Juan D Ávila

Louis Dupaigne

AND Ignacio Guerra

Marcelo Montenegro

STABLE SOLUTIONS FOR THE BILAPLACIAN WITH EXPONENTIAL NONLINEARITY

Keywords: 1991 Mathematics Subject Classification. Primary 35J65, Secondary 35J40 Biharmonic, singular solutions, stability. 1 2

Let λ * > 0 denote the largest possible value of λ such that

has a solution, where B is the unit ball in R N and n is the exterior unit normal vector. We show that for λ = λ * this problem possesses a unique weak solution u * . We prove that u * is smooth if N ≤ 12 and singular when N ≥ 13, in which case u * (r) = -4 log r + log(8(N -2)(N -4)/λ * ) + o(1) as r → 0. We also consider the problem with general constant Dirichlet boundary conditions.

Introduction

We study the fourth order problem

       ∆ 2 u = λe u in B u = a on ∂B ∂u ∂n = b on ∂B (1)
where a, b ∈ R, B is the unit ball in R N , N ≥ 1, n is the exterior unit normal vector and λ ≥ 0 is a parameter. Recently higher order equations have attracted the interest of many researchers. In particular fourth order equations with an exponential non-linearity have been studied in 4 dimensions, in a setting analogous to Liouville's equation, in [START_REF] Baraket | Singular limits for 4-dimensional semilinear elliptic problems with exponential nonlinearity[END_REF][START_REF] Djadli | Existence of conformal metrics with constant Q-curvature[END_REF][START_REF] Wei | Asymptotic behavior of a nonlinear fourth order eigenvalue problem[END_REF] and in higher dimensions by [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF][START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF][START_REF] Berchio | Some remarks on biharmonic elliptic problems with positive, increasing and convex nonlinearities[END_REF][START_REF] Berchio | Positivity preserving property for a class of biharmonic problems[END_REF][START_REF] Ferrero | The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity[END_REF].

We shall pay special attention to [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF] in the case a = b = 0, as it is the natural fourth order analogue of the classical Gelfand problem -∆u = λe u in Ω u = 0 on ∂Ω

(Ω is a smooth bounded domain in R N ) for which a vast literature exists [START_REF] Brezis | Blow-up for ut -∆u = g(u) revisited[END_REF][START_REF] Brezis | Blow-up solutions of some nonlinear elliptic problems[END_REF][START_REF] Cohen | Some positone problems suggested by nonlinear heat generation[END_REF][START_REF] Crandall | Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems[END_REF][START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources[END_REF][START_REF] Martel | Uniqueness of weak extremal solutions of nonlinear elliptic problems[END_REF][START_REF] Mignot | Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe[END_REF][START_REF] Mignot | Solution radiale singulière de -∆u = λe u[END_REF].

From the technical point of view, one of the basic tools in the analysis of ( 2) is the maximum principle. As pointed out in [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF], in general domains the maximum principle for ∆ 2 with Dirichlet boundary condition is not valid anymore. One of the reasons to study [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF] in a ball is that a maximum principle holds in this situation, see [START_REF] Boggio | Sulle funzioni di Freen d'ordine m[END_REF]. In this simpler setting, though there are some similarities between the two problems, several tools that are well suited for [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF] no longer seem to work for [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF].

where the constant N 2 (N -4) 2 /16 is known to be optimal. The proof of Theorem 1.3 is based on the observation that if u * is singular then λ * e u * ∼ 8(N -2)(N -4)|x| -4 near the origin. But 8(N -2)(N -4) > N 2 (N -4) 2 /16 if N ≤ 12 which would contradict the stability condition [START_REF] Baraket | Singular limits for 4-dimensional semilinear elliptic problems with exponential nonlinearity[END_REF].

In view of Theorem 1.3, it is natural to ask whether u * is singular in dimension N ≥ 13. If a = b = 0, we prove Theorem 1.4. Let N ≥ 13 and a = b = 0. Then the extremal solution u * to (1) is unbounded.

For general boundary values, it seems more difficult to determine the dimensions for which the extremal solution is singular. We observe first that given any a, b ∈ R, u * is the extremal solution of (1) if and only if u * -a is the extremal solution of the same equation with boundary condition u = 0 on ∂B. In particular, if λ * (a, b) denotes the extremal parameter for problem [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF], one has that λ * (a, b) = e -a λ * (0, b). So the value of a is irrelevant. But one may ask if Theorem 1.4 still holds for any N ≥ 13 and any b ∈ R. The situation turns out to be somewhat more complicated : Remark 1.6.

• We have not investigated the case b < -4.

• If follows from item a) that for b ∈ [-4, 0], the extremal solution is singular if and only if N ≥ 13. • It also follows from item a) that there exist values of b for which N min > 13.

We do not know whether u * remains bounded for 13 ≤ N < N min .

Our proof of Theorem 1.4 is related to an idea that Brezis and Vázquez applied for the Gelfand problem and is based on a characterization of singular energy solutions through linearized stability (see Theorem 3.1 in [START_REF] Brezis | Blow-up solutions of some nonlinear elliptic problems[END_REF]). In our context we show Proposition 1.7. Assume that u ∈ H 2 (B) is an unbounded weak solution of (1) satisfying the stability condition

λ B e u ϕ 2 ≤ B (∆ϕ) 2 , ∀ϕ ∈ C ∞ 0 (B). ( 7
)
Then λ = λ * and u = u * .

