Gabriele Mancini
email: gabriele.mancini@math.unipd.it

Pierre-Damien Thizy
email: pierre-damien.thizy@univ-lyon1.fr

GLUEING A PEAK TO A NON-ZERO LIMITING PROFILE FOR A CRITICAL MOSER-TRUDINGER EQUATION

Keywords: Mathematics Subject Classication. 35B33, 35B44, 35J15, 35J61

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Let Ω be a smooth bounded domain in R 2 . In this work we are interested in investigating the behavior of sequences of solutions of nonlinear critical elliptic problems of the form ∆u = f γ (x, u) , u > 0 in Ω , u = 0 on ∂Ω , (0.1)

where ∆ = -div(∇•) and (f γ ) γ is a sequence of Moser-Trudinger type nonlinearities.

A typical but very specic example of such a sequence is given by

         f γ (x, u) = β γ h γ (x)u exp(u 2 ) , lim γ→+∞ β γ = β ∞ ≥ 0 , lim γ→+∞ h γ = h ∞ in C 2 ( Ω) , h ∞ > 0 in Ω , (0.2)
where the β γ 's are positive numbers and the h γ 's are positive functions in C 2 ( Ω).

Recall that nonlinearities as in (0.2) arise when looking for critical points of the Moser-Trudinger functional any sequence (u γ ) γ satisfying (0.1) and converging weakly in H 1 0 to some u ∞ , the Dirichlet energy is quantied (see also [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities I-Quantication and location of concentration points[END_REF]Section 2]). Namely, up to a subsequence, there exists an integer N ≥ 0 such that

lim γ→+∞ Ω |∇u γ | 2 dy = 4πN + Ω |∇u ∞ | 2 dy . (0.3)
Observe that such a sequence (u γ ) γ is compact in H 1 0 , if and only if it is uniformly bounded, and if and only if N = 0 in (0.3). As a consequence of the very strong interaction generated by an exponentially critical nonlinearity, it is not clear in general whether it is possible to have loss of compactness, i.e. N ≥ 1 in (0.3), together with u ∞ ≡ 0. For instance, for the typical nonlinearities f γ given by (0.2), in order to understand globally the bifurcation diagram of (0.1) and the associated questions of existence of solutions (see [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities II-Existence of solutions of high energies[END_REF]), Druet-Thizy [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities I-Quantication and location of concentration points[END_REF] pushed further the analysis and proved that, for such a noncompact sequence (u γ ) γ , there necessarily holds that u ∞ ≡ 0, so that the limit of the Dirichlet energy in (0.3) has to be 4π times an integer N > 0. In contrast, the purpose of this paper is to show that for dierent families of exponentially critical nonlinearities it is possible to construct bubbling sequences of solutions with non-trivial weak limit in H 1 0 . Let Ω be the unit disk of R 2 centered at 0 and let 0 < λ 1 < λ 2 < ... be the sequence of the simple radial eigenvalues of ∆ in Ω, with zero Dirichlet boundary condition. Let v 1 , v 2 ... be the associated radial eigenfunctions uniquely determined by Ω v 2 k dy = 1 and v k (0) > 0 for all k. Our rst result shows that there exists a sequence (f γ ) γ with Moser-Trudinger type growth for which (0.1) admits a sequence (u γ ) γ of positive radial solutions converging weakly to a multiple of v 1 and with Dirichlet energy approaching any xed value in (4π, +∞).

Theorem 0.1 (Positive case). Let l > 0 be given and let Ω ⊂ R 2 be the unit disk centered at 0. Let λγ be given by λγ = λ 1 -ε γ , (0.4) where ε γ = 4πv1(0) γ λ1 l . Then there exists β γ > 0 such that the equation ∆u = λγ u + β γ u exp(u 2 ) , u > 0 in Ω , u = 0 on ∂Ω , (0.5) admits a smooth solution u γ satisfying u γ (0) = γ, for all γ > 0 suciently large for λγ to be positive. Moreover, for any sequences (β γ ) γ , (u γ ) γ with the above properties, we have that β γ → 0, that

u γ u ∞ weakly in H 1 0 , (0.6)
where u ∞ = v 1 l λ1 ≡ 0, and that the quantication Observe that (0.5) can be seen as a particular case of (0.1) with f γ (x, u) = λγ u + β γ u exp(u 2 ), (0.8) and it arises when looking at the Euler-Lagrange equation of the Adimurthi-Druet inequality [START_REF] Adimurthi | Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality[END_REF]. We also refer to [START_REF] Mancini | Non-existence of extremals for the Adimurthi-Druet innequality[END_REF], where this Euler-Lagrange equation was studied in this tricky regime λγ → λ 1 , but only in the minimal energy case, where l equals 0 in (0.7). When considering the typical case (0.2), i.e. the Euler-Lagrange equation of the standard Moser-Trudinger inequality, existence results have been obtained using radial analysis [START_REF] Carleson | On the existence of an extremal function for an inequality of J. Moser[END_REF][START_REF] Malchiodi | Critical points of the Moser-Trudinger functional on a disk[END_REF] (see also [START_REF] Mancini | The Moser-Trudinger inequality and its extremals on a disk via energy estimates[END_REF]), variational [START_REF] Flucher | Extremal functions for the Trudinger-Moser inequality in 2 dimensions[END_REF][START_REF] Struwe | Critical points of embeddings of H 1,n 0 into Orlicz spaces[END_REF], perturbative [START_REF] Manuel Del Pino | New solutions for Trudinger-Moser critical equations in R 2[END_REF] or topological methods [START_REF] Druet | Multi-bumps analysis for Trudinger-Moser nonlinearities II-Existence of solutions of high energies[END_REF][START_REF] Lamm | The heat ow with a critical exponential nonlinearity[END_REF][START_REF]Positive solutions of critical semilinear elliptic equations on non-contractible planar domains[END_REF]. According to the previous discussion, contrary to those built in Theorem 0.1, the blow-up solutions obtained in these results always have a zero weak limit in H 1 0 . As a by-product of Theorem 0.1 and its proof below, we also obtain the following result.

Theorem 0.2 (Nodal case). Let l > 0 be xed and Ω ⊂ R 2 be the unit disk centered at 0. Let k ≥ 2 be a xed integer. Then there exist positive real numbers γ = γ(k, l), λγ = λγ (k, l) and

β γ = β γ (k, l) such that λ k -λγ = (1 + o(1)) 4πv k (0) γ λ k l , (0.9)
as γ → +∞, and such that the equation

∆u = λγ u + β γ u exp(u 2 ) in Ω , u = 0 on ∂Ω , (0.10)
admits a smooth solution u γ = u γ (k, l) satisfying u γ (0) = γ, for all γ > γ. Moreover, we have that β γ → 0, that (0.6) holds true for

u ∞ = v k l λ k ≡ 0 ,
and that the quantication (0.7) holds true, as γ → +∞.

As v k for k ≥ 2, the solutions u γ 's in Theorem 0.2 are sign-changing and have exactly k nodal regions in Ω. Theorem 0.2 provides new examples of non-compact sequences of nodal solutions for a Moser-Trudinger critical type equation for which the quantication in (0.7) holds true. We mention that Grossi and Naimen [START_REF] Grossi | Blow-up analysis for nodal radial solutions in Moser-Trudinger critical equations in R 2[END_REF] obtained recently a nice example of a quantized sequence in the sign-changing case. In [START_REF] Grossi | Blow-up analysis for nodal radial solutions in Moser-Trudinger critical equations in R 2[END_REF], the results of [START_REF] Adimurthi | Multiplicity results for semilinear elliptic equations in a bounded domain of R 2 involving critical exponents[END_REF][START_REF]Nonexistence of nodal solutions of elliptic equations with critical growth in R 2[END_REF] are used as a starting point and the point of view is completely dierent from that of Theorem 0.2.

While the nonlinearities of the form (0.8) clearly have Moser-Trudinger type growth, it should be pointed out that they do not have uniformly critical growth in the sense of the denition of Druet [START_REF] Druet | Multibumps analysis in dimension 2: quantication of blow-up levels[END_REF], if β γ → 0 and λγ → λ - k , k ≥ 1 as γ → +∞ as in Theorem 0.1 and Theorem 0.2. However, our techniques can be applied to the study of dierent kinds of nolinearities. For a xed real parameter a > 0, let g : [0, +∞) → R be such that

g(t) = e t 2 -at for t ≥ c 0 > 0, (G1) and g ∈ C 0 ([0, +∞)) with g > 0 in [0, +∞).
(G2) Then, we get the following result.

Theorem 0.3. Let Ω⊂ R 2 be the unit disk centered at 0 and let a > 0 and g ∈ C 0 ([0, +∞)) be given so that (G1) and (G2) hold. For any γ > 0 there exists a unique β γ > 0 such that the equation

∆u = β γ ug(u) , u > 0 in Ω , u = 0 on ∂Ω , (0.11)
admits a (unique) smooth radially symmetric solution u γ satisfying u γ (0) = γ. Moreover, as γ → +∞ we have

β γ → β a 2 > 0, that u γ u a 2 weakly in H 1 0 ,
(0.12) and that the following quantication holds true

Ω |∇u γ | 2 dy = 4π + Ω |∇u a 2 | 2 dy. (0.13)
It is interesting to notice that the value at 0 of the weak limit of the sequence (u γ ) γ in Theorem 0.3 depends only on the choice of a, that is on the asymptotic behavior of the function g. We stress that for any a > 0 one can easily construct a function g satisfying (G1) and (G2) such that g(t) = 1 for t ≤ a 2 . For such function g, the nonlinearities f γ (x, u) = β γ ug(u) have uniformly critical growth according to Druet's denition in [START_REF] Druet | Multibumps analysis in dimension 2: quantication of blow-up levels[END_REF]. Moreover, since the u γ 's are positive and radially decreasing, one has β a 2 = λ 1 and u a 2 = a 2v1(0) v 1 so that the quantication result of Theorem 0.3 reads as

Ω |∇u γ | 2 dy → 4π + a 2 λ 1 4v 1 (0) 2 .
(0.14) Note that the value in the RHS of (0.14) can be arbitrarily large or arbitrarily close to 4π depending on the choice of a.

