavefunction of the condensate [START_REF] Pitaevskii | Bose-Einstein condensation[END_REF][START_REF] Erdős | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF]. Here V (x) models a magnetic trap and the nonlinear potential κ|u| 2 describes a mean-field interaction between particles. The parameter κ is the microscopic 2-body scattering length. Another important field of application of NLS/GP is nonlinear optics, namely in the description of electromagnetic interference of beams in photorefractive crystals [START_REF] Moloney | Nonlinear optics[END_REF]. In this case V (x) is determined by the spatial variations of the background linear refractive index of the medium, while the nonlinear potential accounts for the fact that regions of higher electric field intensity have a higher refractive index (the so-called Kerr nonlinear effect). In this case κ < 0 represents the Kerr nonlinearity coefficient. In the latter situation, the variable t ∈ R denotes the distance along the direction of propagation and x ∈ R 2 the transverse dimensions.

In the above systems honeycomb structures can be realized and tuned through suitable optical induction techniques based on laser or light beam interference [START_REF] Peleg | Conical diffraction and gap solitons in honeycomb photonic lattices[END_REF]. They are encoded in the properties of the periodic potential V = V per .

Let u 0 (x) = u ε 0 (x) be a wave packet spectrally concentrated around a Dirac point, that is:

u ε 0 (x) = √ ε(α 0,1 (εx)Φ 1 (x) + α 0,2 (εx)Φ 2 (x)) (4)
where Φ j , j = 1, 2 are Bloch functions at the Dirac point [START_REF] Ammann | A spinorial analogue of Aubin's inequality[END_REF], and the functions α 0,j are some (complex) amplitudes to be determined.

Then one expects the solution to (3) to evolve, at leading order in , as a modulation of Bloch functions:

u ε (t, x) ∼ →0 + √ ε (α 1 (εt, εx)Φ 1 (x) + α 2 (εt, εx)Φ 2 (x) + O(ε)) , (5)
As explained in [START_REF] Arbunich | Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures[END_REF], performing a multiscale expansion one obtains the following effective Dirac system for the modulation coefficients α j :

∂ t α 1 + λ(∂ x 1 + i∂ x 2 )α 2 = -iκ(2β 2 |α 1 | 2 + β 1 |α 2 | 2 )α 1 ∂ t α 2 + λ(∂ x 1 -i∂ x 2 )α 1 = -iκ(β 1 |α 1 | 2 + 2β 2 |α 2 | 2 )α 2 , (6)
where

β 2 := ˆY |Φ 1 (x)| 2 |Φ 2 (x)| 2 dx, β 1 := ˆY |Φ 1 (x)| 4 dx = ˆY |Φ 2 (x)| 4 dx. (7)
Here λ ∈ C \ {0} is a coefficient related to the potential V (see [START_REF]Wave packets in honeycomb structures and two-dimensional Dirac equations[END_REF], [START_REF] Fefferman | Honeycomb lattice potentials and dirac points[END_REF]). We remark that the system (6) has been first formally derived by Fefferman and Weinstein in [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF].

The large, but finite, time-scale validity of the Dirac approximation has been proved in [START_REF]Wave packets in honeycomb structures and two-dimensional Dirac equations[END_REF] for the linear case and for Schwartz class intial data (4). The case of cubic nonlinearities [START_REF] Boussaïd | On spectral stability of the nonlinear Dirac equation[END_REF], corresponding to the NLS/GP (3), is treated in [START_REF] Arbunich | Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures[END_REF] for high enough Sobolev regularity H s (R 2 ), with s > 3.

Remark 2. The factor √ appearing in [START_REF] Berthier | On the point spectrum of Dirac operators[END_REF][START_REF] Borrelli | Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity[END_REF] is of course irrelevant in the linear case, but it is exactly the critical scaling such that in the cubic case the nonlinearity and the Dirac appear together at first order in the multiscale expansion, as shown in [START_REF] Arbunich | Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures[END_REF].

We are interested in studying zero-modes of (6) in the focusing case κ < 0, that is, we look for particular stationary solutions of the form

α(t, x) = ψ(x), (t, x) ∈ R × R 2 .
It will turn out that they are in general weakly localized, as they are not even square-integrable, in contrast to the results mentioned for the evolution problem. We expect those zero-modes to be useful to prove approximation results for stationary solutions to (3), analogous to the ones proved in [START_REF]Wave packets in honeycomb structures and two-dimensional Dirac equations[END_REF], [START_REF] Arbunich | Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures[END_REF] for the evolution problem, somehow in the spirit of [START_REF] Ilan | Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations, and effective media[END_REF]. However, the absence of a gap at the Dirac point is a serious problem to deal with. This will be the object of a future investigation and will be addressed elsewhere. It is not restrictive to choose κ = -1 in [START_REF] Boussaïd | On spectral stability of the nonlinear Dirac equation[END_REF]. This leads to the following system:

λ(∂ x 1 + i∂ x 2 )ψ 2 = i(2β 2 |ψ 1 | 2 + β 1 |ψ 2 | 2 )ψ 1 λ(∂ x 1 -i∂ x 2 )ψ 1 = i(β 1 |ψ 1 | 2 + 2β 2 |ψ 2 | 2 )ψ 2 (8)
Moreover, we can easily get rid of λ = 0. Indeed, setting

ψ 1 (x) = 1 |λ| ψ 1 (x), ψ 2 (x) = λ |λ| 2 ψ 2 (x), x ∈ R 2
and defining β j := β j |λ| 3 , j = 1, 2, one ends up (dropping superscripts) with the system:

(∂ x 1 + i∂ x 2 )ψ 2 = i(2β 2 |ψ 1 | 2 + β 1 |ψ 2 | 2 )ψ 1 (∂ x 1 -i∂ x 2 )ψ 1 = i(β 1 |ψ 1 | 2 + 2β 2 |ψ 2 | 2 )ψ 2 (9)
where 0 < β 2 β 1 .

For simplicity, we state our main result in terms of equation [START_REF] Cazenave | Existence of localized solutions for a classical nonlinear Dirac field[END_REF].

Theorem 3. Equation ( 9) admits infinitely many solutions

ψ ∈ C ∞ (R 2 , C 2 ) of the form ψ(r, ϑ) = iu(r)e iϑ v(r) (10)
with u, v : [0, +∞) -→ R, (r, ϑ) being polar coordinates in R 2 . Moreover, the spinor components satisfy u(r)v(r) > 0, ∀r > 0, [START_REF] Ekeland | Convexity methods in Hamiltonian mechanics[END_REF] and there holds

|u(r)| ∼ 1 r , |v(r)| ∼ 1 r 2 , as r → +∞, (12)
In particular,

ψ ∈ L p (R 2 , C 2 ), ∀p > 2, but ψ / ∈ L 2 (R 2 , C 2 ).
For this reason, we say that those solutions are weakly localized.

Remark 4. Heuristically, weak localization is expected as the L 2 -spectrum of the massless Dirac operator is equal to R, as it is easily seen using the Fourier transform (see [START_REF] Thaller | The Dirac equation[END_REF] for more details). As shown in Theorem 3, in general stationary solutions in the massless case only exhibit a polynomial decay at infinity. This is in striking contrast with the massive case, where stationary solutions (of arbitrary form) are exponentially localized (see, e.g., [START_REF] Boussaïd | On spectral stability of the nonlinear Dirac equation[END_REF] where the method of [START_REF] Berthier | On the point spectrum of Dirac operators[END_REF] has been generalized to deal with nonlinear bound states in any dimensions).