We do not use Proposition 1.7 directly but some variants of it -see Lemma 2.6 and Remark 2.7 in Section 2 -because we do not have at our disposal an explicit solution to the equation [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF]. Instead, we show that it is enough to find a sufficiently good approximation to u * . When N ≥ 32 we are able to construct such an approximation by hand. However, for 13 ≤ N ≤ 31 we resort to a computer assisted generation and verification.

Only in very few situations one may take advantage of Proposition 1.7 directly. For instance for problem [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF] with a = 0 and b = -4 we have an explicit solution ū(x) = -4 log |x| associated to λ = 8(N -2)(N -4). Thanks to Rellich's inequality [START_REF] Boggio | Sulle funzioni di Freen d'ordine m[END_REF] the solution ū satisfies condition [START_REF] Brezis | Blow-up for ut -∆u = g(u) revisited[END_REF] when N ≥ 13. Therefore, by Theorem 1.3 and a direct application of Proposition 1.7 we obtain Theorem 1.4 in the case b = -4.

In [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF] the authors say that a radial weak solution u to (1) is weakly singular if lim r→0 ru ′ (r) exists.

For example, the singular solutions U σ of [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF] verify this condition. As a corollary of Theorem 1.2 we show Proposition 1.8. The extremal solution u * to (1) with b ≥ -4 is always weakly singular.

A weakly singular solution either is smooth or exhibits a log-type singularity at the origin. More precisely, if u is a non-smooth weakly singular solution of (1) with parameter λ then (see [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF])

lim r→0 u(r) + 4 log r = log 8(N -2)(N -4) λ , lim r→0 ru ′ (r) = -4.
In Section 2 we describe the comparison principles we use later on. Section 3 is devoted to the proof of the uniqueness of u * and Propositions 1.7 and 1.8. We prove Theorem 1.3, the boundedness of u * in low dimensions, in Section 4. The argument for Theorem 1.4 is contained in Section 5 for the case N ≥ 32 and Section 6 for 13 ≤ N ≤ 31. In Section 7 we give the proof of Proposition 1.5.

Notation.

• B R : ball of radius R in R N centered at the origin. B = B 1 .

• n: exterior unit normal vector to B R • All inequalities or equalities for functions in L p spaces are understood to be a.e.

Comparison principles

Lemma 2.1. (Boggio's principle, [START_REF] Boggio | Sulle funzioni di Freen d'ordine m[END_REF])

If u ∈ C 4 (B R ) satisfies    ∆ 2 u ≥ 0 in B R u = ∂u ∂n = 0 on ∂B R then u ≥ 0 in B R . Lemma 2.2. Let u ∈ L 1 (B R ) and suppose that BR u∆ 2 ϕ ≥ 0 for all ϕ ∈ C 4 (B R ) such that ϕ ≥ 0 in B R , ϕ| ∂BR = 0 = ∂ϕ ∂n | ∂BR . Then u ≥ 0 in B R . Moreover u ≡ 0 or u > 0 a.e. in B R .
For a proof see Lemma 17 in [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF].

Lemma 2.3. If u ∈ H 2 (B R ) is radial, ∆ 2 u ≥ 0 in B R in the weak sense, that is BR ∆u∆ϕ ≥ 0 ∀ϕ ∈ C ∞ 0 (B R ), ϕ ≥ 0 and u| ∂BR ≥ 0, ∂u ∂n | ∂BR ≤ 0 then u ≥ 0 in B R .
Proof. We only deal with the case R = 1 for simplicity. Solve

   ∆ 2 u 1 = ∆ 2 u in B 1 u 1 = ∂u 1 ∂n = 0 on ∂B 1 in the sense u 1 ∈ H 2 0 (B 1 ) and B1 ∆u 1 ∆ϕ = B1 ∆u∆ϕ for all ϕ ∈ C ∞ 0 (B 1 ). Then u 1 ≥ 0 in B 1 by Lemma 2.2. Let u 2 = u -u 1 so that ∆ 2 u 2 = 0 in B 1 . Define f = ∆u 2 .
Then ∆f = 0 in B 1 and since f is radial we find that f is constant. It follows that u 2 = ar 2 + b. Using the boundary conditions we deduce a + b ≥ 0 and a ≤ 0, which imply u 2 ≥ 0.

Similarly we have

Lemma 2.4. If u ∈ H 2 (B R ) and ∆ 2 u ≥ 0 in B R in the weak sense, that is BR ∆u∆ϕ ≥ 0 ∀ϕ ∈ C ∞ 0 (B R ), ϕ ≥ 0 and u| ∂BR = 0, ∂u ∂n | ∂BR ≤ 0 then u ≥ 0 in B R .
The next lemma is a consequence of a decomposition lemma of Moreau [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]. For a proof see [START_REF] Gazzola | Critical dimensions and higher order Sobolev inequalities with remainder terms[END_REF][START_REF] Gazzola | Hardy inequalities with optimal constants and remainder terms[END_REF]. Lemma 2.5. Let u ∈ H 2 0 (B R ). Then there exist unique w, v ∈ H 2 0 (B R ) such that u = w + v, w ≥ 0, ∆ 2 v ≤ 0 in B R and BR ∆w∆v = 0.

We need the following comparison principle.