1. Proof of Theorem 0.1

In the whole paper, Ω = B 0 (1) is the unit disk centered at 0 in R 2 . If f is a radially symmetric function, since no confusion is possible, we will often write

f (|x|) instead of f (x) .
(1.1) In the sequel, we let λ 1 and v 1 be as in Theorem 0.1. For all R > 0, it is known that the smallest eigenvalue λR of ∆ in B 0 (R) with zero Dirichlet boundary condition is given by

λR = λ 1 R 2 .
(1.2)

Let ṽR ≥ 0 be the (radial) eigenfunction associated to λR such that Ω ṽ2 R dy = 1.

We get rst the following existence result.

Lemma 1.1. Let λ ∈ (0, λ 1 ) and γ > 0 be given. Then there exists β > 0 and a smooth function u in Ω such that u(0) = γ and such that u solves ∆u = λu + βu exp(u 2 ) , u > 0 in Ω , u = 0 on ∂Ω .

(1.3)

Proof of Lemma 1.1. Let λ ∈ (0, λ 1 ) and γ > 0 be xed. For all given β ≥ 0, there exists a unique smooth and radially symmetric solution u β of

∆u = λu + βu exp(u 2 ) , u(0) = γ , (1.4)
well dened in R 2 . Indeed, if [0, T β ) is the maximal interval of existence for u β , we know from rather standard theory of radial elliptic equations that T β ∈ (0, +∞], and that T β < +∞ implies that

lim sup t→T - β |u β (t)| + |u β (t)| = +∞ .
(1.5)

Now observe that E(u β ) : [0, T β ) → R given by E(u β ) = (u β ) 2 + λu 2 β + β exp(u 2 β )
is nonincreasing in (0, T β ) by (1.4). Then, (1.5) cannot occur and T β = +∞ as claimed. Now, given β > 0, there exists an ε β > 0 such that u β (s) > 0 for all s ∈ [0, ε β ] by continuity. Then, since the RHS of the rst equation of (1.4) is positive for u β > 0, and since

-ru β (r) = r 0 (∆u β )(s)sds , (1.6)
we get that

u β (r) ≤ γ + log ε β r ε β 0 (∆u β )(s)sds , (1.7)
for all r ≥ ε β such that u β > 0 in [ε β , r]. Setting now

R β = sup {r > 0 s.t. u β (s) > 0 for s ∈ [0, r]} , (1.8)
we clearly get from (1.7) that 0 < R β < +∞. Then we have u β (R β ) = 0 and u β > 0 in [0, R β ). Now, since (β, r) → u β (r) is continuous, and since we have that u β (R β ) < 0 by (1.6), we get that

β → R β is continuous in [0, +∞) . (1.9)
It is clear that R 0 = λ 1 /λ > 1 by (1.2) and (1.4). Independently, multiplying (1.4) by ṽR β > 0 and integrating by parts in B 0 (R β ), we get that

λR β B0(R β ) u β ṽR β dy = B0(R β ) ∇u β ∇ṽ R β dy , = B0(R β ) (λ + β exp(u 2 β ))u β ṽR β dy , > β B0(R β )
u β ṽR β dy , so that λR β → +∞ and then that R β → 0 as β → +∞, using (1.2). Thus by (1.9), we get that there exists β > 0 such that R β = 1, which concludes the proof of Lemma 1.1.

Proof of Theorem 0.1. Let l > 0 be a xed real number. Let λγ , λ 1 and v 1 > 0 in Ω be as in the statement of Theorem 0.1. By Lemma 1.1, there exists β γ > 0 such that the equation (0.5) admits a smooth solution u γ satisfying u γ (0) = γ, for all γ > 0 suciently large for λγ to be positive. First, we check that

β γ = O 1 γ , (1.10)
which implies that β γ → 0 as γ → +∞, as claimed in Theorem 0.1. For this, it is sucient to multiply the rst equation in (0.5) by v 1 and to integrate by parts:

(λ 1 -λγ ) Ω v 1 u γ dy = β γ Ω v 1 exp(u 2 γ )u γ dy ≥ β γ Ω v 1 u γ dy .
(1.11)

In view of (0.4), this clearly implies (1.10). Now, we perform the blow-up analysis of the u γ 's as γ → +∞, in order to get (0.6)-(0.7). Observe that we do not assume here that (u γ ) γ is bounded in H 1 0 , as in [START_REF] Mancini | Non-existence of extremals for the Adimurthi-Druet innequality[END_REF]. But since we are in a radially symmetric setting, we are able to start the analysis and to prove that the u γ 's can be rescaled around 0 in order to detect a bubble of Moser-Trudinger critical type. Observe that our choice of λγ in (0.4) plays a key role for this to be true. Let µ γ > 0 be given by

β γ γ 2 exp(γ 2 )µ 2 γ = 4 , (1.12)
and τ γ be given by

u γ (µ γ y) = γ - τ γ (y) γ .
(

Observe that, since ∆u γ > 0 in (0.5), u γ is radially decreasing in Ω, so that

u γ (r) ≤ u γ (0) = γ , for all r ∈ [0, 1] . (1.14)
Here and often in the sequel, we use the identications of (1.1).

Step 1.1. We have that

lim γ→+∞ µ γ γ = 0 (1.15)
and

lim γ→+∞ τ γ = T 0 := log(1 + | • | 2 ) in C 2 loc (R 2 ) .
(1.16)

Proof of Step 1.1. By (1.12), (1.15) is equivalent to

lim γ→+∞ β γ exp(γ 2 ) = +∞ .
(1.17)

In order to prove (1.17), assume by contradiction that

β γ exp(γ 2 ) = O(1) , (1.18)
up to a subsequence. Then, if w γ is given by u γ = γw γ , we have that 0 ≤ w γ ≤ 1 and w γ (0) = 1. Moreover, by (0.4), (0.5), (1.14), (1.18) and standard elliptic theory, we get that

lim γ→+∞ w γ = w ∞ in C 1 ( Ω) , (1.19)
where w ∞ (0) = 1. Then, since w ∞ > 0 in a neighborhood of 0, we have that

∆w γ ≥ λ 1 (1 + o(1))w ∞ > 0
around 0, so that there exists ε 0 ∈ (0, 1) such that

ε 0 |x| 2 ≤ 1 -w γ (x) , for all x ∈ Ω , (1.20)
using that w γ is radially symmetric and decreasing. Thus, using (1.18), (1.20) and 0 ≤ w γ ≤ 1, we get that

β γ Ω v 1 exp(u 2 γ )u γ dx = β γ exp(γ 2 )γ Ω v 1 exp(-γ 2 (1 -w 2 γ ))w γ dx , = O γ Ω exp(-ε 0 γ 2 |x| 2 )dx , = o (1)
.

(1.21) Independently, since w ∞ ≥ 0 and w ∞ ≡ 0, we get from (1.19) that

γ = O Ω v 1 u γ dx .
( Next, we prove that β γ does not converge to zero too fast. In the sequel it is useful to denote

t γ (r) := T 0 r µ γ = log 1 + r 2 µ 2 γ .
(1.23)

Step 1.2. We have that

| log β γ | = o(γ 2 ) .
(1.24)

Proof of Step 1.2. In view of (1.10), assume by contradiction that

lim γ→+∞ log 1 βγ γ 2 > 0 .
(1.25)

Here and in the sequel, we argue up to subsequences. Let r γ ∈ (0, 1) be given by

β γ exp(u γ (r γ ) 2 ) = 1 .
(1.26) By (1.10), (1.17), and since u γ is radially decreasing and zero on ∂Ω, r γ is well dened. In particular, we have that

β γ exp(u 2 γ ) > 1 in [0, r γ ) , β γ exp(u 2 γ ) ≤ 1 in [r γ , 1] .
(1.27) By (1.25) and (1.26), we have that

lim γ→+∞ u 2 γ (r γ ) γ 2 > 0 .
(1.28)

Observe also that r γ µ γ by (1.13), (1.16), (1.17) and (1.26). Now, we prove that

u γ = γ - t γ (1 + o(1)) γ uniformly in [0, r γ ] , (1.29)
as γ → +∞, where t γ is given by (1.23). For this, for any given η ∈ (0, 1), we let r γ be given by

r γ = sup r ∈ (0, r γ ] s.t. u γ -γ - t γ γ ≤ η t γ γ in (0, r] .
(1.30)

In order to get (1.29), since η may be arbitrarily small, it is sucient to prove that

r γ = r γ , (1.31)
for all γ 1. Observe that (1.28) and (1.30) imply that

lim γ→+∞ t γ (r γ ) γ 2 < 1 , (1.32)
if η was chosen small enough. By the denition of r γ in (1.30), for any r ∈ [0, r γ ],

we have that

u γ (r) 2 ≤ γ 2 + t γ (r) -2 + 2η + t γ (r) γ 2 (1 -2η + η 2 ) .
(1.33)