Remark 5. Equation ( 9) is invariant by scaling. Indeed, it can be easily checked that if ψ is a solution, then the same holds for the rescaled spinor

ψ δ (•) := √ δψ(δ•), ∀δ > 0. ( 13
)
Thus it suffices to prove the existence of one (non-trivial) solution, to get multiplicity. Observe also that if ψ solves the equation, then

ψ(•) := -ψ(•) (14)
is another solution.

Remark 6. Theorem 3 is in some sense suggested by the literature on the spinorial Yamabe problem. A particular family of test spinors is used to study conformal invariants or nonlinear Dirac equations on spin manifolds (see e.g. [START_REF] Ammann | A spinorial analogue of Aubin's inequality[END_REF], [START_REF] Isobe | Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds[END_REF] and references therein). It is given by

ϕ(y) = f (y)(1 -y) • ϕ 0 y ∈ R 2 (15)
where ϕ 0 ∈ C 2 , f (y) = 2 1+|y| 2 and the dot represents the Clifford product. It can be easily checked that they are H 1 2 (R 2 , C 2 )-solutions to the following "isotropic" Dirac equation (corresponding to

β 1 = 1, β 2 = 1 2 ) Dϕ = |ϕ| 2 ϕ ( 16
)
The spin structure of euclidean spaces is quite explicit and the spinors in (15) can be rewritten in matrix notation as

ϕ(y) = f (y)(1 2 + iy 1 σ 1 + iy 2 σ 2 ) • ϕ 0 y ∈ R 2
1 2 and σ i being the identity and the Pauli matrices, respectively. See [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF] for more details. One can show that ( 15) is of the form [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF] and has the decay properties stated in Theorem 3.

The present paper is organized as follows. In (Section 2) we prove (Theorem 3), exploiting a particular radial ansatz. The proof follows by direct dynamical systems arguments. Then we show in (Section 3) that the solutions found in the first part of the paper admit a variational characterization. This is done using duality, combined with standard concentration compactness theory and Nehari manifold arguments. The last section (Section 4) is devoted to some remarks concerning the massive case, where we quickly explain how the results of the present paper allow to extend those of [START_REF] Borrelli | Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity[END_REF]. Acknowledgment. I wish to thank Michael I. Weinstein for his encouragement.

Existence and asymptotics

In this section we prove Theorem (3), providing the existence and the exact asymptotic behavior of (non-trivial) solutions of ( 9) satisfying the ansatz [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF]. The latter allows us to convert the PDE (9) into a dynamical system. Indeed, passing to polar coordinates in R 2 , (x 1 , x 2 ) → (r, ϑ), the equation reads as:

       -e iϑ i∂ r - ∂ ϑ r ψ 2 = -2β 2 |ψ 1 | 2 + β 1 |ψ 2 | 2 ψ 1 , -e -iϑ i∂ r + ∂ ϑ r ψ 1 = β 1 |ψ 1 | 2 + 2β 2 |ψ 2 | 2 ψ 2 . ( 17
)
Plugging the ansatz

ψ(r, ϑ) = iu(r)e iϑ v(r) (18)
into [START_REF] Fefferman | Honeycomb schrödinger operators in the strong binding regime[END_REF] gives:

   u + u r = v(2β 2 u 2 + β 1 v 2 ) v = -u(β 1 u 2 + 2β 2 v 2 ) (19)
Thus we are lead to study the flow of the above system.

In particular, since we are looking for localized states, we are interested in solutions to [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF]

such that (u(r), v(r)) -→ (0, 0) as r → +∞
In order to avoid singularities and to get non-trivial solutions, we choose as initial conditions u(0) = 0 , v(0) = λ = 0 (20) Moreover, the symmetry of the system allows us to consider only the case λ > 0. Thus (Theorem 3) reduces to the following Proposition 7. For any λ > 0 there exists a unique solution

(u λ , v λ ) ∈ C ∞ ([0, +∞), R 2 )
of the Cauchy problem [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF][START_REF]Wave packets in honeycomb structures and two-dimensional Dirac equations[END_REF].

Moreover, there holds

u λ (r), v λ (r) > 0, ∀r > 0, (21)
and

u λ (r) ∼ 1 r , v λ (r) ∼ 1 r 2 , as r → +∞, (22)
In particular,

ψ ∈ L p (R 2 , C 2 ), ∀p > 2, but ψ / ∈ L 2 (R 2 , C 2 ). Figure 1.
The trajectory of a representative solution of [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] with λ > 0.

The proof of (Prop. 7) will be achieved in several intermediate steps.

Local existence and uniqueness of solutions of ( 19) are guaranteed by the following

Lemma 8. Let λ > 0. There exist 0 < R λ +∞ and (u, v) ∈ C 1 ([0, R λ ), R 2 )
unique maximal solution to [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF], which depends continuously on λ and uniformly on [0, R] for any 0 < R < R λ .

Proof. We can rewrite the system in integral form as

       u(r) = 1 r ˆr 0 sv(s)(2β 2 u 2 (s) + β 1 v 2 (s))ds v(r) = λ - ˆr 0 u(s)(β 1 u 2 (s) + 2β 2 v 2 (s))ds (23)
where the r.h.s. is a Lipschitz continuous function with (u, v) ∈ C 1 . Then the claim follows by a contraction mapping argument, as in [START_REF] Cazenave | Existence of localized solutions for a classical nonlinear Dirac field[END_REF].

Given λ > 0, we will denote by (u λ , v λ ) the corresponding (maximal) solution. Dropping the singular term in [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] we obtain a hamiltonian system

u = v(2β 2 u 2 + β 1 v 2 ) v = -u(β 1 u 2 + 2β 2 v 2 ) ( 24
)
whose hamiltonian is given by

H(u, v) = β 1 4 (u 4 + v 4 ) + β 2 u 2 v 2 (25)
Consider H λ (r) := H(u λ (r), v λ (r)) (26) then a simple computation gives

Ḣλ (r) = - u λ (r) r (β 1 u 2 λ (r) + 2β 2 v 2 λ (r)) 0 ( 27
)
so that the energy H is non-increasing along the solutions of [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF]. This implies that ∀r ∈ [0, R x ), (u λ (r), v λ (r)) ∈ {H(u, v) H(0, λ)}, the latter being a compact set. Thus there holds Lemma 9. Every solution to [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] is global.

Remark 10. Smoothness of solutions follows by basic ODE theory.