Lemma 2.6. Let u 1 , u 2 ∈ H 2 (B R ) with e u1 , e u2 ∈ L 1 (B R ). Assume that ∆ 2 u 1 ≤ λe u1 in B R in the sense BR ∆u 1 ∆ϕ ≤ λ BR e u1 ϕ ∀ϕ ∈ C ∞ 0 (B R ), ϕ ≥ 0, (8)
and ∆ 2 u 2 ≥ λe u2 in B R in the similar weak sense. Suppose also

u 1 | ∂BR = u 2 | ∂BR and ∂u 1 ∂n | ∂BR = ∂u 2 ∂n | ∂BR .
Assume furthermore that u 1 is stable in the sense that

(9) λ BR e u1 ϕ 2 ≤ BR (∆ϕ) 2 , ∀ϕ ∈ C ∞ 0 (B R ).
Then

u 1 ≤ u 2 in B R .
Proof.Let u = u 1 -u 2 . By Lemma 2.5 there exist w, v ∈ H 2 0 (B R ) such that u = w + v, w ≥ 0 and ∆ 2 v ≤ 0. Observe that v ≤ 0 so w ≥ u 1 -u 2 .

By hypothesis we have for all

ϕ ∈ C ∞ 0 (B R ), ϕ ≥ 0, BR ∆(u 1 -u 2 )∆ϕ ≤ λ BR (e u1 -e u2 )ϕ ≤ λ BR∩[u1≥u2]
(e u1 -e u2 )ϕ and by density this holds also for w:

(10) BR (∆w) 2 = BR ∆(u 1 -u 2 )∆w ≤ λ BR∩[u1≥u2]
(e u1 -e u2 )w = λ BR (e u1 -e u2 )w, where the first equality holds because BR ∆w∆v = 0. By density we deduce from ( 9):

(11) λ BR e u1 w 2 ≤ BR (∆w) 2 .
Combining [START_REF] Crandall | Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems[END_REF] and [START_REF] Dávila | Comparison results for PDEś with a singular potential[END_REF] we obtain BR e u1 w 2 ≤ BR (e u1 -e u2 )w.

Since u 1 -u 2 ≤ w the previous inequality implies

(12) 0 ≤ BR (e u1 -e u2 -e u1 (u 1 -u 2 ))w.
But by convexity of the exponential function e u1 -e u2 -e u1 (u 1 -u 2 ) ≤ 0 and we deduce from [START_REF] Djadli | Existence of conformal metrics with constant Q-curvature[END_REF] that (e u1 -e u2 -e u1 (u 1 -u 2 ))w = 0. Recalling that u 1 -u 2 ≤ w we deduce that u 1 ≤ u 2 .

Remark 2.7. The following variant of Lemma 2.6 also holds:

Let u 1 , u 2 ∈ H 2 (B R ) be radial with e u1 , e u2 ∈ L 1 (B R ). Assume ∆ 2 u 1 ≤ λe u1 in B R in the sense of (8) and ∆ 2 u 2 ≥ λe u2 in B R . Suppose u 1 | ∂BR ≤ u 2 | ∂BR and ∂u1 ∂n | ∂BR ≥ ∂u2
∂n | ∂BR and that the stability condition (9) holds. Then

u 1 ≤ u 2 in B R .
Proof. We solve for ũ ∈ H 2 0 (B R ) such that

BR ∆ũ∆ϕ = BR ∆(u 1 -u 2 )∆ϕ ∀ϕ ∈ C ∞ 0 (B R ).
By Lemma 2.3 it follows that ũ ≥ u 1 -u 2 . Next we apply the decomposition of Lemma 2.5 to ũ, that is ũ = w + v with w, v ∈ H 2 0 (B R ), w ≥ 0, ∆ 2 v ≤ 0 in B R and BR ∆w∆v = 0. Then the argument follows that of Lemma 2.6.

Finally, in several places we will need the method of sub and supersolutions in the context of weak solutions. Lemma 2.8. Let λ > 0 and assume that there exists ū ∈ H

2 (B R ) such that e ū ∈ L 1 (B R ), BR ∆ū∆ϕ ≥ λ BR e ūϕ for all ϕ ∈ C ∞ 0 (B R ), ϕ ≥ 0 and ū = a, ∂ ū ∂n ≤ b on ∂B 1 .
Then there exists a weak solution to (1) such that u ≤ ū.

The proof is similar to that of Lemma 19 in [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF].

3. Uniqueness of the extremal solution: proof of Theorem 1.2

Proof of Theorem 1.2. Suppose that v ∈ H 2 (B) satisfies ( 4), ( 5) and v ≡ u * . Notice that we do not need v to be radial. The idea of the proof is as follows :

Step 1. The function

u 0 = 1 2 (u * + v)
is a super-solution to the following problem

       ∆ 2 u = λ * e u + µηe u in B u = a on ∂B ∂u ∂n = b on ∂B (13)
for some

µ = µ 0 > 0, where η ∈ C ∞ 0 (B), 0 ≤ η ≤ 1 is a fixed radial cut-off function such that η(x) = 1 for |x| ≤ 1 2 , η(x) = 0 for |x| ≥ 3 4 .
Step 2. Using a solution to [START_REF] Ferrero | The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity[END_REF] we construct, for some λ > λ * , a super-solution to [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF]. This provides a solution u λ for some λ > λ * , which is a contradiction. Proof of Step 1. Observe that given 0 < R < 1 we must have for some

c 0 = c 0 (R) > 0 v(x) ≥ u * (x) + c 0 |x| ≤ R. (14)
To prove this we recall the Green's function for ∆ 2 with Dirichlet boundary conditions