In particular,by (1.12), (1.14), (1.32) and (1.33), for any suciently small η there exists κ = κ(η) > 1 such that

β γ exp(u 2 γ )u γ ≤ 4 γµ 2 γ exp(-κt γ ) , (1.34) in [0, r γ ],
for suciently large γ. Let (s γ ) γ be an arbitrary sequence such that s γ ∈ [0, r γ ], for all γ. If s γ = O(µ γ ), then, arguing as in (1.6), we get from (1.13), (1.16), and (1.23) that

u γ (s γ ) + t γ (s γ ) γ = 1 γs γ sγ µγ 0 ((∆τ γ )(s) -(∆T 0 )(s))sds = o s γ µ 2 γ γ . (1.35) If instead s γ µ γ , then given R 1 we compute sγ 0 (∆u γ )(s)2πsds = O Rµγ 0 γsds + O sγ Rµγ (β γ exp(u 2 γ )) ≥1 u γ sds + β γ Rµγ 0 exp(u 2 γ )u γ 2πsds , = o 1 γ + 1 γ O +∞ R exp(-κT 0 )rdr + 1 γ sγ /µγ 0 8πr (1 + r 2 ) 2 dr ,
(1.36) using (0.5), (1.12), (1.13), (1.15), (1.16), (1.27) and (1.34). Then, as R may be arbitrarily large in (1.36), we get that sγ 0

(∆u γ )(s)2πsds = o 1 γ + 1 γ sγ /µγ 0 8πr (1 + r 2 ) 2 dr , = o 1 γ - 1 γ sγ 0 (∆t γ )(s)2πsds .
(1.37)

In any case we obtain that

u γ (s γ ) + t γ (s γ ) γ = o t γ (s γ ) γ , (1
u γ (r) - log 1 r 2 γ = o log 1 r 2 γ + O ( u γ 2 ) , (1.41)
for all r ∈ [r γ , 1] and all γ, using the fundamental theorem of calculus and u γ (1) = 0. Now, we evaluate u γ (r γ ) in both formulas (1.29) and (1.41). Since (1.12) and (1.23) imply that

γ - t γ (r) γ = 1 γ log 1 β γ (µ 2 γ + r 2 ) + O log γ γ (1.42)
for any r ∈ [0, 1], and that log 1 (1.43)

In order to conclude the proof, we now show that (1.43) contradicts our choice of ε γ in (0.4). First, observe that r γ → 0, as γ → +∞. Otherwise, since (1.28) implies

γ = O(u γ ) in [0,
u γ (r γ ) -u γ (r) γ(r -r γ ) 2 ∈ (0, +∞] .
(1.45) From (0.5), (1.14), (1.27), (1.43) and standard elliptic theory, we get that there exists a function v such that

lim γ→+∞ u γ u γ 2 → v in C 1 loc ( Ω\{0}) , v ≥ 0 in Ω\{0} . ( 1
u γ (r γ ) -u γ (r) γ(r 2 -r 2 γ ) ∈ (0, +∞],
(1.47) which concludes the proof of (1.45). Now, we compute

1 0 v 1 β γ exp(u 2 γ )u γ 2πsds = rγ 0 v 1 β γ exp(u 2 γ )u γ 2πsds + 1 rγ v 1 exp(u 2 γ -u γ (r γ ) 2 )u γ 2πsds , = O 1 γ + O u γ ∞ 1 rγ exp(-u γ (r γ )(u γ (r γ ) -u γ ))sds = O 1 γ + O γ 1-rγ 0 exp(-ε 0 γ 2 r 2 )(r + r γ )dr = O 1 γ + O(r γ ) .
(1.48)

The rst equality in (1.48) uses (1.26); the second one uses (0.5), (1.39), and u γ ≥ 0; in the third one, the existence of such a positive ε 0 is given by (1.28) and (1.45). Independently, (1.43) and (1.46) with v ≡ 0 imply that

γ = O 1 0 v 1 u γ rdr .
(1.49)

But the equality in (1.11), and (1.48)-(1.49) clearly contradict our denition of ε γ in (0.4). Then (1.25) cannot be true. This concludes the proof of (1.24) and that of Step 1.2.

Let t γ be as in (1.23) and let ρ 1,γ , ρ γ , ρ 2,γ > 0 be given by

t γ (ρ 1,γ ) = γ , t γ (ρ γ ) = γ 2 2
and t γ (ρ 2,γ ) = γ 2 -γ .

(1.50)

Since we have now (1.24), resuming verbatim the argument in [14, Step 3.2], there exists C > 0

u γ -γ - t γ γ ≤ C 1 + t γ γ 3 , (1.51) in [0, ρ γ ],
and that

β γ u γ exp(u 2 γ ) - 4 exp(-2t γ ) γµ 2 γ ≤ 4 C 3 exp(-2t γ )(1 + t 2 γ ) γ 3 µ 2 γ , (1.52) in [0, ρ 1,γ ],
for all γ. Observe that (1.50) and (1.24) imply

ρ γ ρ 1,γ = µ γ exp(γ(1/2 + o(1))) µ γ , (1.53)
and that

ρ 2 1,γ γ = o 1 γ 3 .
(1.54)

Note that (1.52), (1.53) and (1.54) with u γ ≤ γ imply that

ρ1,γ 0 (∆u γ )2πrdr - 4π γ ≤ 4π C + o(1) γ 3 .
(1.55)

Let r γ ∈ (0, 1] be given by

β γ exp(u γ (r γ ) 2 ) = 1 γ 2 if β γ = o 1 γ 2 , r γ = 1 otherwise .
(1.56)

Using u γ (0) = γ, u γ (1) = 0 and (1.24), r γ is well dened and positive. We have in addition that

β γ exp(u 2 γ ) ≥ 1 γ 2 in [0, r γ ] , β γ exp(u 2 γ ) ≤ 1 γ 2 in (r γ , 1] .
(1.57)

Moreover, since (1.12) and (1.24) give

t γ ≤ γ 2 (1 + o(1)) in [0, 1], (1.58)
we get from the denition of ρ γ in (1.50) and (1.51) that u γ (ρ γ ) 2 = γ 2 4 + O(1), so that (1.24) and (1.56) give

r γ ≥ ρ γ ,
(1.59) for γ large enough, this inequality being obvious if r γ = 1. Finally, we dene

ρ γ = min(ρ 2,γ , r γ ) .
(1.60)

Step 1.3. Let C be as in (1.51), (1.52) and (1.55). For any R 0 > C we have

u γ -γ - t γ γ ≤ R 0 γ in [0, ρ γ ] , (1.61)
for suciently large γ. Moreover, (1.62)

Proof. For a given R 0 > C, let ρ γ > 0 be given by

ρ γ = sup r ∈ (0, ρ γ ] s.t. u γ -γ - t γ γ ≤ R 0 γ in [0, r] . ( 1
β γ exp(u 2 γ )u γ rdr ≤ ρ2,γ ρ1,γ exp(-2t γ + (t 2 γ /γ 2 ) + O(1)) γµ 2 γ rdr = o 1 γ 5 .
(

u γ -γ - t γ γ s ρ1,γ ≤ C + o(1) γ 3 log s 2 ρ 2 1,γ
, so that

u γ (s) -γ - t γ (s) γ ≤ C + o(1) 1 + t γ (s) γ 3 , (1.68)
for all s ∈ [ρ 1,γ , ρ γ ], by (1.51) at ρ 1,γ and (1.53). Then we get

ρ γ = ρ γ ,
(1.69) using R 0 > C and ρ γ > ρ 1,γ . The proof of (1.61) is complete. Finally, the estimate in (1.62) follows from (0.5), (1.57), (1.65) and (1.69).

Step 1.4. We have that

lim inf γ→+∞ u γ 2 ∈ (0, +∞] .
(1.70) Moreover, we have

ρ γ → 0 , (1.71)
and

u γ (r) = log 1 r 2 γ + o( u γ 2 ) + v 1 (r) u γ 2 in [ρ γ , 1] (1.72) as γ → +∞ .
Proof. First, coming back to (1.11), we get that

4πv 1 (0) + o(1) γ = β γ ρ1,γ 0 v 1 u γ exp(u 2 γ )2πrdr ≤ λ 1 -λγ 1 0 v 1 u γ 2πrdr = O u γ 2 γ ,
by (1.52), the Cauchy-Schwarz inequality and our denition of ε γ below (0.4). This, clearly implies (1.70).