Heuristically, [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] should reduce to (24) in the limit r → +∞ (u being bounded), that is, dropping the singular term in the first equation. The following lemma indeed shows that the solutions to [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] are close to the hamiltonian flow [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF] as r → +∞. The proof is the same as in [START_REF] Cazenave | Existence of localized solutions for a classical nonlinear Dirac field[END_REF]. Lemma 11. Let (f, g) be the solution of [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF] with initial data (f 0 , g 0 ). Let (u 0 n , v 0 n ) and ρ n be such that

ρ n n→+∞ -----→ +∞ and (u n , v n ) n→+∞ -----→ (f 0 , g 0 ) Consider the solution of    un + u n r + ρ n = (2β 2 u 2 n + β 1 v 2 n )v n vn = -(β 1 u 2 n + 2β 2 v 2 n )u n such that u n (0) = u 0 n and v n (0) = v 0 n . Then (u n , v n ) converges to (f, g) uniformly on bounded intervals.
Proposition 12. For any λ > 0, we have

u λ (r), v λ (r) > 0, ∀r > 0. ( 28
)
and

lim r→+∞ (u λ (r), v λ (r)) = (0, 0). ( 29
)
Proof. Using the equations in [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] one can compute

d dr (ru λ (r)v λ (r)) = β 1 r(v 4 λ -u 4 λ ), (30)
and

d dr (r 2 H λ (r)) = β 1 2 r(v 4 λ -u 4 λ ). (31)
Combining ( 30) and ( 31) and integrating gives

u λ (r)v λ (r) = 2rH λ (r) (32)
and ( 28) follows, H λ being positive definite. Combining [START_REF] Moloney | Nonlinear optics[END_REF] and the second equation in [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] one sees that vλ (r) 0 for all r > 0, and then

∃ lim r→+∞ v λ (r) =: µ 0. (33)
Moreover, since u is bounded, there exists a sequecnce r n ↑ +∞ such that

∃ lim n→+∞ u λ (r n ) = δ 0. ( 34
)
We claim that lim

r→+∞ u λ (r) = δ. ( 35
)
By contradiction, suppose that (35) does not hold. Then there exist ε > 0 and another sequence s n ↑ +∞ such that

|u λ (s n ) -δ| ε 0, ∀n ∈ N. (36)
Up to subsequences, we can suppose that

lim n→+∞ u λ (s n ) = γ = δ, (37)
for some γ 0. Recall that H decreases along the flow of ( 19), as shown in [START_REF] Mawhin | Critical point theory and Hamiltonian systems[END_REF], and then

∃ lim r→+∞ H λ (r) = h 0. ( 38
)
Then it follows that

(δ, µ), (γ, µ) ∈ {H(u, v) = h} . ( 39
)
It is easy to see that the algebraic equation for u

H(u, µ) = h, (40)
has (at most) one non-negative solution and thus δ = γ, reaching a contradiction. This proves the claim [START_REF] Wang | A remark on nonlinear dirac equation[END_REF], and then there holds

lim r→+∞ (u λ (r), v λ (r)) = (δ, µ). (41) Let (ρ n ) n ⊆ R be a sequence such that lim n→+∞ ρ n = +∞ , lim n→+∞ (u λ (ρ n ), v λ (ρ n )) = (δ, µ) (42)
and consider the solution (U, V ) to [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF] such that (U (0), V (0)) = (δ, µ).

By (Lemma 11), it follows that (u

λ (ρ n + •), v λ (ρ n + •)) converges uniformly to (U, V ) on bounded intervals. But since lim n→+∞ (u λ (ρ n + r), v λ (ρ n + r)) = (δ, µ), ∀r > 0, (43)
this implies that

(U (r)), V (r)) = (δ, µ), ∀r > 0 (44)
and thus (δ, µ) = (0, 0) as the latter is the only equilibrium of the hamiltonian system [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF]. This proves [START_REF] Peleg | Conical diffraction and gap solitons in honeycomb photonic lattices[END_REF].

The above proposition shows that the solutions of ( 19) actually correspond to localized solutions of the PDE [START_REF] Cazenave | Existence of localized solutions for a classical nonlinear Dirac field[END_REF]. The aim of the rest of the section is then to provide the exact asymptotic behavior.

Proposition 13. For large r > 0, there holds The second part of (45) follows by [START_REF] Struwe | Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]46), using the elementary inequality

1 r 2 u 2 λ (r) + v 2 λ (r) 1 r . ( 45
) Proof. Remark that (u 2 λ (r) + v 2 λ (r)) 2 ∼ H λ (r). ( 46
2u λ (r)v λ (r) u 2 λ (r) + v 2 λ (r), ∀r > 0.
The first equation in [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] can be rewritten as

d dr (ru λ (r)) = rv λ (r)(2β 2 u 2 λ (r) + β 1 v 2 λ (r)) (47)
Since v λ > 0, we deduce from (47) that the function f (r) := (ru λ (r)) is strictly increasing and thus

lim r→+∞ f (r) =: l ∈ (0, +∞] (48)
Suppose that l = +∞.

(49) This implies that

u λ (r) 1 r (50)
for r > 0 large. Combining (50) and (45), using the second equation in [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] we deduce that vλ (r) -

1 r 3 . ( 51
)
Using again the comparison principle, we conclude that

v λ (r) 1 r 2 (52)
for r > 0 large. By (45,52), integrating (47) gives

f (r) = ˆr 0 v λ (s) (2β 2 u 2 λ (s) + β 1 v 2 λ (s))s bounded ds ˆ+∞ 0 ds s 2 < +∞, ∀r > 0,
(53) thus contradicting (49). Then 0 < l < +∞, and this implies that

u λ (r) ∼ 1 r (54)
for large r > 0. Since (52) holds, using the second equation in [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] and (54) one gets

vλ (r) ∼ - 1 r 3 . (55)
and then for large r > 0, we have

v λ (r) ∼ 1 r 2 . ( 56
)
The integrability properties of the solution follow by the fact that

|ψ(r)| 2 = u 2 λ (r) + v 2 λ (r) ∼ 1 r 2
, as r -→ +∞. This concludes the proof of (Prop. 7), and thus of (Theorem 3).

Variational characterization

The solutions of (9) found in the previous section by dynamical systems methods admit a variational characterization. Indeed, one can prove that they are critical points of a suitable action functional. More precisely one can show that they are least action critical points of the corresponding action. In this sense they can be considered as ground state solutions. Our variational argument also provides an alternative, more sophisticated, existence proof. This is not only interesting in itself, but also gives more informations about the properties of those solutions. Remark 14. The argument presented in this section works for H 1 2 -solutions of (57) of arbitrary form. However, we focus on symmetric solutions of the form [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF] as in that case we can also provide the exact asymptotic behavior of solutions, by the method described in the previous section.

Theorem [START_REF] Evans | Partial differential equations[END_REF]. Equation ( 9) admits a family of smooth solutions in H 1 2 (R 2 , C 2 ), of the form [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF] and satisfying the decay estimates [START_REF] Erdős | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF]. Moreover, they coincide with the solutions found in the previous section (Theorem 3). This section is devoted to the proof of the above theorem. Some preliminary definitions are in order.

The system (9) can be written in a more compact form as:

Dψ = ∇G β 1 ,β 2 (ψ), ( 57
) with ψ = ψ 1 ψ 2 : R 2 -→ C 2 , where G β 1 ,β 2 (ψ) := β 1 4 (|ψ 1 | 4 + |ψ 2 | 4 ) + β 2 |ψ 1 | 2 |ψ 2 | 2 . ( 58
)
To simplify notations, in the sequel we omit the indices β j .

Here

D := -i( σ • ∇) (59) is the Dirac operator and σ • ∇ := σ 1 ∂ 1 + σ 2 ∂ 2 ,
where

σ 1 := 0 1 1 0 , σ 2 := 0 i -i 0 ( 60
)
are Pauli-type matrices 1 .