       ∆ 2 x G(x, y) = δ y x ∈ B G(x, y) = 0 x ∈ ∂B ∂G ∂n (x, y) = 0 x ∈ ∂B,
where δ y is the Dirac mass at y ∈ B. Boggio gave an explicit formula for G(x, y) which was used in [START_REF] Grunau | Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions[END_REF] to prove that in dimension N ≥ 5 (the case 1 ≤ N ≤ 4 can be treated similarly)

G(x, y) ∼ |x -y| 4-N min 1, d(x) 2 d(y) 2 |x -y| 4 (15)
where

d(x) = dist(x, ∂B) = 1 -|x|.
and a ∼ b means that for some constant C > 0 we have C -1 a ≤ b ≤ Ca (uniformly for x, y ∈ B). Formula (15) yields

G(x, y) ≥ cd(x) 2 d(y) 2 (16)
for some c > 0 and this in turn implies that for smooth functions ṽ and ũ such that ṽ -ũ ∈ H 2 0 (B) and ∆ 2 (ṽ -ũ) ≥ 0,

ṽ(y) -ũ(y) = ∂B ∂∆ x G ∂n x (x, y)(ṽ -ũ) -∆ x G(x, y) ∂(ṽ -ũ) ∂n dx + B G(x, y)∆ 2 (ṽ -ũ) dx ≥ cd(y) 2 B (∆ 2 ṽ -∆ 2 ũ)d(x) 2 dx.
Using a standard approximation procedure, we conclude that

v(y) -u * (y) ≥ cd(y) 2 λ * B (e v -e u * )d(x) 2 dx. Since v ≥ u * , v ≡ u * we deduce (14). Let u 0 = (u * + v)/2.
Then by Taylor's theorem

e v = e u0 + (v -u 0 )e u0 + 1 2 (v -u 0 ) 2 e u0 + 1 6 (v -u 0 ) 3 e u0 + 1 24 (v -u 0 ) 4 e ξ2 (17)
for some u 0 ≤ ξ 2 ≤ v and

e u * = e u0 + (u * -u 0 )e u0 + 1 2 (u * -u 0 ) 2 e u0 + 1 6 (u * -u 0 ) 3 e u0 + 1 24 (u * -u 0 ) 4 e ξ1 (18)
for some u * ≤ ξ 1 ≤ u 0 . Adding ( 17) and ( 18) yields

1 2 (e v + e u * ) ≥ e u0 + 1 8 (v -u * ) 2 e u0 . ( 19
)
From ( 14) with R = 3/4 and [START_REF] Martel | Uniqueness of weak extremal solutions of nonlinear elliptic problems[END_REF] we see that u 0 = (u * + v)/2 is a super-solution of (13) with µ 0 := c 0 /8. Proof of Step 2. Let us show now how to obtain a weak super-solution of (1) for some λ > λ * . Given µ > 0, let u denote the minimal solution to [START_REF] Ferrero | The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity[END_REF]. Define ϕ 1 as the solution to

       ∆ 2 ϕ 1 = µηe u in B ϕ 1 = 0 on ∂B ∂ϕ 1 ∂n = 0 on ∂B,
and ϕ 2 be the solution of

       ∆ 2 ϕ 2 = 0 in B ϕ 2 = a on ∂B ∂ϕ 2 ∂n = b on ∂B.
If N ≥ 5 (the case 1 ≤ N ≤ 4 can be treated similarly), relation ( 16) yields

ϕ 1 (x) ≥ c 1 d(x) 2 for all x ∈ B, (20)
for some c 1 > 0. But u is a radial solution of ( 13) and therefore it is smooth in

B \ B 1/4 . Thus u(x) ≤ M ϕ 1 + ϕ 2 for all x ∈ B 1/2 , (21)
for some M > 0. Therefore, from ( 20) and ( 21), for λ > λ * with λ -λ * sufficiently small we have

( λ λ * -1)u ≤ ϕ 1 + ( λ λ * -1)ϕ 2 in B. Let w = λ λ * u -ϕ 1 -( λ λ * -1)ϕ 2 .
The inequality just stated guarantees that w ≤ u. Moreover

∆ 2 w = λe u + λµ λ * ηe u -µηe u ≥ λe u ≥ λe w in B and w = a ∂w ∂n = b on ∂B.
Therefore w is a super-solution to (1) for λ. By the method of sub and supersolutions a solution to (1) exists for some λ > λ * , which is a contradiction.

Proof of Proposition 1.7. Let u ∈ H 2 (B), λ > 0 be a weak unbounded solution of (1). If λ < λ * from Lemma 2.6 we find that u ≤ u λ where u λ is the minimal solution. This is impossible because u λ is smooth and u unbounded. If λ = λ * then necessarily u = u * by Theorem 1.2.

Proof of Proposition 1.8. Let u denote the extremal solution of (1) with b ≥ -4.

If u is smooth, then the result is trivial. So we restrict to the case where u is singular. By Theorem 1.3 we have in particular that N ≥ 13. We may also assume that a = 0. If b = -4 by Theorem 1.2 we know that if N ≥ 13 then u = -4 log |x| so that the desired conclusion holds. Henceforth we assume b > -4 in this section. For ρ > 0 define

u ρ (r) = u(ρr) + 4 log ρ, so that ∆ 2 u ρ = λ * e uρ in B 1/ρ . Then du ρ dρ ρ=1,r=1 = u ′ (1) + 4 > 0.
Hence, there is δ > 0 such that

u ρ (r) < u(r) for all 1 -δ < r ≤ 1, 1 -δ < ρ ≤ 1.
This implies

u ρ (r) < u(r) for all 0 < r ≤ 1, 1 -δ < ρ ≤ 1. ( 22
)
Otherwise set

r 0 = sup { 0 < r < 1 | u ρ (r) ≥ u(r) }.