In order to prove (1.71), we observe that (1.50), (1.60) and (1.61) imply

u γ ≥ u γ (ρ γ ) = γ - t γ (ρ γ ) γ + o(1) ≥ γ - t γ (ρ 2,γ ) γ + o(1) = 1 + o(1) ,
in [0, ρ γ ] and then that

ρ 2 γ = O ργ 0 u γ (s)2πsds = O ργ 0 ∆u γ (s)2πsds = O 1 γ ,
by (1.55) and (1.62). Now, we turn to the proof of (1.72). Note that

β γ exp(u 2 γ ) = o(1) in (ρ γ , 1] . ( 1
(∆u γ )(r)2πrdr = ργ 0 (∆u γ )(r)2πrdr + O(s γ u γ 2 ) = 4π γ + o 1 γ 2 + O(s γ u γ 2 ) , (1.75)
for any sequence s γ ∈ [ρ γ , 1]. Estimating u γ by (1.75) in the spirit of (1.6), using the fundamental theorem of calculus and u γ (1) = 0, and noting that 1

γ 2 log 1 ρ 2 γ = O( u γ 2 ), we obtain that u γ - log 1 r 2 γ = O ( u γ 2 ) , (1.76
log 1 βγ γ = u γ 2 v 1 (0) + o ( u γ 2 ) ,
(1.79) so that in particular

β γ = o 1 γ , (1.80)
by (1.70). Then, arguing as in (1.73) but using (1.80) instead of (1.10) if ρ γ = ρ 2,γ , we get that

β γ exp(u 2 γ ) = o 1 γ in (ρ γ , 1] . (1.81)
Now, on the one hand we have (1.83)

β γ 1 0 v 1 exp(u 2 γ )u γ 2πrdr = 4πv 1 (0) + o(1) γ + o u γ 2 γ (1.82) using (1.52) for r ≤ ρ 1,γ , (1.62) for ρ 1,γ ≤ r ≤ ρ γ ,
Then, by the equality in (1.11), (1.82), (1.83) and our denition of ε γ below (0.4), we get that

u γ 2 = l λ 1 + o(1) .
(1.84) Now, by (0.5), we have that

Ω |∇u γ | 2 dx = λγ Ω u 2 γ dx + β γ Ω u 2 γ exp(u 2 γ )dx,
(1.85) and, using (1.51), (1.52), (1.62) and (1.81), that 1) by (0.4), we prove that (0.7) holds true. Moreover, by (0.7) with (1.71) and (1.72), there must be the case that (0.6) holds true. Thus, Theorem 0.1 is proved.

β γ Ω u 2 γ exp(u 2 γ )dx =γ(1 + o(1))β γ ρ1,γ 0 exp(u 2 γ )u γ 2πrdr + O γ ργ ρ1,γ ∆u γ 2πrdr + o u γ 2 γ =4π + o 1 γ (1.86) By (1.83)-(1.86), since λγ = λ 1 + o(

Proof of Theorem 0.2

Let Ω ⊂ R 2 be the unit disk centered at 0. Let λ 1 , λ 2 , . . . and v 1 , v 2 , . . . be as above Theorem 0.1. By Bessel functions' theory, v 1 extends in a unique way to a radial function v1 , satisfying ∆v 1 = λ 1 v1 in R 2 . It is known that v1 vanishes exactly for |x| = r n := λ n /λ 1 with v 1 (r n ) = 0 (see (1.1)) for any n ≥ 1. Moreover, we have that vn := v1 (r n •) is proportional to v n in Ω, namely

v n = r n √ α n v1 (r n •) where α n = B0(rn) v2 1 dx , (2.1)
for all integer n ≥ 1. Let l > 0 be a xed real number and k ≥ 2 be a xed integer.

Let us dene

λγ := λ 1 - 4πv 1 (0) γ λ 1 l with l = l α k .
(2.2) By Theorem 0.1 (with l = l) for all γ large enough, there exist βγ > 0 and a smooth radial function ũγ such that

     ∆ũ γ = λγ ũγ + βγ ũγ exp(ũ 2 γ ), ũγ > 0 in Ω , ũγ = 0 on ∂Ω , ũγ (0) = γ , (2.3)
and we have

βγ = o(1) and ũγ → ũ∞ := v 1 l λ 1 in C 1 loc ( Ω\{0}) ∩ L 2 (Ω) ,
(2.4) as γ → +∞. Moreover, as discussed in Lemma 1.1, we have that ũγ is globally dened on R 2 and, as a consequence of (2.4) and standard ODE theory, we nd that

ũγ → v1 l/λ 1 in C 1 loc (R 2 \{0}) ∩ L 2 loc (R 2 ).
(2.5) Besides, by the implicit function theorem and ODE theory, there exists ε 0 > 0 and γ 1 large such that ũγ vanishes exactly once in I k := (r k -ε 0 , r k + ε 0 ) at some r k,γ , for all γ ≥ γ (indeed ũγ vanishes exactly k-times in [0, r k + ε 0 ) for γ 1). By construction, we also have that r k,γ = r k + o(1) as γ → +∞. Setting u γ := ũγ (r k,γ •), we get from the above discussion that u γ solves (0.10) with ). Besides (2.5) implies

β γ = r 2 k,γ βγ and λγ = r 2 k,γ λγ = λ k + o(1
u γ → v1 (r k •) l λ 1 = v k l λ k in C 1 loc ( Ω\{0}) ∩ L 2 (Ω) , (2.6)
as γ → +∞. Then, using the invariance of the L 2 -norm of the gradient under dilation in dimension 2, (0.7) for ũγ if |x| ≤ 1/r k,γ , and (2.6) if |x| ≥ 1/r k,γ , we get that

Ω |∇u γ | 2 dx = Ω |∇ũ γ | 2 dy + Ω\B0(1/r k,γ ) |∇u γ | 2 dx , = 4π + l + l B0(r k )\Ω v2 1 dy + o(1) , = 4π + l + o(1) , (2.7)
so that (0.7) holds true. Clearly (2.6) and (2.7) give also (0.6). Finally, we shall prove that (0.9) holds. In order to do this, we may multiply (0.10) by v k , integrate by parts and use (2.6) to get

λ k -λγ (1 + o(1)) l λ k = λ k -λγ Ω u γ v k dy = β γ Ω u γ exp(u 2 γ )v k dy . (2.8)
Arguing as in (1.81)-(1.83) for the ũ γ s, we nd that

β γ B0( 1 r k,γ ) u γ exp(u 2 γ )v k dy = βγ Ω ũγ exp(ũ 2 γ )v k dy = 4πv k (0)(1 + o(1)) γ , (2.9)
and that βγ = o( 1 γ ), so that (2.6) implies

β γ Ω\B0( 1 r k,γ ) u γ exp(u 2 γ )v k dy = o 1 γ .
(2.10) Using (2.8)-(2.10) we conclude the proof of (0.9), and Theorem 0.2 is proved.

Proof of Theorem 0.3

As in Section 1, we rst prove the existence part of Theorem 0.3. The argument is similar to the one of [12, Proof of Theorem 1]. Lemma 3.1. Let g be as in Theorem 0.3. For any γ > 0 there exist a unique real number β γ > 0 and a unique smooth radial function u γ in Ω solving

     ∆u = β γ ug(u) , u > 0 in Ω , u = 0 on ∂Ω , u(0) = γ . (3.1)
Moreover, we have

0 < β γ < λ 1 inf [0,+∞] g -1 . (3.2)
Proof. Let f : R → R be the continuous odd extension of the function t → tg(t), t ≥ 0. For any γ > 0, let w γ be the solution the ODE

∆w γ = f (w γ ) , w γ (0) = γ . (3.3)
Arguing as in the proof of Lemma 1.1, we get that w γ is dened on R 2 , since the function

E(w γ ) = 1 2 |w γ | 2 + F (w γ ) with F (t) = t 0 f (s)
ds is nonincreasing in the existence interval for w γ , and since F (t) → +∞ as t → +∞. Let now

R γ = sup {r > 0 s.t. w γ > 0 in [0, r]} .
(3.4) Clearly R γ > 0 and for any xed ε γ ∈ (0, R γ ) and and r ∈

[ε γ , R γ ) have -rw γ (r) = r (∆w γ )(s)sds ≥ εγ 0 f (w γ (s))sds > 0 . (3.5)
Using the fundamental theorem of calculus as in (1.7) we get that R γ < +∞, so that R γ is the rst zero of w γ . Then, the function u γ = w γ (R γ •) satises (3.1) with

β γ = R 2 γ .
Multiplying the equation in (3.1) by v 1 and integrating by parts, we get that

λ 1 Ω v 1 u γ dx = β γ Ω v 1 u γ g(u γ )dx ≥ β γ inf [0,+∞] g Ω v 1 u γ dx ,
which gives (3.2). Note that (G1) and (G2) imply inf [0,+∞] g > 0. Finally, we observe that β γ and u γ are uniquely determined by γ and g. Indeed if β > 0 and ū ∈ C 1 ( Ω) are such that

         ∆ū = β ūg(ū) , ū > 0, in Ω ū = 0 on ∂Ω, ū(0) = γ,
ū is radially symmetric in in Ω , then uniqueness theory for solution of ODEs implies ū( • √ β ) = w γ . In particular

β = β γ = R 2 γ and ū = u γ .
For any γ > 0 let β γ and u γ be as in Lemma 3.1. As in Section 1 we shall rescale around 0 and prove that the u γ 's are close to a Moser-Trudinger bubble up to a suciently large scale. However, here more precise expansions (see (3.13) and (3.31)) are needed in order to detect the eect of the term e -au on the shape of such bubble.

Let us dene µ γ > 0 such that

µ 2 γ β γ γ 2 g(γ) = 4.
(3.6) Note that µ γ → 0 as γ → +∞.

(3.7) Otherwise, by (3.6), we could nd a subsequence such that β γ γ 2 g(γ) = O(1). Since u γ ≤ u γ (0) = γ in Ω and since g(γ) → +∞ as γ → +∞, the assumptions (G1) and (G2) imply that g(u γ ) ≤ g(γ) for large values of γ. Hence, we would have

∆u γ ≤ β γ γg(γ) = O(γ -1 ),
which contradicts u γ (0) = γ → +∞ by standard elliptic estimates. Now let τ γ be dened as in (1.13) with µ γ as in (3.6). Then, we will show that

τ γ → T 0 and γ(τ γ -T 0 ) → aS 0 in C 2 loc (R 2 ),
where T 0 is as in (1.16) and

S 0 (x) = - 1 2 T 0 (x) + 1 2 |x| 2 1 + |x| 2 .
(3.8) Note that we have

∆T 0 + 4 exp(-2T 0 ) = 0 in R 2 , (3.9) and ∆S 0 -8 exp(-2T 0 )S 0 = 4T 0 exp(-2T 0 ) in R 2 .
(3.10) More precisely, setting

t γ = T 0 • µ γ and S γ = S 0 • µ γ , (3.11)
and letting ρ γ > 0 be dened by

t γ (ρ γ ) = γ 2 2 , (3.12)
we get the following expansion in [0, ρ γ ].