It is easy to see that (57) is, formally, the Euler-Lagrange equation of the action functional

L(ψ) := 1 2 ˆR2 ψ, Dψ dx - ˆR2 G(ψ)dx. (61)
We look for critical points of (61) belonging to the Sobolev space H 1 2 (R 2 , C 2 ), as this is a natural choice in view of the continuous embedding

H 1 2 (R 2 , C 2 ) → L 4 (R 2 , C 2 ). ( 62
)
given by the Gagliardo-Nirenberg inequality (see, e.g. [START_REF] Evans | Partial differential equations[END_REF]). Moreover, it is not hard to see that L ∈ C 1 ( H 1 2 (R 2 , C 2 )). More precisely, we will work with the closed subspace of functions satisfying [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF]:

E := ψ ∈ H 1 2 (R 2 , C 2 ) : ψ(r, ϑ) = iu(r)e iϑ v(r) , u, v : [0, +∞) -→ R , (63)
(r, ϑ) being polar coordinates in R 2 . To simplify the presentation, we will sometimes adopt the notation

ψ = (u, v) (64)
for ψ ∈ E, and more generally for spinors satisfying [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF]. We will often identify ψ with the pair (u, v). If ψ ∈ E, the action functional on E reads as

S(u, v) = L(ψ) 2π = ˆ+∞ 0 1 2 uv + uv r -u v -H(u, v) rdr ( 65
)
1 We could rewrite the equation ( 9) in terms of standard Pauli matrices σ1 = 0 1 1 0 , σ2 = 0 -i i 0 . This amounts to an unitary transformation on the spinor space C 2 and does not affect our argument. However we prefer not to do so, in order to remain consistent with the notations of [START_REF] Arbunich | Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures[END_REF].

where H is the hamiltonian defined in [START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF]. It is not hard to see that the Euler-Lagrange equation for (65) is given by the ODE [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF].

Looking for critical points of (65) one may try to prove that it has a linking geometry ( see e.g. in [START_REF] Esteban | Stationary states of the nonlinear Dirac equation: a variational approach[END_REF]). However, since this may not be straightforward we rather exploit the convexity of the hamiltonian H in order to use duality techniques. This allows us to easily define a minimax level, the dual functional possessing a mountain pass structure. Duality is a classical tool in the study of hamiltonian systems (see [START_REF] Mawhin | Critical point theory and Hamiltonian systems[END_REF][START_REF] Ekeland | Convexity methods in Hamiltonian mechanics[END_REF]), which turns out to be useful also for elliptic PDEs as shown, for instance, in [START_REF] Isobe | Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds[END_REF][START_REF] Ambrosetti | A note on the problem -∆u = λu + u|u| 2 * -2[END_REF].

Lemma 16. The function

H : (u, v) ∈ R 2 -→ H(u, v) ∈ R is convex. Proof. A simple computation gives det D 2 H(u, v) = 6β 1 β 2 (u 4 + v 4 ) + (9β 1 2 -12β 2 2 )u 2 v 2 . ( 66
)
Recall that 0 < β 2 β 1 , and then by (66)

det D 2 H(u, v) > β 2 2 6(u 4 + v 4 ) -3u 2 v 2 > 9 2 β 2 2 (u 4 + v 4 ) (67)
thanks to the elementary inequality 2u 2 v 2 u 4 + v 4 , and the claim follows.

We can thus define the Legendre transform H * : R 2 -→ R ∪ {+∞} of H as the function

H * (w, z) = sup{ (u, v), (w, z) R 2 -H(u, v) : (u, v) ∈ R 2 } (68)
The hamiltonian H is a homogeneous polynomial of degree 4. This implies that H * is everywhere finite and, thanks to basic scaling properties of the Legendre transform, it is homogeneous of degree 4 3 . Moreover, since H(0, 0) = 0, it immediately follows from (68) that H * is positive definite. We collect those remarks in the following Proposition 17. The function H * is everywhere finite, positive definite and homogeneous of degree 4 3 . Consider the functional, defined for (u, v) ∈ L 4 (R + , rdr) 2 as

H(u, v) := ˆ+∞ 0 H(u, v)rdr (69)
Its Legendre transform (or dual) is the functional

H * : L 4 3 (R + , rdr) 2 -→ R (70)
defined (with an abuse of notation) as

H * (w, z) : = sup{ (u, v), (w, z) L 4 ×L 4 3 -H(u, v) : (u, v) ∈ L 4 (R + , rdr) 2 } = ˆ+∞ 0 H * (w, z)rdr (
Consider the following isomorphism

D : E -→ E * , (73)
and its inverse

A := D -1 : E * -→ E. (74) where E * is the dual of E. Let j : E -→ L 4 (R + , rdr) 2 (75)
be the Sobolev embedding. Consider the following sequence of maps

K : L 4 3 (R + , rdr) 2 E * E L 4 (R + , rdr) 2 , j * A j (76)
The action functional (65) can be rewritten as

S(u, v) = 1 2 (w, z), D(w, z) E×E -H(j(u, v)), (u, v) ∈ E. ( 77
)
Then for ψ = (u, v) ∈ E the differential of S reads as

dS(ψ) = Dψ -j * dH(j(ψ)) ∈ E * . ( 78
)
We finally define the dual action functional Proof. Let ψ ∈ E be a critical point of S. Then by (78), we have Dψ = j * dH(j(ψ)). Define ϕ = dH(j(ψ)) ∈ L 4 3 (R + , rdr) 2 , so that Dψ = j * (ϕ). This implies that ψ = A • j * (ϕ) and

S * (w, z) : = H * (w, z) - 1 2 K(w, z), (w, z) L 4 ×L
j(ψ) = j • A • j * (ϕ) = K(ϕ). ( 80
)
On the other hand, by (72) we have

j(ψ) = dH * (ϕ). ( 81
)
Combining ( 80) and ( 81) we obtain

dS * (ϕ) = dH * (ϕ) -K(ϕ) = 0, ( 82
)
and then ϕ is a critical point of S * . Conversely, suppose ϕ ∈ L 4 3 (R + , rdr) 2 is a critical point of S * , and define

ψ = A • j * (ϕ) ∈ E. Since ϕ is a critical point, we have dH * (ϕ) -K(ϕ) = 0. Then (72) implies that ϕ = dH • K(ϕ) = dH • j • A • j * (ϕ) = dH(j(ψ)). ( 83
)
We have j * (ϕ) = j * • dH(j(ψ)) and Dψ = j * • dH(j(ψ)), and thus ψ is a critical point of S.

Remark 19. More generally, S and S * have the same compactness properties and there is a one-to-one correspondence between their Palais-Smale sequences (see, e.g. [START_REF] Isobe | Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds[END_REF] for more details).

Since finding a critical point of S is equivalent to finding a critical point of the dual functional S * we will focus on the latter, which has a simpler structure. More precisely, we will exploit the homogeneity properties of S * using a Nehari-manifold argument (see, e.g. [START_REF] Szulkin | The method of Nehari manifold[END_REF] and references therein). However, the fact that the second integral in (79) is not positive definite must be taken into account. We remark that a Nehari-type argument has been previously used by Ding and Ruf [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF], in the study of semiclassical states for critical Dirac equations.