This definition yields

u ρ (r 0 ) = u(r 0 ) and u ′ ρ (r 0 ) ≤ u ′ (r 0 ). ( 23
) Write α = u(r 0 ), β = u ′ (r 0 ). Then u satisfies      ∆ 2 u = λe u on B r0 u(r 0 ) = α u ′ (r 0 ) = β. (24)
Observe that u is an unbounded H 2 (B r0 ) solution to [START_REF] Wei | Asymptotic behavior of a nonlinear fourth order eigenvalue problem[END_REF], which is also stable. Thus Proposition 1.7 shows that u is the extremal solution to this problem. On the other hand u ρ is a supersolution to [START_REF] Wei | Asymptotic behavior of a nonlinear fourth order eigenvalue problem[END_REF], since u ′ ρ (r 0 ) ≤ β by [START_REF] Rellich | Halbbeschränkte Differentialoperatoren höherer Ordnung[END_REF]. We may now use Theorem 1.2 and we deduce that u(r) = u ρ (r) for all 0 < r ≤ r 0 , which in turn implies by standard ODE theory that u(r) = u ρ (r) for all 0 < r ≤ 1, a contradiction with [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]. This proves estimate [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF].

From [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF] we see that

du ρ dρ ρ=1 (r) ≥ 0 for all 0 < r ≤ 1. (25) But du ρ dρ ρ=1 (r) = u ′ (r)r + 4 for all 0 < r ≤ 1
and this together with (25) implies

du ρ dρ (r) = 1 ρ (u ′ (ρr)ρr + 4) ≥ 0 for all 0 < r ≤ 1 ρ , 0 < ρ ≤ 1. ( 26
)
which means that u ρ (r) is non-decreasing in ρ. We wish to show that lim ρ→0 u ρ (r) exists for all 0 < r ≤ 1. For this we shall show

u ρ (r) ≥ -4 log(r) + log 8(N -2)(N -4) λ * for all 0 < r ≤ 1 ρ , 0 < ρ ≤ 1. (27) Set u 0 (r) = -4 log(r) + log 8(N -2)(N -4) λ * .
and suppose that (27) is not true for some 0 < ρ < 1. Let

r 1 = sup { 0 < r < 1/ρ | u ρ (r) < u 0 (r) }.
Observe that

λ * > 8(N -2)(N -4). ( 28
)
Otherwise w = -4 ln r would be a strict supersolution of the equation satisfied by u, which is not possible by Theorem 1.2. In particular, r 1 < 1/ρ and

u ρ (r 1 ) = u 0 (r 1 ) and u ′ ρ (r 1 ) ≥ u ′ 0 (r 1 ). It follows that u 0 is a supersolution of        ∆ 2 u = λ * e u in B r1 u = A on ∂B r1 ∂u ∂n = B on ∂B r1 , (29)
with A = u ρ (r 1 ) and B = u ′ ρ (r 1 ). Since u ρ is a singular stable solution of (29), it is the extremal solution of the problem by Proposition 1.7. By Theorem 1.2, there is no strict supersolution of (29) and we conclude that u ρ ≡ u 0 first for 0 < r < r 1 and then for 0 < r ≤ 1/ρ. This is impossible for ρ > 0 because u ρ (1/ρ) = 4 log ρ and u 0 (1/ρ) < 4 log ρ + log( 8(N -2)(N -4) λ *

) < u ρ (1/ρ) by ( 28). This proves (27). By ( 26) and ( 27) we see that v(r) = lim ρ→0 u ρ (r) exists for all 0 < r < +∞, where the convergence is uniform (even in C k for any k) on compact sets of R N \{0}.

Moreover v satisfies ∆ 2 v = λ * e v in R N \ {0}. ( 30
)
Then for any r > 0

v(r) = lim ρ→0 u ρ (r) = lim ρ→0 u(ρr) + 4 log(ρr) -4 log(r) = v(1) -4 log(r).
Hence, using equation (30) we obtain

v(r) = -4 log r + log 8(N -2)(N -4) λ * = u 0 (r).
But then We will show first Lemma 4.1. Suppose that the extremal solution u * to (1) is singular. Then for any σ > 0 there exists 0 < R < 1 such that

u ′ ρ (r) = u ′ (ρr)ρ → -4,
u * (x) ≥ (1 -σ) log 1 |x| 4 , ∀ |x| < R. (32)
Proof. Assume by contradiction that (32) is false. Then there exists σ > 0 and a sequence x k ∈ B with x k → 0 such that

u * (x k ) < (1 -σ) log 1 |x k | 4 . (33) Let s k = |x k | and choose 0 < λ k < λ * such that max B u λ k = u λ k (0) = log 1 s 4 k . (34) Note that λ k → λ * , otherwise u λ k would remain bounded. Let v k (x) = u λ k (s k x) log( 1 s 4 k ) x ∈ B k ≡ 1 s k B. Then 0 ≤ v k ≤ 1, v k (0) = 1, ∆ 2 v k (x) = λ k s 4 k log( 1 s 4 k ) e u λ k (s k x) ≤ λ k log( 1 s 4 k ) → 0 in B k by (34). By elliptic regularity v k → v uniformly on compact sets of R N to a function v satisfying 0 ≤ v ≤ 1, v(0) = 1, ∆ 2 v = 0 in R N
.By Liouville's theorem for biharmonic functions [START_REF] Huilgol | On Liouville's theorem for biharmonic functions[END_REF] we conclude that v is constant and therefore v ≡ 1.