Step 3.1. As γ → +∞, we have that

u γ = γ - t γ γ + aS γ γ 2 + O t γ γ 3 in [0, ρ γ ] , (3.13)
with T γ and S γ as in (3.11). Moreover, we have that

B0(ργ ) ∆u γ dy = 4π γ + 2πa γ 2 + O 1 γ 3 , (3.14)
and that

B0(ργ )
u γ ∆u γ dy = 4π + o(1) .

(3.15)

Proof. The proof of Step 3.1 is similar to the one of [START_REF] Mancini | Non-existence of extremals for the Adimurthi-Druet innequality[END_REF]Step 3.2]. Observe that (G1), (3.2) and (3.6) imply that µ -2 γ = O(γ 2 exp(γ 2 )). Then, we get

t γ (r) ≤ log 1 + 1 µ 2 γ = O(γ 2 ) , (3.16)
for any r ∈ [0, 1]. This will be used several times in the sequel. Let w γ be dened by

u γ = γ - t γ γ + aw γ γ 2 ,
(3.17) and let

ρ γ = sup{r ∈ [0, ρ γ ] : |w γ -S γ | ≤ 1 + t γ }.
(3.18) First, by (3.12) and (3.16)-(3.18), one has that u γ ≥ γ 2 + O(1) in [0, ρ γ ], so that (G1) gives g(u γ ) = exp(u 2 γ -au γ ) for γ suciently large. Moreover, (3.16), (3.17) and (3.18) imply that

u 2 γ -au γ = γ 2 -aγ -2t γ + ψ γ + O 1 + t γ γ 2 , (3.19) in [0, ρ γ ],
where

ψ γ = a 2w γ + t γ γ + t 2 γ γ 2 - 2at γ w γ γ 3 .
Using the simple inequality |e x -1 -x| ≤ e |x| |x| 2 for x ∈ R, we get that

exp (ψ γ ) = 1 + ψ γ + O exp (|ψ γ |) ψ 2 γ = 1 + a 2w γ + t γ γ + O 1 + t 2 γ γ 2 + O exp(|ψ γ |) 1 + t 4 γ γ 2 = 1 + a 2w γ + t γ γ + O exp(|ψ γ |) 1 + t 4 γ γ 2 , (3.20) in [0, ρ γ ]. Since u γ = γ 1 + O 1+tγ γ 2
by (3.16), (3.17) and (3.18), using (3.1), (3.16), (3.19), (3.20) and the denition of µ γ in (3.6), we obtain that

∆u γ = β γ γg(γ) exp(-2t γ + ψ γ ) 1 + O 1 + t γ γ = 4 γµ 2 γ exp (-2t γ ) 1 + a 2w γ + t γ γ + O exp(|ψ γ |) 1 + t 4 γ γ 2 , (3.21) in [0, ρ γ ].
In particular, (3.9), (3.10), (3.17) and (3.21) yield

∆(w γ -S γ ) = 4 µ 2 γ exp(-2t γ ) 2(w γ -S γ ) + O exp(|ψ γ |) 1 + t 4 γ γ .
(3.22)

Note that, since t γ ≤ γ 2 2 in [0, ρ γ ] by (3.12), we get

-2t γ + |ψ γ | = t γ -2 + t γ γ 2 + O 1 γ + o(1) , ≤ t γ - 3 2 + O 1 γ + o(1) .
Hence, there exists κ > 1 such that

(1 + t 4 γ ) exp(-2t γ + |ψ γ |) = O(exp(-κt γ )) in [0, ρ γ ] , (3.23
≤ C 1 r r 2 µ 2 γ 1 + r 3 µ 3 γ w γ -S γ L ∞ ([0,ρ γ ]) + C 1 γ r 2 µ 2 γ 1 + r 2 µ 2 γ ,
for any r ∈ [0, ρ γ ]. Therefore we have that

|(w γ -S γ ) (r)| ≤ C 2 r µγ 1 + r 2 µ 2 γ w γ -S γ L ∞ ([0,ρ γ ]) + 1 µ γ γ , (3.24)
for some constant C 2 > 0. We claim now that

(w γ -S γ ) L ∞ ([0,ρ γ ]) = O 1 µ γ γ .
(3.25)

Otherwise there exists 0 < ρ γ ≤ ρ γ such that (3.27)

µ γ γ (w γ -S γ ) L ∞ ([0,ρ γ ]) = µ γ γ|(w γ -S γ ) (ρ γ )| → +∞.
We dene

z γ = (w γ -S γ )(µ γ •) µ γ (w γ -S γ ) L ∞ ([0,ρ γ ])
, so that we have

|z γ | ≤ 1 in [0,
γ → z 0 in C 1 loc (B 0 (δ 0 )), but we also get that z γ (ρ γ ) → z 0 (δ) as γ → +∞, where z 0 ∈ C 1 (B 0 (δ)) satises ∆z 0 -8 exp(-2T 0 )z 0 = 0 in B 0 (δ 0 ) ∩ B 0 (δ) , z 0 (0) = 0 .
But, since z 0 is radially symmetric, we get z 0 ≡ 0 and then z γ (ρ γ ) → 0. But this contradicts our denition of ρ γ in (3.26), which gives that |z γ (ρ γ )| = 1. Hence, (3.26) cannot hold and (3.25) is proved. Then (3.24) reads as

|(w γ -S γ ) | ≤ 2C 3 µ γ γ r µγ 1 + r 2 µ 2 γ = C 3 γ t γ (r) , in [0, ρ γ ].
In particular, using the fundamental theorem of calculus, we get (3.28)

Note that the expansion in (3.13), (3.8), and (3.28) imply that

u γ (ρ γ ) = γ - t γ (ρ γ ) γ 1 + a 2γ + o t γ (ρ γ ) γ 2 , (3.29)
as γ → +∞. Let us now x any c 1 > max{c 0 , a 2 }, where c 0 is as in (G1). Let r γ > 0 be such that

u γ (r γ ) = c 1 .
(3.30) We shall prove that an expansion similar to (3.29) holds uniformly in [ρ γ , r γ ].

Step 3.2. As γ → +∞, we have that

u γ = γ - t γ γ 1 + a 2γ + o t γ γ 2 (
T γ = O(γ 2 ) (3.33) in [0, 1]. For any xed η ∈ (0, 1) we set r γ = sup r ∈ [ρ γ , r γ ] : u γ -γ - T γ γ ≤ ηT γ γ 2 .
Note that r γ > ρ γ by (3.29). We claim that there exists δ > 0 such that

r γ ργ ∆u γ 2πr dr = O exp - δ 2 γ .
(3.34)

As in the proof of Step 3.1, we write

u γ = γ - T γ γ + ϕ γ γ 2 with |ϕ γ | ≤ ηT γ in [ρ γ , r γ ] . (3.35)
In particular, using (3.33) and (3.35), we can write

u 2 γ -au γ ≤ γ 2 -aγ -2T γ + aT γ γ + T 2 γ γ 2 + C 1 ηT γ γ + C 2 ,
where C 1 ,C 2 > 0 do not depend on the choice of η. Since c 1 > c 0 and u γ is radially decreasing, (G1), (3.1) and (3.6) imply that

∆u γ = β γ u γ exp(u 2 γ -au γ ) , β γ γ exp(γ 2 -aγ) exp -2T γ + aT γ γ + T 2 γ γ 2 + C 1 η T γ γ , 4 µ 2 γ γ exp -2T γ + aT γ γ + T 2 γ γ 2 + C 1 η T γ γ .
Integrating in the interval [ρ γ , r γ ] and using the change of variable τ = T γ (r) so that

r dr µ 2 γ = exp( τ αγ )dτ 2α γ , we get that r γ ργ ∆u γ 2πr dr τ (r γ ) τ (ργ ) 1 γ exp -2τ + aτ γ + τ 2 γ 2 + C 1 ητ γ + τ α γ dτ .
Since

1 αγ = 1 -a 2γ + O( 1 γ 2 ) and τ γ 2 ≤ T (r γ ) γ 2
= O(1) by (3.33), we nd that

r γ ργ ∆u γ 2πr dr 1 γ τ (r γ ) τ (ργ ) exp -τ 1 - a 2γ - τ γ 2 - C 1 η γ dτ .
(3.36) Now, by denition of r γ and r γ , we know that

c 1 = u γ (r γ ) ≤ u γ (r γ ) ≤ γ - T γ (r γ ) γ + η T γ (r γ ) γ 2 ≤ γ - T γ (r γ ) γ + C 2 η .
Since c 1 > a 2 and C 1 , C 2 > 0 do not depend on η, we can nd δ > 0 such that

1 - a 2γ - τ γ 2 - C 1 η γ ≥ c 1 - a 2 -η(C 1 + C 2 ) 1 γ ≥ δ γ ,
for any τ ≤ T γ (r γ ) and any suciently small η. Thus, (3.36) implies that

r γ ργ ∆u γ 2πr dr 1 γ τ (r γ ) τ (ργ ) exp -τ δ γ dτ , 1 γ ∞ γ 2 2 exp -τ δ γ dτ , = O exp - δ 2 γ ,
where we have also used that

T γ (ρ γ ) = αγ γ 2 2 ≥ γ 2
2 . This completes the proof of (3.34). Now, observe that (3.14) and (3.34) imply that In particular we nd that Then we must have r γ = r γ for any large γ (and in particular (3.32) follows from (3.34)). Since η can be arbitrarily small, we get (3.31).