We have pointed out (Remark 5) that the equation (57), and thus the functional (61), is scale-invariant. The same holds, of course, for the dual action S * . Indeed, one can verify that it is invariant with respect to the following scaling

ψ(•) → ψ δ (•) := δ 3 2 ψ(δ•), δ > 0.
(84) Moreover, even if the functional (61) is invariant by translation, this is no longer true for (79) thanks to the ansatz [START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF]. Thus scaling is the only (local) symmetry which may prevent strong convergence in our variational procedure. In what follows we only sketch the rest of the proof, as it is based on standard arguments from concentration-compactness theory [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF][START_REF] Struwe | Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF].

First of all, it easy to see that the functional S * possesses a mountain-pass geometry.

Lemma 20. There exists ρ > 0 such that α := inf{S * (w, z) : (w, z) ∈ L Proof. Recall that the dual functional is defined as

S * (w, z) = ˆ+∞ 0 H * (w, z)rdr - 1 2 ˆ+∞ 0 K(w, z), (w, z) rdr for (w, z) ∈ L 4 3 (R + , rdr) 2
. Since H * is homogeneous of degree4 3 , as already remarked, the first assertion follows is ρ > 0 is sufficiently small, the other term being quadratic. The second part of the claim follows immediately, for the same reason.

In view of the above lemma it is natural to define the mountain-pass level for S * as

c := inf max t 0 S * (t(w, z)) : (w, z) ∈ L 4 3 (R + , rdr) 2 , ˆ+∞ 0 K(w, z), (w, z) rdr > 0 .
Remark that there holds

max t 0 S * (t(w, z)) α > 0, ∀(w, z) ∈ L
where α > 0 is as in Lemma 20. Then we have

c α > 0. (86)
Moreover, the homogeneity properties of the terms appearing in S * imply that c = inf

(w,z)∈N S * (w, z) > 0, ( 87
)
where N is the Nehari manifold

N := {(w, z) ∈ L 4 3 (R + , rdr) 2 \ {0} : dS * (w, z), (w, z) = 0}. ( 88
)
We are thus led to study the minimization problem (87). Let (w n , z n ) n∈N ⊆ N be a minimizing sequence for S * . By Ekeland's variational principle (see [START_REF] Ekeland | Convexity methods in Hamiltonian mechanics[END_REF]), we can assume that it actually is a Ceramisequence, that is:

S * (w n , z n ) -→ c, (1 + (w n , z n ) L 4 3 )dS * (w n , z n ) L 4 -→ 0, as n -→ ∞. ( 89
) Proposition 21. The sequence (w n , z n ) n∈N ⊆ N is bounded in L 4 3 (R + , rdr) 2 .
Proof. We have

dS * (w n , z n ) = ∇H * (w n , w n ) -K(w n , z n ).
Since the function H * is 4 3 -homogeneous there holds

∇H * (w n , w n ), (w n , z n ) = 4 3 H * (w n , z n ).
Then by the definition of the Nehari manifold (88), it follows that

S * (w n , z n ) = 1 3 ˆ+∞ 0 H * (w n , z n )rdr. ( 90
)
The claim thus follows because

ˆ+∞ 0 H * (w n , z n )rdr ∼ (w n , z n ) 4 3 L 4 3 , (91)
and (w n , z n ) n∈N is a minimizing sequence.

By the above lemma we may assume that

(w n , z n ) (w, z), weakly in L 4 3 (R + , rdr) 2 , (92)
as n → +∞. One needs to study the concentration behavior of the minimizing sequence in order to prove strong L 4 3 -convergence. We already remarked that scaling invariance may prevent strong convergence, as Cerami sequences may blow-up around some points. Since we are essentially working with radial functions, concentration may only occur at the origin. More precisely (recall (91)), there holds

H * (w n , z n )rdr =: ν n ν := H * (w, z)rdr + α 0 δ 0 , (93)
weakly in the sense of measures, where δ 0 is a Dirac mass concentrated at the origin and α 0 0.

Recall that ˆ+∞

0 H * (w n , z n )rdr = 3S * (w n , z n ) → 3c, as n → +∞. ( 94
)
Suppose that the minimizing sequence (w n , z n ) n∈N splits into two bumps, one of them centered around the origin and the other one carrying a positive part of the "mass" at infinity (the dichotomy case [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF]). More precisley, assume that there exist 0 < b < 3c, and two sequences of radii r n , r n → +∞, with rn

r n → 0 such that ˆrn 0 H * (w n , z n )rdr → b, ˆr n rn H * (w n , z n )rdr → 0,
as n → +∞.

(95) Take a cutoff function θ ∈ C ∞ c ([0, ∞)), 0 θ 1 such that θ ≡ 1 on [0, 1] and θ ≡ 0 on [2, ∞), and define

(w 1 n , z 1 n )(r) := θ r r n (w n , z n ), (w 2 n , z 2 n ) := 1 -θ r r n (w n , z n )(r).
(96) There holds

S * (w n , z n ) -S * (w 1 n , z 1 n ) -S * (w 2 n , z 2 n ) → 0, as n → +∞,
and both sequences in (96) are Cerami sequences for the functional S * , that is 0

< S * (w k n , z k n ) → c k < c, (97) and 1
+ (w k n , z k n ) L 4 3 dS * (w k n , z k n ) L 4 -→ 0, (98)
as n → +∞, with k = 1, 2.

Remark 22. The above estimates can be worked out (along the same lines as in [26, Section 2.1]) recalling that the operator K in (79) acts as D -1 and exploiting the decay of the corresponding Green kernel G(x, y) = -1 2π

x -y |x -y| 2 •, where the dot indicates the Clifford product (see Remark 6).

Consider, for instance, the sequence (w 1 n , z 1 n ) n∈N . Then the condition (98) implies that

dS * (w 1 n , z 1 n ), (w 1 n , z 1 n ) L 4 ×L 4 3 -→ 0, as n → +∞, (99)
that is, the sequence is asymptotically on the Nehari manifold N . Moreover, there exists a sequence t n > 1 such that t n (w 1 n , z 1 n ) ∈ N , ∀n ∈ N ( see, e.g. [START_REF] Szulkin | The method of Nehari manifold[END_REF] ). Recalling that H * is 4 3 -homogeneous, the definition (88) of N then gives 4 3t

2 3 n ˆ∞ 0 H * (w 1 n , z 1 n )rdr = ˆ∞ 0 K(w 1 n , z 1 n ), (w 1 n , z 1
Up to suitably rescaling the sequence, we may assume that the weak limit is non-trivial, that is, vanishing is also excluded [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF]. Indeed, one can find a sequence λ n > 0, n ∈ N such that for the rescaled spinor

( w n , z n )(•) := λ 3 2 n (w n , z n )(λ n •) (104)
there holds

Q n (1) = ˆ1 0 H * ( w n , z n )rdr = c, ∀n ∈ N, (105)
where Q n (•) is the concentration function of ( w n , z n ) [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The locally compact case. I[END_REF][START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF]. Assume that (w, z) = (0, 0). Then by (93) we have ν = α 0 δ 0 , and the normalization (105) gives α 0 c. Then (93) and (103) imply

3c = ˆR2 dν = α 0 c, (106)
which is clearly absurd. This allows us to conclude that (w, z) = (0, 0),

that is, the above normalization (105) rules out the vanishing case. The last step in order to conclude the strong convergence of the minimizing sequence (w n , z n ) n∈N is to show that actually α 0 = 0 in (93). If this is not the case, since by (105) there holds 0 < α 0 c, this property and the tightness of the sequence (dν n ) n∈N imply that the sequence (w n , z n ) n∈N splits into two parts, one blowing up at the origin, as n → +∞, and concentrating a portion α 0 of the mass at that point, and another non-trivial part carrying the rest of the mass, essentially localized in an interval of the form [1, R] (corresponding to an annulus in R 2 ), for some R > 0. Exploiting again the scale-invariance of the problem, one can suitably rescale the sequence, as in (104), removing the blowup at the origin, and at the same time "sending at infinity" the bump localized in [1, R]. In this way we have created a sequence for which dichotomy holds (see (95)), and this is not possible, as already shown.