Since

|x k | = s k we deduce that u λ k (x k ) log( 1 s 4 k ) → 1,
which contradicts (33).

Proof of Theorem 1.3. We write for simplicity u = u * , λ = λ * . Assume by contradiction that u * is unbounded and 5 ≤ N ≤ 12. If N ≤ 4 the problem is subcritical, and the boundedness of u * can be proved by other means : no singular solutions exist for positive λ (see [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF])-though in dimension N = 4 they can blow up as λ → 0, see [START_REF] Wei | Asymptotic behavior of a nonlinear fourth order eigenvalue problem[END_REF].

For ε > 0 let ψ = |x| 4-N 2 +ε and let η ∈ C ∞ 0 (R N ) with η ≡ 1 in B 1/2 and supp(η) ⊆ B. Observe that (∆ψ) 2 = (H N + O(ε))|x| -N +2ε , where H N = N 2 (N -4) 2 16
.

Using a standard approximation argument as in the proof of Lemma 2.6, we can use ψη as a test function in [START_REF] Cohen | Some positone problems suggested by nonlinear heat generation[END_REF] and we obtain

B (∆ψ) 2 + O(1) ≥ λ B e u ψ 2 ,
since the contribution of the integrals outside a fixed ball around the origin remains bounded as ε → 0 (here O(1) denotes a bounded function as ε → 0). This implies

λ B e u |x| 4-N +2ε ≤ (H N + O(ε)) B |x| -N +2ε = ω N H N 2ε + O(1) (35)
where ω N is the surface area of the unit N -1 dimensional sphere S N -1 . In particular B e u |x| 4-N +2ε < +∞.

For ε > 0 we define ϕ = |x| 4-N +2ε . Note that away from the origin

∆ 2 ϕ = εk N |x| -N +2ε , where k N = 4(N -2)(N -4) + O(ε). (36) Let ϕ j solve    ∆ 2 ϕ j = εk N min(|x| -N +2ε , j) in B ϕ j = ∂ϕ j ∂n = 0 on ∂B. ( 37
)
Then ϕ j ↑ ϕ as j → +∞. Using (35) and (37)

εk N B u min(|x| -N +2ε , j) = B u∆ 2 ϕ j = λ B e u ϕ j ≤ λ B e u ϕ ≤ ω N H N 2ε + O(1)
where O( 1) is bounded as ε → 0 independently of j. Letting j → +∞ yields

εk N B u |x| -N +2ε ≤ ω N H N 2ε + O(1), (38)
showing that the integral on the left hand side is finite. On the other hand, by (32)

εk N B u |x| -N +2ε ≥ εk N ω N (1 -σ) 1 0 log( 1 r 4 )r -1+2ε dr = k N ω N (1 -σ) 1 ε . (39)
Combining ( 38) and (39) we obtain

(1 -σ)k N ≤ H N 2 + O(ε).
Letting ε → 0 and then σ → 0 we have

8(N -2)(N -4) ≤ H N = N 2 (N -4) 2 16 .
This is valid only if N ≥ 13, a contradiction.

Remark 4.2. The conclusion of Theorem 1.3 can be obtained also from Proposition 1.8. However that proposition depends crucially on the radial symmetry of the solutions, while the argument in this section can be generalized to other domains.

The extremal solution is singular in large dimensions

In this section we take a = b = 0 and prove Theorem 1.4 for N ≥ 32. The idea for the proof of Theorem 1.4 is to to estimate accurately from above the function λ * e u * , and to deduce that the operator ∆ 2 -λ * e u * has a strictly positive first eigenvalue (in the H 2 0 (B) sense). Then, necessarily, u * is singular. Upper bounds for both λ * and u * are obtained by finding suitable sub and supersolutions. For example, if for some λ 1 there exists a supersolution then λ * ≥ λ 1 . If for some λ 2 one can exhibit a stable singular subsolution u, then λ * ≤ λ 2 . Otherwise λ 2 < λ * and one can then prove that the minimal solution u λ2 is above u, which is impossible. The bound for u * also requires a stable singular subsolution.

It turns out that in dimension N ≥ 32 we can construct the necessary subsolutions and verify their stability by hand. For dimensions 13 ≤ N ≤ 31 it seems difficult to find these subsolutions explicitly. We adopt then an approach that involves a computer assisted construction of subsolutions and verification of the desired inequalities. We present this part in the next section. Proof. Define ū(x) = -4 log |x|. Then ū satisfies

       ∆ 2 ū = 8(N -2)(N -4)e ū in R N ū = 0 on ∂B 1 ∂ ū ∂n = -4 on ∂B 1
Observe that since ū is a supersolution to (1) with a = b = 0 we deduce immediately that λ * ≥ 8(N -2)(N -4).