Step 3.3. As γ → +∞ we have β γ → β a 2 > 0 and u γ → u a 2 in C 1 loc ( Ω \ {0}).

Proof. Let c 1 , ρ γ and r γ be as in (G1), (3.12), and (3.30). Since u γ ≤ c 1 in B 0 (1) \ B 0 (r γ ), (3.1), (3.2), (3.14) and (3.32) give that ∆u γ is bounded in L 1 (Ω).

Hence, we have that

u γ (r) = O log 1 r
for r ∈ (0, 1], so that u γ is locally bounded in Ω \ {0}. By (3.1), (3.2) and elliptic estimates, up to a subsequence we have that where the last inequality follows from (3.14) and (3.32). Then necessarily r γ → 0, otherwise we would have u ∞ ≡ c 1 in B 0 (δ) \ {0}, for some δ > 0, which contradicts (3.37). Then, since u γ ≤ c 1 in [r γ , 1] implies u ∞ ≤ c 1 in (0, 1], by (3.37) and standard elliptic regularity we get that u ∞ ∈ C 1 ( Ω) and that u ∞ solves (3.44)

β γ → β ∞ ∈ [0, +∞) and u γ → u ∞ in C 1 loc ( Ω \ {0}), where u ∞ solves ∆u ∞ = β ∞ u ∞ g(u ∞ ), in Ω \ {0} , u ∞ = 0, in ∂Ω . ( 3
∆u ∞ = β ∞ u ∞ g(u ∞ ) in Ω , u ∞ = 0 on ∂Ω. ( 3
Note that one cannot have β ∞ = 0, otherwise (3.38) would imply u ∞ ≡ 0 in B 1 (0) and in particular u ∞ (0) = 0, which contradicts (3.44). Then, we have β ∞ > 0, so that (3.44) implies u ∞ (0) = a 2 . The uniqueness result of Lemma 3.1 implies that

β ∞ = β a
2 and u ∞ = u a 2 . In view of Step 3.3, in order to prove Theorem 0.3 it remains to prove that the quantication in (0.13) holds true. Indeed, at that stage, this also implies that u γ → u a 2 weakly in H 1 0 (Ω), as claimed in (0.12). By Step 3.3, we can nd (σ γ ) γ such that u γ -u a

Ω |∇u γ | 2

 2 dy = 4π + l + o(1) , (0.7) holds true as γ → +∞.

r 2 γ= O log 1 µ 2 γ=

 22 O(γ 2 ), we get from (1.25), (1.29), (1.41), and (1.42) that lim γ→+∞ u γ 2 γ > 0 .

1 0v 1 u

 11 and (1.81) with the Cauchy-Schwarz inequality to estimate the integral in [ρ γ , 1]. On the other hand, using (1.54) with u γ ≤ γ for r ∈ [0, ρ 1,γ ], (1.62) with ∆u γ ≥ λγ u γ in [ρ 1,γ , ρ γ ], and (1.70)-(1.72) in [ρ γ , 1], we obtain that γ 2πrdr = u γ 2 (1 + o(1)) .

  ) for suciently large γ. By (3.22) and (3.23), we can nd C 1 > 0 such that r|(w γ -S γ ) (r)| ≤ r 0 |∆(w γ -S γ )|sds , ≤ 8 w γ -S γ L ∞ ([0,ρ γ ]) (-κt γ (s)))s ds ,

( 3 .

 3 26)Note that (3.24) and (3.26) imply ρ γ = O(µ γ ) and µ γ = O(ρ γ ) = O(ρ γ ), so that, up to a subsequence we getρ γ µ γ → δ ∈ (0, +∞) and ρ γ µ γ → δ 0 ∈ [δ, +∞] .

  ρ γ µγ ]. Using (3.27), z γ (0) = 0 the fundamental theorem of calculus, and then the ODE (3.22) and (3.26), we observe thatz γ L ∞ ([0,ρ γ /µγ ]) γ and ∆z γ L ∞ ([0,ρ γ /µγ ]) γare both bounded sequences. Then by radial elliptic estimates, up to a subsequence we do not only get that z

2 γ = µ 2 γ exp γ 2 2 - 1 .

 221 w γ -S γ = O( tγ γ ) in [0, ρ γ ], so that ρ γ = ρ γ for large γ and (3.13) holds. Finally, (3.14) and (3.15) follow from (3.21), (3.23) and ρ

γ = 4π γ α γ + O 1 γ 3 ,

 3 B0(r) ∆u γ dy = B0(ργ ) ∆u γ dy + O exp -δ 2 for any r ∈ [ρ γ , r γ ]. Moreover, by (3.11) and (3.28), we have that B0(r)∆t γ dy = -2πrt γ (r) = -

2 ,

 2 for any r ∈ [ρ γ , r γ ]. Applying the fundamental theorem of calculus and using (3.29), we nd thatu γ (r) -γ -T γ (r) γ = u γ (ρ γ ) -γ -T γ (ρ γ )

log r 2 γ r 2 ,

 22 .37) Note that for r ∈ [0, r γ ], we have u γ (r) ≥ c 1 andu γ (r) = c 1 + rγ r 1 2πs B0(s)∆u γ dy ds ≤ c 1 + 1 + o(1) γ

  .38) Now, let us take a sequence (σ γ ) γ such that r γ ≤ σ γ → 0,u γ -u ∞ C 1 ( Ω\B0(σγ )) → 0 ,

( 3 .

 3 40)Applying the fundamental therorem of calculus and using (3.14), (3.32), (3.39) and (3.40) we get thatu γ (r γ ) = u γ (σ γ ) + σγ rγ 1 2πr B0(r)∆u γ dy dr , = u γ (σ γ ) +

( 3 .

 3 41)Note that, since u γ (r γ ) = c 1 , one has necessarily that

  can compute u γ (r γ ) according to the expansion in (3.31) and ndu γ (r γ ) = γ -1 + a 2γ t γ (r γ ) γ + o(1) ,

  .38) either by(1.35) if s γ = O(µ γ ), or by (1.37) arguing as in (1.6) if s γ µ γ . Then, using that u γ (0) = γ, t γ (0) = 0, and the fundamental theorem of calculus, we get (1.31) from (1.38) and then conclude the proof of (1.29). Observe further that Now, let (s γ ) γ be such that s γ ∈ (r γ , 1] for all γ. We have that

	0	rγ	(∆u γ )(s)2πsds =	4π + o(1) γ	,	(1.39)
	by (1.37). sγ					rγ
	(∆u γ )(s)2πsds -	(∆u γ )(s)2πsds
	0					0
	≤ (λ 1 + 1)	sγ	u γ (s)2πsds ,	(1.40)
	rγ ≤ s γ (λ 1 + 1) √	π u γ 2 ,
	by (0.5), (1.27) in [r

γ , s γ ], and the Cauchy-Schwarz inequality, where • p stands for the L p norm in Ω. Then, estimating u γ as in (1.6), we get from (1.39) and (1.40) that

  .46) Moreover, by(1.14) and (1.43), we get that u γ / u γ 2 → v strongly in L 2 , so that

	lim inf γ→+∞	inf r∈(rγ ,1]

Ω v 2 dx = 1 and then that v ≡ 0 in Ω. Using (0.5), (1.43),

(1.46

) and the radial decay of the u γ 's, there exists δ ∈ (0, 1) such that γ u γ ∆u γ in [0, δ]. Then, estimating u γ as in (1.6) we nd -u γ (r) γr in [0, 1], and from the fundamental theorem of calculus that

  +∞ exponentially fast as γ → +∞ by (1.53), we also get that

	Since ργ µγ → sγ 0	(∆t γ )2πrdr = -4π + o	1 γ 2 .	(1.67)
	Now (1.66) and (1.67) give an estimate on u γ + by the fundamental theorem of calculus, we get	t γ γ in [ρ 1,γ , ρ γ ] as in (1.6) so that,
									.65)
	Then, from (1.55), (1.64) and (1.65) we get that
	0	sγ	(∆u γ )2πrdr -	4π γ	≤	4π C + o(1) γ 3	.	(1.66)

  .73) Indeed (1.73) follows directly from (1.57) if ρ γ = r γ , while if ρ γ = ρ 2,γ then (1.61) gives u γ = O(1) in [ρ γ ,[START_REF] Adimurthi | Blow-up analysis in dimension 2 and a sharp form of Trudinger-Moser inequality[END_REF] and (1.73) follows from (1.10).