Finally, we conclude that α 0 = 0 in (93) and then

(w n , z n ) L 4 3 --→ (w, z) ∈ N as n → +∞. ( 108
)
Thus S * (w, z) = min N S * , and correspondingly

(u, v) = (A • j * ) (w, z), is a critical point of S.
Remark 23. Since the Nehari manifold N contains all critical points of S * and S * (w, z) = min N S * , we conclude that (w, z) is a least action critical point of S * . In this sense it can be considered a sort of ground state. The same remark holds for (u, v), as a critical point of S.

Since we are dealing with a critical equation, smoothness of solutions is not authomatic as standard bootstrap arguments do not apply. Anyway, the regularity result proven in [START_REF] Wang | A remark on nonlinear dirac equation[END_REF] (which holds for weak solutions in L 4 ) ensures that (u, v) actually is of class C ∞ . Being a critical point of S, (u, v) solves the Euler-Lagrange equation [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF], and smoothness forces

u(0) = 0, (109)
as we cannot have singularities. Moreover, since (u, v) is a non-trivial solution, necessarily v(0) = 0. (110) Assume, for instance, v(0) = λ > 0. Since the equation is scale-invariant and odd, as anticipated in Remark (5), we get a continuous family of (nontrivial) solutions (u λ , v λ ) parametrized by λ = 0, applying those symmetries. Uniqueness for [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] then allows to conclude the proof.

The massive case

Dirac points are protected by particular symmetries [START_REF] Cayssol | Introduction to dirac materials and topological insulators[END_REF], and this is the case for PT -symmetry (parity + time inversion) for honeycomb Schrödinger operators. Indeed, in [START_REF] Fefferman | Honeycomb schrödinger operators in the strong binding regime[END_REF][START_REF] Fefferman | Honeycomb lattice potentials and dirac points[END_REF] it is proved that a suitable perturbation breaking such symmetry creates a gap, thus lifting the conical degeneracy. This results in a mass term in the effective Dirac operator (59) 2 :

D m := -i σ • ∇ + m σ 3 , σ 3 := -1 0 0 1 , (111)
as it can be seen (at least formally) using a multiscale expansion analogous to the one done in [START_REF] Fefferman | Topologically protected states in one-dimensional continuous systems and Dirac points[END_REF][START_REF] Ilan | Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations, and effective media[END_REF]. In [START_REF] Borrelli | Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity[END_REF] we proved the existence of solitary waves for the massive variant of ( 9), with a pure cubic nonlinearity, corresponding to the choice

β 1 = 1, β 2 = 1 2 .
This particular case is a model example as it exhibits all the analytical difficulties, being H 1 2 -critical. Our proof in [START_REF] Borrelli | Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity[END_REF] takes advantages of the ansatz (10) in order to solve compactness issues, and it is based on dynamical systems techniques. More precisely, after a suitable rescaling we exploit the properties of the explicit solution to the (massless) limiting problem. The analysis carried out in the present paper allows us to deal with the general case: 0 < β 2 β 1 , 112) is odd and thus there actually are two distinct solutions, due to this symmetry property. Notice that the mass term in (111) breaks the conformal invariance of the equation, so that the existence of infinitely many solutions does not follow authomatically in this case. This still is an open problem, to our knowledge.

The theorem can be proved studying (113) thanks to the shooting argument given in [START_REF] Borrelli | Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity[END_REF]. The delicate part consists in controlling the error committed in approximating the solution with that of the limiting problem [START_REF]Waves in honeycomb structures, Journées équations aux dérivées partielles[END_REF] on a suitable interval. However, thanks to the asymptotic behavior [START_REF] Erdős | Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems[END_REF], the proof follows with minor modifications (see Section 2.2 and Appendix in [START_REF] Borrelli | Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity[END_REF]).

  ) Moreover, by[START_REF]The concentration-compactness principle in the calculus of variations. The limit case. I[END_REF][START_REF] Mawhin | Critical point theory and Hamiltonian systems[END_REF] one gets Ḣλ (r) -4 H λ (r) r , and the comparison principle for ODEs implies that H λ (r) 1 r 4 and thus by (46), we get the first inequality in (45).

4 3 4 3

 44 stands for the duality product. There holdsdH • dH * = id L , dH * • dH = id L 4 .

4 3 4 3 2 . 18 . 4 3

 442184 (R + , rdr) 2 ,which is of class C 1 on L (R + , rdr)Proposition There is a one-to-one correspondence between critical points of S in E and critical points of S * in L (R + , rdr) 2 .

4 3 ( 3 = 4 3

 334 R + , rdr) 2 , (w, z) 4 ρ} > 0.Moreover, for (w, z) ∈ L (R + , rdr) 2 such that ´+∞ 0 K(w, z), (w, z) rdr > 0, there holds lim t→+∞ S * (t(w, z)) = -∞.

2Theorem 24 .

 24 As already remarked, one can express everything in term of standard Pauli matrices by exchanging spinor components. and thus with the equationi∂ t α + D m α = ∇G β 1 ,β 2 (α),with G β 1 ,β 2 is defined as in (58). Looking for stationary solutionsα(t, x) = e iωt ψ(x), (t, x) ∈ R × R 2 ,gives to the following equation for ψ(•)(D m -ω)ψ = ∇G β 1 ,β 2 (ψ) (112)where the frequency lies in the spectral gap ω ∈ (-m, m), as the spectrum of D m is given byσ(D m ) = (-∞, -m] ∪ [m, +∞),see e.g.[START_REF] Thaller | The Dirac equation[END_REF].Exploiting the ansatz[START_REF] Ding | Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities[END_REF] one is lead to study the flow of the following system 2 u 2 + β 1 v 2 )v -(m -ω)v v = -(β 1 u 2 + 2β 2 v 2 )u -(m + ω)u(113)complemented with initial data u(0) = 0, v(0) = λ = 0.The following result holds: For each ω ∈ (-m, m), equation (112) admits a smooth localized solution, with exponential decay at infinity.Remark 25. Equation (

  for n large, contradicting the minimality of c = inf N S * . This means that dichotomy cannot occur, and then the sequence of measures (dν n ) n∈N in (93) is tight. Consequently, up to extraction, (94) gives ˆR2 dν = 3c.

	and thus the first condition in (95) implies that
		lim n→+∞	t n = 1.	(102)
	Then since t n (w 1 n , z 1 n ) ∈ N , by (95) and (102) there holds
	S * (t n (w 2 n , z 2 n )) =	t n 4 3 3	ˆ+∞ 0	H * (w 1 n , z 1 n )rdr ∈ (0, c),
					n ) rdr,	∀n ∈ N. (100)
	Combining (99) and (100) one gets 1 -t -2 3 n 4 3 ˆ∞ 0 H * (w 1 n , z 1 n )rdr -→ 0,	as n → +∞,	(101)

3 (R + , rdr) 2 ,(85)


Giuseppe Iurato
email: giuseppe.iurato@community.unipa.it

Discussing on the origins of symbolism from the latest paleoanthropological research: the case of Homo naledi

From an historical-critical comparison of some data and certain results coming from paleoanthropology and neurosciences, we would like to do some possible remarks and putting forward some simple suggestions about the early origins of symbolic function starting from the recent discovery of a new species of the genus Homo, called Naledi.