In the case λ * = 8(N -2)(N -4) we have u λ ≤ ū for all 0 ≤ λ < λ * because ū is a supersolution, and therefore u * ≤ ū holds. Alternatively, one can invoke Theorem 3 in [START_REF] Arioli | A semilinear fourth order elliptic problem with exponential nonlinearity[END_REF] to conclude that we always have λ * > 8(N -2)(N -4).

Suppose now that λ * > 8(N -2)(N -4). We prove that u λ ≤ ū for all 8(N -2)(N -4) < λ < λ * . Fix such λ and assume by contradiction that u λ ≤ ū is not true. Note that for r < 1 and sufficiently close to 1 we have u λ (r) < ū(r) because

u ′ λ (1) = 0 while ū′ (1) = -4. Let R 1 = inf{ 0 ≤ R ≤ 1 | u λ < ū in (R, 1) }. Then 0 < R 1 < 1, u λ (R 1 ) = ū(R 1 ) and u ′ λ (R 1 ) ≤ ū′ (R 1 ). So u λ is a super-solution to the problem (40)        ∆ 2 u = 8(N -2)(N -4)e u in B R1 u = u λ (R 1 ) on ∂B R1 ∂u ∂n = u ′ λ (R 1 ) on ∂B R1
while ū is a subsolution to (40). Moreover it is stable for this problem, since from Rellich's inequality (6) and 8(N -2)(N -4) ≤ N 2 (N -4) 2 /16 for N ≥ 13, we have

8(N -2)(N -4) BR 1 e ūϕ 2 ≤ N 2 (N -4) 2 16 R N ϕ 2 |x| 4 ≤ R N (∆ϕ) 2 ∀ϕ ∈ C ∞ 0 (B R1 ).
By Remark 2.7 we deduce ū ≤ u λ in B R1 which is impossible.

An upper bound for λ * is obtained by considering again a stable, singular subsolution to the problem (with another parameter, though): Lemma 5.2. For N ≥ 32 we have

λ * ≤ 8(N -2)(N -4)e 2 . ( 41
)
Proof. Consider w = 2(1 -r 2 ) and define

u = ū -w where ū(x) = -4 log |x|. Then ∆ 2 u = 8(N -2)(N -4) 1 r 4 = 8(N -2)(N -4)e ū = 8(N -2)(N -4)e u+w ≤ 8(N -2)(N -4)e 2 e u .
Also u(1) = u ′ (1) = 0, so u is a subsolution to (1) with parameter λ 0 = 8(N -2)(N -4)e 2 .

For N ≥ 32 we have λ 0 ≤ N 2 (N -4) 2 /16. Then by [START_REF] Boggio | Sulle funzioni di Freen d'ordine m[END_REF] u is a stable subsolution of (1) with λ = λ 0 . If λ * > λ 0 = 8(N -2)(N -4)e 2 the minimal solution u λ0 to (1) with parameter λ 0 exists and is smooth. From Lemma 2.6 we find u ≤ u λ0 which is impossible because u is singular and u λ0 is bounded. Thus we have proved (41) for N ≥ 32.

Proof of Theorem 1.4 in the case N ≥ 32.

Combining Lemma 5.1 and 5.2 we have that if N ≥ 32 then λ * e u * ≤ r -4 8(N -2)(N -4)e 2 ≤ r -4 N 2 (N -4) 2 /16. This and [START_REF] Boggio | Sulle funzioni di Freen d'ordine m[END_REF] show that inf

ϕ∈C ∞ 0 (B) B (∆ϕ) 2 -λ * B e u * ϕ 2 B ϕ 2 > 0 which is not possible if u * is bounded.
6. A computer assisted proof for dimensions 13 ≤ N ≤ 31 Throughout this section we assume a = b = 0. As was mentioned in the previous section, the proof of Theorem 1.4 relies on precise estimates for u * and λ * . We present first some conditions under which it is possible to find these estimates. Later we show how to meet such conditions with a computer assisted verification.

The first lemma is analogous to Lemma 5.2.

Lemma 6.1. Suppose there exist ε > 0, λ > 0 and a radial function

u ∈ H 2 (B) ∩ W 4,∞ loc (B \ {0}) such that ∆ 2 u ≤ λe u for all 0 < r < 1 |u(1)| ≤ ε, ∂u ∂n (1) ≤ ε u ∈ L ∞ (B) λe ε B e u ϕ 2 ≤ B (∆ϕ) 2 for all ϕ ∈ C ∞ 0 (B). (42) Then λ * ≤ λe 2ε . Proof. Let ψ(r) = εr 2 -2ε (43) so that ∆ 2 ψ ≡ 0, ψ(1) = -ε, ψ ′ (1) = 2ε and -2ε ≤ ψ(r) ≤ -ε for all 0 ≤ r ≤ 1. It follows that ∆ 2 (u + ψ) ≤ λe u = λe -ψ e u+ψ ≤ λe 2ε e u+ψ .
On the boundary we have u(1) + ψ(1) ≤ 0, u ′ (1) + ψ ′ (1) ≥ 0. Thus u + ψ is a singular subsolution to the equation with parameter λe 2ε . Moreover, since ψ ≤ -ε we have λe 2ε e u+ψ ≤ λe ε e u and hence, from (42) we see that u + ψ is stable for the problem with parameter λe 2ε . If λe 2ε < λ * then the minimal solution associated to the parameter λe 2ε would be above u + ψ, which is impossible because u is singular. Lemma 6.2. Suppose we can find ε > 0, λ > 0 and u ∈ H

2 (B) ∩ W 4,∞ loc (B \ {0}) such that ∆ 2 u ≥ λe u for all 0 < r < 1 |u(1)| ≤ ε, ∂u ∂n (1) ≤ ε.
Then

λe -2ε ≤ λ * .
Proof. Let ψ be given by ( 43). Then u -ψ is a supersolution to the problem with parameter λe -2ε .