	Then, by (0.4), (0.5) and
	(1.73), we always get that
	∆u γ = (λ 1 + o(1))u γ in [ρ γ , 1] .	(1.74)
	By elliptic estimates, this implies that (1.46) holds true with v satisfying ∆v = λ 1 v
	in Ω \ {0}. We shall now prove that v = v 1 . By (1.55), (1.62) and (1.74), we get
	that
	sγ
	0

  Now, we conclude the proof of Theorem 0.1. Comparing (1.61) (with (1.42)) and (1.72) at ρ γ , we obtain that

			u γ u γ 2	-v	C 0 ( Ω\B0(σγ ))	→ 0 ,	(1.77)
	min(σγ ,sγ )			ργ
			(∆u γ )(r)2πrdr =		(∆u γ )(r)2πrdr
	0					0
						min(σγ ,sγ )
				+ =	ργ 4π γ	+ o	(∆u γ )(r)2πrdr , γ 2 + o(min(σ (1.78) 1
	In order to get the second equality in (1.78), we estimate the integral up to ρ γ by
	(1.55) and (1.62), and the integral for ρ γ ≤ r ≤ min{σ γ , s γ } by (1.74) and (1.76).
	Using (0.4), (1.77) and (1.78), we nd that
	0	sγ	(∆u γ )(r)2πrdr =	4π γ	+ o(s γ u 2 ) + λ 1 u 2	0	sγ	v 1 (r)2πrdr,

) for all r ∈ [ρ γ

, 1]

. By (1.70),

(1.71

) and (1.76) we get that v is bounded with bounded laplacian around {0}, and then v ∈ C 1 ( Ω). Take now a sequence (σ γ ) γ such that σ γ ≥ ρ γ , σ γ = o(1) and as γ → +∞. Using u γ ≤ γ with (1.54) for r ≤ ρ 1,γ , (1.76) for ρ 1,γ ≤ r ≤ σ γ and (1.77) otherwise, we get that (u γ / u γ 2 ) γ converges to v in L 2 on the whole disk, so that Ω v 2 dx = 1, v > 0 in Ω and v = v 1 . Finally, we observe that for any sequence s γ ∈ [ρ γ , 1] we have that γ , s γ ) u 2 ) . for any sequence s γ ∈ [ρ γ , 1]. Then, (1.72) follows, using again the fundamental theorem of calculus, with (1.6) and u γ (1) = 0.

  3.31) uniformly in [ρ γ , r γ ]. Moreover, there exists δ > 0 such that Proof. Let us denote α γ = 1 + a 2γ and T γ = α γ t γ . Arguing as in (3.16), we get that

	(3.32)

B0(rγ )\B0(ργ )

∆u γ dy = O(exp(-δγ)).

C 1 (B0(1)\B0(σγ )) → 0 and r γ ≤ σ γ → 0 ,

Acknowledgements

The rst author was supported by Swiss National Science Foundation, projects nr. PP00P2-144669 and PP00P2-170588/1. The second author was supported by BQR Accueil EC 2018, funding the new researchers in the university of Lyon 1.

as γ → +∞, where r γ is as in (3.30). Then, by (3.1) we have

and, since


Alexandre Galdeano
email: agaldean@liris.cnrs.fr

Alix Gonnot
email: agonnot@liris.cnrs.fr

Clément Cottet
email: clement.cottet@etu.univ-lyon1.fr

Salima Hassas
email: shassas@liris.cnrs.fr

Mathieu Lefort
email: mlefort@liris.cnrs.fr

Amélie Cordier
email: amelie.cordier@hoomano.com

Developmental Learning for Social Robots in Real-World Interactions

Keywords: Developmental Learning, Social Robotics, Interactions, Intrinsic Motivation, Hierarchical Learning, Enaction

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Developmental learning is an artificial intelligence approach drawing inspiration from psychology, and more specifically from the way children learn and build incrementally their own representation of the world by experimenting their environment through interactions [START_REF] Piaget | La construction du réel chez l'enfant[END_REF], [START_REF] Cangelosi | Developmental robotics: From babies to robots[END_REF]. In robotics, developmental learning has been used to address problems such as reaching and grasping [START_REF] Savastano | A robotic model of reaching and grasping development[END_REF], exploration and curiosity [START_REF] Oudeyer | What is intrinsic motivation? a typology of computational approaches[END_REF], navigation [START_REF] Arkin | Motor schema-based mobile robot navigation[END_REF], speech acquisition [START_REF] Moulin-Frier | The role of intrinsic motivations in learning sensorimotor vocal mappings: A developmental robotics study[END_REF], [START_REF] Cangelosi | Integration of action and language knowledge: A roadmap for developmental robotics[END_REF], and adaptation to arm and leg damage [START_REF] Cully | Robots that can adapt like animals[END_REF].

In the field of social robotics, developmental learning appears to be an interesting approach to consider, notably because of its potential to bring answers to problems such as life-long learning, unsupervised learning without prior knowledge, and, last but not least, its incremental capacity to learn from few data.

In this paper, we present preliminary investigations on applying developmental learning for improving humanmachine interaction in the field of social robotics. The work reported here was performed within the project BE-HAVIORS.AI (an Engine enHancing verbAL and non-Verbal InteractiOns of RobotS, based on Artificial Intelligence). BEHAVIORS.AI is a joint laboratory 1 involving academic researchers and social robotics engineers which aim is to design and develop solutions to make human-robot interaction more instinctive and empathic. For that purpose, we explore the opportunities offered by developmental learning, we propose implementations on social robots, and we use the social robots field to implement and improve this framework.

Our goal is to implement a form of "interactional intelligence" in social robots. For that purpose, we focus on several aspects of social interactions: emotional intelligence, timing of the interaction, adaptability to the changing context, preference learning, etc. In the context of this project, we focus on how developmental learning could enable continuous learning of interaction skills and, ultimately, lead to the emergence of appropriate (and unique) behaviors in social robots. In this work, our goal was to set up a common framework for implementing, experimenting and evaluating developmental learning algorithms with various social robots. For bootstrapping our research, we decided to start with a very simple developmental learning algorithm that we chose because it is easy to implement and to analyze, and it offers interesting challenges (hierarchical learning, implementation of motivational systems, etc.). In the future, we will focus on more advanced algorithms.

This paper is organized as follows. In the next section, we briefly introduce developmental learning and we describe some of the theoretical hypotheses we rely on. In section III, we describe our work in progress and report on two distinct contributions. First, we present a first implementation. This implementation validates the potential of developmental learning algorithms to enhance the intelligence of interactions in social robots and highlights the next challenges to face. Next, we present a visualization tool that we have designed in order to help researchers better design developmental algorithms for social robots. The paper ends with a discussion on the challenges and perspectives we foresee for the next steps of the project.

II. DEVELOPMENTAL LEARNING

Developmental learning is a constructivist approach, inspired by theories on the cognitive development of human being (learn like a baby) [START_REF] Piaget | La construction du réel chez l'enfant[END_REF], [START_REF] Cangelosi | Developmental robotics: From babies to robots[END_REF], [START_REF] Guerin | Learning like a baby: A survey of artificial intelligence approaches[END_REF], that aims to address the "Symbol Grounding problem" [START_REF] Harnad | The symbol grounding problem[END_REF] by making an agent interacting with its environment "constructs sensorimotor transformation knowledge rather than an internal mirror of Fig. 1. Traditionally, the agent receives an input (observation) and computes the output (action). However, in our approach, the agent initiates the cycle by acting (experiment) then perceives the consequences (result). This is a way to implement an intrinsically motivated learning algorithm. Reprinted from [19, p. 246]. some external reality" [11, p. 164]. As such, the learning process occurs through the experience developed by the agent along its interaction with its environment [START_REF] Mazac | On bootstrapping sensori-motor patterns for a constructivist learning system in continuous environments[END_REF], and, according to some authors, learning these sensorimotor contingencies may even lead to some kind of consciousness [START_REF] O'regan | A sensorimotor account of vision and visual consciousness[END_REF]. Moreover, the agent's behavior is driven by an intrinsic motivation and its value function depends on the agent's behavior rather than of the environment states. We claim that such an approach has the necessary ingredients to develop autonomous robots that are able to adapt to highly dynamic environments that include humans. Using active perception may help the agent to deal with complex environments [START_REF] O'regan | A sensorimotor account of vision and visual consciousness[END_REF] and autonomously adapt to new situations thanks to its incremental learning and its intrinsic motivation. Learning from its experience of interaction with humans, including imitating them or drawing inspiration from their behaviors, allows for more natural interactions.

This learning paradigm may be implemented in several ways. For instance, in [START_REF] Pitti | Contingency perception and agency measure in visuo-motor spiking neural networks[END_REF], the authors use an implementation based on neural networks to detect contingencies in sensorimotor networks. In [START_REF] Drescher | Made-up minds: A constructivist approach to artificial intelligence[END_REF] developmental learning is based on a set of "tripartite structure[s] comprising a context, action, and result" called schema, this structure is then completed in [START_REF] Guerin | A piagetian model of early sensorimotor development[END_REF] by adding a target value, and in [START_REF] Holmes | Schema learning: Experiencebased construction of predictive action models[END_REF] the original schemas have been extended to "improve the original learning criteria to handle POMDP domains".

In this preliminary work, we chose to rely on a hierarchical implementation of schemas, drawing inspiration from the intrinsically motivated schema mechanism proposed in [START_REF] Georgeon | Implementation of DEvelopmentAl learning (IDEAL) course, Implementation of DEvelopmentAl Learning (IDEAL) MOOC[END_REF]. We made this choice because it is a simple and intuitive way to envision developmental learning. Our goal is to demonstrate that rudimentary implementations of developmental learning algorithms could be performed on social robots and could lead, in the long run, to new ways of implementing unsupervised learning abilities on these robots. More complex approaches and implementations are kept for future work.

This model is grounded on several key concepts:

• An interaction is an atomic element defined as a couple composed of the experiment performed by the agent and the result obtained for this experiment (see figure 1). Focusing on the interaction implements a form of active Fig. 2. Example of a composite interaction with its weight and valence, and its sub-interactions. Here, a complex move performed by the robot is in fact a sequence of primitive moves. One can observe that the background color of the symbols changes: it reflects the result perceived by the agent, i.e., blue for "my head has been touched" and white for "my head has not been touched".

perception. Moreover, the agent does not require any direct access to the state of the environment.