The recent, interesting paleoanthropological discovery of the species Homo naledi, allows to open a fruitful discussion on the possible origins of human symbolic function. Indeed, if we accept all the hypotheses put forward by the paleoanthropologist Lee R. Berger and his team (Berger et al., 2015), which we deem quite reasonable 1 , then new insights may come from the outcomes emerging from the analysis of the historical reconstruction around Naledi.

), this remote chamber contained a large bone bed of an extinct population, finding more than 1,500 fossil specimens, one of the most significant finds in the history of human evolution. That remarkable discovery has triggered a fiery discussion within anthropological community, above all because these new species' traits seem do not fit well into the human evolutionary tree. So, it has been hypothesized that these be the remnants of a new species of the genus Homo, said to be Homo naledi, featured by a bodily mass and stature similar to those of modern species of the genus Homo, but with a smaller endocranial volume similar to that of Australopithecus species, with a skull shape conformally similar to early Homo species, but with a volume approximately half that of modern human skulls, quite equal to that of the latest hominins. The skeletal anatomic pattern of the specimens of this find is quite unique, with a confusing mix of seemingly primitive and derived traits that, in evolutionary biology, is known as anatomic mosaic. This species has several anatomical details already known in early species of our genus, such as Homo habilis and early members of Homo erectus, but it has a much smaller brain size with respect to that typical of these latter species, as in several aspects of its teeth Naledi resembles some species that branched from our family tree much earlier in the timeline, such as the 3.2-million-year-old Australopithecus afarensis, i.e., the species of the famous "Lucy" fossil skeleton. Until now, however, the fossils have not been chronometrically dated, but estimates, derived from a first statistical analysis of cranial traits, have yielded a range of 912,000 years to 2 million years (with ± 0.5 million years). However, more accurate chronological datings are in progress.

Paleoanthropologists have already learned, since the late of the past century, that human evolution was not a gradual progression from a very apelike ancestor to modern humans, even to suppose a kind of shrub growth of species 2 . For instance, our small canine teethes and a more upright posture evolved very early in our lineage, while the bipedalism pattern came next. At the midpoint of the human evolutionary tree, in our ancestors evolved larger molar and premolar teethes, a trend that appeared around when the first members of the genus Homo, just started hunting and making stone

tools. Only in this time, they developed that first forms of social sharing which led to the language, characterizing today all the people around the world. The brain evolved late, the legs early, with every species in our ancestry equipped with its own set of characterizing features.

Nevertheless, Naledi's mosaic seems to be in conflict with this storyline, with wrists and fingertips more humanlike than those of Homo habilis, but combined with very curved fingers, comparable to those of both the very earliest hominins and to living apes. At the same time, Naledi seems to show a shoulder canted upward on the trunk typical of a climbing species, but with an upper arm twisted in a way unlike other human relatives. With a narrowed upper thorax cage, Naledi appears to have been a climber and a possible toolmaker, even if any stone artifacts, into the Dinaledi chamber, have not been found out. Then, legs are long and comparatively slender, with some evidence for an elongation of the lower leg, but the hips substantially share an anatomical pattern with the muchshorter-statured Lucy's skeleton. Adult individuals of Naledi weighed between 40 and 55 kilograms, and were between 140 and 160 centimetres in stature, with males and females differing only slightly in size. But to be emphasized is that Naledi had a very small brain for this size. The overall picture provided by all the results so recollected, is that Naledi walked more or less like humans do, seems to have had hands well made for handling and manipulating objects, and had teethes that indicate a high-quality diet, all elements these that link this species to our genus. Yet, it had a trunk, hips, shoulders and fingers that contrast with this picture, and, above all, it had a brain similar in size to that of some of the latest branches of the hominin lineage, like Australopithecines. In many respects, Naledi was adapted like a human, but without anything like a human brain.

Berger's team has not revealed in Naledi's bones any trace of carnivore activity, as well as noted that the sediments in Dinaledi chamber represent an isolated depositional environment, different from the nearby chambers, which were not carried into the chamber by water action, so Berger and colleagues hypothesised that Naledi were using this chamber just to deposit their dead. Therefore, according to this hypothesis, Naledi would have already had ritualistic behaviours highly symbolic, so that they had gained a certain degree of the symbolic function notwithstanding the reduced brain size, inasmuch as sacred rituals, like the funerary ones, require a certain development of symbolic capability socially shared 3 .

This hypothesis would entail a certain numbers of interesting historical consequences. Firstly, the birth of symbolic function would date back even the earliest steps of human evolution, just at the beginnings of the first species of the genus Homo 4 , contrarily to the current hypothesis according to which symbolic culture was born with the rising of Homo sapiens, around Neolithic 5 . Secondly, the symbolic function precedes every other abstract mental functions, like writing and counting, as, for example, first countings bring back to the late Paleolithic when the coming of agriculture gives rise to the crucial passage from hunter-gatherer societies to farmer ones, almost at the same time of the rising of writing, but independently of each other 6 .

Furthermore, according to Jean Piaget 7 , symbolic function should also precede language as meant in its widest sense, in that, any possible form of language (oral, spoken, mythological, gestural, etc.)

3 Cf. (Barnard 2012) and (Segalen 1998, 2002). 4 As is well-known, funerary rituals are present in many mammal species but with an increasing complexity gradually we proceed along primate evolution lineage. Thus, we might infer that symbolic function (in its early mythological expression) has its origins from ritualistic practices and usages which embed their roots into primate evolution lineage stages prior to hominid ramification. So, we may also infer that, since hominins, first raw defence psychic mechanisms (mainly related with anguish or anxiety, due to death fear) were already present, at least, in latest hominin species, like Naledi, and responsible, therefore, of the origins of symbolic function, for example, by means of disavowal mechanism, as outlined in (Iurato 2016b), which is a psychic mechanism just based on anxiety or anguish. 5 Although, as early in Homo erectus we may find first expressions of the symbolic function (according to Claude Levi-Strauss), in earliest cooking practices and clothing usages (starting from underwear: e.g., koteka). 6 See (Iurato 2016a) and references therein. 7 Already (Hallpike 1979, 1984) had applied, with success, Piagetian theories on symbolism to the study of primitive societies, reaching to interesting and original conclusions coherent with many points of our presentation.

always presupposes the preliminary existence of a symbolic underpinning as a main communication support or substratum 8 . In turn, language should precede writing and counting 9 in that there exist languages devoid of any writing system, and, on the other hand, as cognitive neurosciences say 10 , there not exist precise counting systems which have not a corresponding verbal system of counting. As recent neurosciences say us 11 , one of the chief neuronal circuits of consciousness seems to involve mainly frontal and parietal cerebral zones, above all in the exploitation of symbolic function. Now, Berger and colleagues (Berger et al. 2015) have inferred that Nadeli's skull is much closer, in cranial volume, to Australopithecine one, but with a cranial structure more similar to that of genus Homo than to that of Australopithecine, particularly in its slender features, with presence of temporal and occipital bossing (which are a weak indicators of the occurrence of such cerebral zones), and with a slight post-orbital constriction 12 . So, we would be tempted to put forward the hypothesis for which, already in Homo naledi, all the cerebral areas were present according to a geometrical-anatomical disposition topographically homotopic but not isometrically homeomorphic to that of the latest species of genus Homo, and that it might have been undergone subsequently to an evolutionary process of non-uniform enlargement of cerebral areas, like the parietal ones (as, it seems that, from recent paleoneurological inferences based on correlations between cranial and cerebral anatomy, just parietal lobe dilation is responsible for the major differences in neurocranial morphology between modern and non-modern humans 13 ) and the occipital 14 ones (in that, it seems too that Neanderthals had relatively larger occipital lobes with respect to modern humans 15 ). Likewise, it has to be considered too the relevant rising of the cerebellum in Sapiens with respect to Neanderthal 16 . Furthermore, since the earliest studies in paleoneurology, the parietal lobes have been recognized to be surprisingly variable among hominids, showing too marked differences among human species when analyzed with traditional or geometrical approaches. It is worth to highlight that, although lower parietal regions, like the supramarginal and the angular gyrus, have a relevant cognitive role in our species (like in speech understanding), until now however the paleoneurological research has evidenced no identified differences, in their rough morphology, between modern humans and other large brained hominids, like Neanderthals. Instead, morphological changes in the parietal regions among hominoids are probably associated with deep and upper parietal volumes, like the precuneus 8 See also (Barnard 2016) and the question he posed (at page 238) about the possible influence of mythology in the birth of language, to which we might answering, on the basis of this latest paleoanthropological discovery, in favour of a possible net precedence of symbolic function with respect to any other abstract psychic function (like language, writing, counting, etc.). All this, in agreement with Piaget's theory, already called into question in (Hallpike 1979, 1984) about functioning of primitive mind, as well as with philosophical thought which has always given a precedence to symbolism as a primary psychic function since primitive epoch. Indeed, also with the support of psychoanalytic thought (cf. (Iurato 2016b)), it seems that, in human being, the aesthetic moment has a predominant role with respect to any other psychic manifestation. Some philosophical trends have always supported the thesis of a pre-eminence of the symbolic function with respect to the other mental functions: for example, as early Giambattista Vico claimed the pre-eminence of symbolism in primitive humans (in this regard, see also (Mazur 2014, Introduction)), then followed by both modern anthropology (for example, with the natural symbolism of M. Douglas) and mythological studies of J.J. Bachofen, till to modern philosophy (with M. Scheler -who considers, in agreement later with E. Fromm, symbolism as a universal grammar of interpersonal communication of the affects -and E. Cassirer) and psychoanalytic theories of Freud and Jung. 9 Which seem to be arisen almost at the same time (Iurato 2016a). 10 See (Dehaene & Brannon 2011). 11 See (Dehaene & Brannon 2011). 12 Which is a physical anthropology parameter estimating the breadth of frontal zone. 13 See (Bruner et al. 2014). 14 This may explain why, in the human brain, about the 50% of the neurons, is more or less promptly correlated with visual images (Maffei 2014). 15 See (Bruner et al. 2014). 16 See (Barton & Venditti 2014), in which it is argued that cerebellum underwent a rapid size increase throughout the evolution of apes, including humans, expanding significantly faster than predicted by the change in neocortex size.

or the intraparietal sulcus (IPS), which are largely involved in visual-spatial integration and, the latter, in symbolic function (to be precise, in numerical symbolism17 ). The evolution of human brain has always represented a challenge in biology, which, in different periods throughout the long phylogenetic history of the human genus, has identified a generalized encephalization process associated with increasing behavioural complexity. So, the variations of the brain volume from about the 500 cc of Naledi and Habilis to the 1,100 cc of Erectus, hence to the 1,500 cc of Neanderthal, until reaching the 1,350 cc of Sapiens, might be, for instance, interpreted as an evolutionary process regarding the architecture of a brain pattern (to be precise, the Naledi's one) with roughly the same anatomic structure, characterized approximately by the same (conformal) geometry of cerebral regions and layers (in its tripartite model), but, above all, subjected to either volume and surface transformations of these latter (besides to changes in biochemical composition), with an increasing complexity in either (above all, neocortex) neuronal connections (as well as neuronal density) and ruffle/folding of cortex surface (with an increasing in number and complexity of cortex sulci/gyri and of the specific cerebral surface) 18 . Coming back to Naledi, with it we have therefore an early species of the genus Homo with a body quite similar to the latest species of the this genus, but with a brain similar, in dimensions, to that of latest hominins notwithstanding its skull be morphologically quite similar to that of genus Homo. This new species, therefore, has deeply revolutionized human evolution lineage, changing too our knowledge about human symbolic function. Just reflecting on the agile and well-developed corporal constitution of Naledi, as well as on its raw capability to make use of symbolic function, we are led toward those psychoanalytic theories which give a pre-eminence to human body (and its perception) in the formation and development of the Ego19 (bodily Ego) and its psychic functions, this being coherent, as has been said just above, with the great development underwent by parietal and occipital cerebral areas, occurred along the main trait of evolution lineage of the species of the genus Homo, i.e., from Habilis to Erectus, until Neanderthal and Sapiens, coherently too with the great growth of cerebellum along this evolutionary pathway, until up to may speak of an embodied cognitive evolution of humans 20 .

Above all, by the geological features of this discovery (Dirks et al.

2015). Indeed, Naledi's fossils were just found in a South Africa cave, called the Dinaledi Chamber, placed at about 1,450 meters underground. It is hardly attainable, and is still today accessed only through a complicated series of steep, twisting, and narrow tunnels. This is quite enough to corroborate a volunteer and intentional aim to reaching this place, not due to other motifs (of protection, shelter, etc.). 2 For the theory of evolutive shrub of humans, see, for example, (Pievani 2016).

See (Dehaene & Brannon 2011).

These conjectures about Naledi's neurobiology, have just been recently supported by the latest paleoanthropological research(Holloway et al., 2018).

See (Iurato

2016b) and references therein. 20 See (Barton 2012).

• Barnard, A. (2012). Genesis of Symbolic Thought. Cambridge, UK: Cambridge University Press.

• Barnard, A. (2016). Language in Prehistory. Cambridge, UK: Cambridge University Press.

• Barton, R.A. (2012). Embodied cognitive evolution and the cerebellum. Philosophical Transactions of the Royal Society of London. Series B. Biological Sciences, 367(1599): 2097-2107.

• Barton, R.A. and Venditti, C. (2014). Rapid evolution of the cerebellum in humans and other great apes. Current Biology, 24(20): 2440-2444. • Berger, L.R. et al. (2015). Homo naledi, a new species of the genus Homo from the Dinaledi Chamber, South Africa. eLife, 4, e09560.

• Bruner, E., de la Cuétara, J.M., Masters, M., Amano, H. and Ogihara, N. ( 2014 Translation: (2002). Riti e rituali contemporanei. Bologna, IT: Società editrice il Mulino).