• An intended interaction is an interaction that the agent tries to perform (i.e., it is an action and an expected result for this action).

• An enacted interaction is an interaction that the agent has actually performed (i.e., an action and its result, that may be different from the expected result).

• The valence of an interaction is the internal value associated by the agent to it. It can be seen as the cost of doing the interaction from the agent's perspective.

• The proclivity is a weighted value computed by the agent, depending on its internal emotional state. At each step, the agent computes the proclivity of all the interaction that it can enact and chooses the best one given its current decision-making strategy (depending on its emotional state). In our approach, we observe the following properties.

• The agent reacts to its active perception of the environment, after experimenting an interaction with it, by attempting a new experiment based on its internal evaluation of its perception (difference between its prediction and the actual result of the experiment).

• The agent creates hierarchical aggregated schemas (a kind of abstraction). It memorizes the enacted interactions and aggregates them hierarchically, thus creating composite interactions. A composite interaction is an interaction composed of several primitive or composite interactions that can be performed in a row (see figure 2 for an example). Composite interactions can be seen as higher level representations of the abilities of the agent.

• The agent implements a strategy to balance exploration and exploitation. The agent chooses what to do next based on the set of enacted interactions using different strategies depending on the agent's internal state, the interactions' proclivity, and the previous interactions. Typically, the agent chooses the interaction with the highest proclivity but the agent's internal state may make it choose another interaction. This mechanism improves the diversity of enacted interactions and allows a better exploration of the interaction space. This approach has been successfully applied on several toy problems. Albeit it seems a promising way to address the problem of unsupervised and intrinsically motivated learning without any prior knowledge on the environment, it also raises numerous concerns. Just to list a few, we could mention the memory consumption (the current naive implementation is very costly as it records all interactions from the start), the sensitivity to noise, the required regularity of the environment, and the convergence of the learning process.

Our goal in the preliminary work reported here was to implement this approach on social robots, first to validate that they could at least provide the same results as in simulation, and second, to better identify the problems related to the reality gap. These contributions are described in the next section.

III. CONTRIBUTIONS

We started with toy implementations and we plan to gradually increase the complexity of our experiments.

A. Simple Experiment on a Social Robot

In this experiment we wanted to demonstrate that developmental learning could be used to make a social robot able to favor specific actions according to the reactions of the human. In the scope of social interaction, the robot's user act as the environment.

Using this approach, we were able to make a Nao robot learn sequences of two actions according to the preferences of the human interacting with it. In this experiment, Nao can do four primitive actions: 1) turn its head to the left; 2) turn its head to the right; 3) rise its left arm; 4) rise its right arm. After each movement, the robot waits a few seconds for the environment's feedback-here the user touching, or not, the robot's head. Each pair (movement, feedback) is an interaction. All the interactions which feedback is "The head sensor has been triggered" have a positive valence and the other interactions have a null valence. Starting from the second interaction enacted, the robot memorizes every distinct pair of interactions it enacts-i.e., if the robot enacts ABC, it memorizes AB and BC, and then associates a weight to it. The weight goes up every time the pair of interactions is enacted and starts at 1 when the pair is enacted for the first time. Each pair of interaction, called composite interactions, is considered as an interaction so that it can be combined with others. The valence of the composite interaction is equal to the valence of its last interaction.

The robot alternates regularly between exploration and exploitation. Exploitation. The robot chooses which simple interaction should enable it to enact the best composite interaction. The best composite interaction is the one with the highest proclivity (in this implementation, proclivity = valence × weight). Exploration. The robot chooses a random action in order to learn a new composite interaction.

With this experiment, we showed that we were able to perform a robot implementation of a previous algorithm performing well in simulation. Besides, when using this algorithm, the robot is able to learn a simple behavior adapted to the user's preferences.

We learned a lot about the problems we will face when increasing the complexity of our algorithms. We identified two obvious obstacles. First, the robot's movement and the user's reactions are a few seconds long which-compared to the near-instantaneous reaction times in simulation-is very long and that makes the execution of the algorithm with the robot a lot slower than in a simulation. Our research question here is to manage timing, i.e., to find a strategy to decide when and how to gather feedback from the environment and to match it with the active perception strategy of the agent. Second, the robot and the environment are less reliable than the computer and some errors may occur during the execution of the algorithm, causing the robot to learn wrong behaviors and to have to deal with noisy information. To go further and to facilitate the research work, we implemented a tool for the researchers for accelerating the design and the tunning of developmental learning algorithms.

B. Web Application for Visualization Experiment Control

The interface enables us to visualize the execution trace of the algorithms and to modify it on the fly (see figure 3). The interface displays the mental state of the robot in real-time by showing every simple interaction the robot ever enacted, the content of its memory (i.e., all the enacted composite interactions with weights and valences) and the type of behavior it is currently in (i.e., exploration or exploitation).

Elements currently active in the decision and memory processes are highlighted on the screen in real-time. This interface also allows the control of the agent's memory by the designer: increasing or decreasing the weight of a composite interaction, modifying the valence of a composite interaction, or add a new composite interaction like it was enacted. Modifying these elements through the interface enables us to observe how the learning process is impacted without reenacting the experiment from scratch.

Next, we created a second version of this interface with additional features (see figure 4). This new version is capable of displaying complex composite interactions and have some control on the algorithms, particularly with a virtual agent: (i) start, pause, and stop the agent's execution, (ii) define the agent's execution rate, i.e., how many milliseconds are between each steps, (iii) define the policy that will have the best valence and (iv) define some of the agent's preferences.

These two interfaces help us to have a better understanding This version is an improvement to the first one. The robot's memory and history of actions, its mood are displayed. The algorithm can be controlled and configured-to some extent-through this interface.

of how our algorithms behave under different and changing configurations. The control features can be used to speed up the robot's learning and therefore constitute useful tools for researchers and interaction designers in an experimentation context. They can quickly test and validate hypotheses through this tool, in order to produce configuration ready to be tested in real-world experiments.

C. Multi-robots implementations and experimentations

So far, we have built an architecture enabling us to experiment different developmental learning algorithms on various robots, including virtual robots (agents in a simulation environment). We have conducted experiments with Nao and Pepper, and we have an ongoing implementation for Cozmo. A virtual agent-combined with a virtual environment-has also been implemented in order to enable us to debug and test the behavior of an algorithm under various-and possibly varying-conditions quickly. We have proposed three different implementations of developmental learning algorithms, with increasing complexity, in different contexts, but their description is out of the scope of this paper. Demonstrating that the same algorithms behave well with different robots and for different application purposes is important as it shows that active perception and intrinsic motivation are key factors to enable learning without prior knowledge on the environment.

IV. DISCUSSION AND FUTURE WORK

We defended the idea that developmental artificial intelligence is a relevant approach for developing social robots that can interact in a more natural way with humans. Indeed, the developmental paradigm is intrinsically defined to provide autonomy, lifelong learning and adaptability to an agent, which are required properties for a social robot. Moreover, taking inspiration from human may provide more natural behaviors to robots. Preliminary experiments show that a developmental architecture, based on interaction, can lead a robot to learn a behavior desired by the human, providing indirect feedback through its actions. However, developmental AI is a recent research field and some major questions still have to be answered. To go beyond, we developed a web application that allows to monitor (and eventually to modify) the internal representations learned by the agent.

Our preliminary work is a starting point, raising several issues that we have to address to reach the objectives of our project. For instance, to go further, we will need to address scalability issues and robustness to noisy environments. Some other issues are related to improving the implementation and optimizing memory usage, choosing more complex-but more efficient-approaches, dealing with forgetting issues, etc. But more broadly, in the context of the BEHAVIORS.AI project, we want to study: a) How to choose the next best action to perform (depending on several parameters such as the intrinsic motivation, the trade-off between exploration and exploitation, the context), and how to take into account the timing in the process, which is of particular importance in social interactions? b) Using high level actions and perceptions may produce more complex behaviors without significantly increasing the search space's size, but how does it affect the algorithm's adaptability and relevance? In addition, the environment is, by definition, noisy. How to deal with noise and the potential lack of consistency of the environment? c) How to include a form of emotional intelligence, through empathy, in the process. Could we envision a way to share representations of the interactional context between humans and robots to globally improve the interactions? d) Could we include multimodality in the process in order to combine verbal and non-verbal interactions in the context understanding process? e) Could we estimate how developmental learning mechanisms implemented in social robot's applications affect the user's experience? Last, we are deeply interested in finding ways to evaluate the performances of developmental learning. Evaluating performances of such algorithms, when we cannot identify an a priori goal, is never easy. How can we assess that a solution performs well, or better than another, when the goal is not defined? How can we measure the impact on the user experience? Proposing tools for evaluating both the technical performances of our contributions and the improvements they bring on the user experience is a major objective for our future work.

Fig. 3 .

 3 Fig. 3. Visualization tool is used for the developmental algorithm presented in section III-A. The robot's memory and history of actions, the current type of behavior it has, and what it is about to do are displayed in real-time. The valence of each sequences may be altered for experimental purposes.

Fig. 4 .

 4 Fig. 4.

BEHAVIORS.AI involves people from the company HOOMANO and LIRIS's SMA research group. See behaviors.ai for more details.

*This work was supported by the French Agence Nationale de la Recherche (ANR) contract ANR-16-LCV2-0003-01.