The next result is the main tool to guarantee that u * is singular. The proof, as in Lemma 5.1, is based on an upper estimate of u * by a stable singular subsolution. Lemma 6.3. Suppose there exist ε 0 , ε > 0, λ a > 0 and a radial function

u ∈ H 2 (B) ∩ W 4,∞ loc (B \ {0}) such that ∆ 2 u ≤ (λ a + ε 0 )e u for all 0 < r < 1 (44) ∆ 2 u ≥ (λ a -ε 0 )e u for all 0 < r < 1 (45) |u(1)| ≤ ε, ∂u ∂n (1) ≤ ε (46) u ∈ L ∞ (B) (47) β 0 B e u ϕ 2 ≤ B (∆ϕ) 2 for all ϕ ∈ C ∞ 0 (B), (48)
where

β 0 = (λ a + ε 0 ) 3 (λ a -ε 0 ) 2 e 9ε . ( 49
)
Then u * is singular and

(λ a -ε 0 )e -2ε ≤ λ * ≤ (λ a + ε 0 )e 2ε . ( 50
)
Proof. By Lemmas 6.1 and 6.2 we have (50). Let

δ = log λ a + ε 0 λ a -ε 0 + 3ε.
and define

ϕ(r) = - δ 4 r 4 + 2δ.
We claim that

u * ≤ u + ϕ in B 1 . (51)
To prove this, we shall show that for λ < λ *

u λ ≤ u + ϕ in B 1 . (52) Indeed, we have ∆ 2 ϕ = -δ2N (N + 2) ϕ(r) ≥ δ for all 0 ≤ r ≤ 1 ϕ(1) ≥ δ ≥ ε, ϕ ′ (1) = -δ ≤ -ε and therefore ∆ 2 (u + ϕ) ≤ (λ a + ε 0 )e u + ∆ 2 ϕ ≤ (λ a + ε 0 )e u = (λ a + ε 0 )e -ϕ e u+ϕ ≤ (λ a + ε 0 )e -δ e u+ϕ . (53)
By (50) and the choice of δ

(λ a + ε 0 )e -δ = (λ a -ε 0 )e -3ε < λ * . ( 54
)
To prove (52) it suffices to consider λ in the interval (λ a -ε 0 )e -3ε < λ < λ * . Fix such λ and assume that (52) is not true. Write

ū = u + ϕ and let R 1 = sup{ 0 ≤ R ≤ 1 | u λ (R) = ū(R) }. Then 0 < R 1 < 1 and u λ (R 1 ) = ū(R 1 ). Since u ′ λ (1) = 0 and ū′ (1) < 0 we must have u ′ λ (R 1 ) ≤ ū′ (R 1 ). Then u λ is a solution to the problem        ∆ 2 u = λe u in B R1 u = u λ (R 1 ) on ∂B R1 ∂u ∂n = u ′ λ (R 1 ) on ∂B R1
while, thanks to (53) and (54), ū is a subsolution to the same problem. Moreover ū is stable thanks to (48) since, by Lemma 6.1,

λ < λ * ≤ (λ a + ε 0 )e 2ε (55)
and hence λe ū ≤ (λ a + ε 0 )e 2ε e 2δ e u ≤ β 0 e u .

We deduce ū ≤ u λ in B R1 which is impossible, since ū is singular while u λ is smooth. This establishes (51).

From (51) and (55) we have λ * e u * ≤ β 0 e -ε e u and therefore inf

ϕ∈C ∞ 0 (B) B (∆ϕ) 2 -λ * e u * ϕ 2 B ϕ 2 > 0.
This is not possible if u * is a smooth solution.

For each dimension 13 ≤ N ≤ 31 we construct u satisfying (44) to (48) of the form

u(r) = -4 log r + log 8(N -2)(N -4) λ for 0 < r < r 0 ũ(r) for r 0 ≤ r ≤ 1 (56)
where ũ is explicitly given. Thus u satisfies (47) automatically.

Numerically it is better to work with the change of variables

w(s) = u(e s ) + 4s, -∞ < s < 0 which transforms the equation ∆ 2 u = λe u into Lw + 8(N -2)(N -4) = λe w , -∞ < s < 0 where Lw = d 4 w ds 4 + 2(N -4) d 3 w ds 3 + (N 2 -10N + 20) d 2 w ds 2 -2(N -2)(N -4)

dw ds

The boundary conditions u(1) = 0, u ′ (1) = 0 then yield

w(0) = 0, w ′ (0) = 4.
Regarding the behavior of w as s → -∞ observe that

u(r) = -4 log r + log 8(N -2)(N -4) λ for r < r 0 if and only if w(s) = log 8(N -2)(N -4) λ for all s < log r 0 .
The steps we perform are the following:

1) We fix x 0 < 0 and using numerical software we follow a branch of solutions to

         L ŵ + 8(N -2)(N -4) = λe ŵ , x 0 < s < 0 ŵ(0) = 0, ŵ′ (0) = t ŵ(x 0 ) = log 8(N -2)(N -4) λ , d 2 ŵ ds 2 (x 0 ) = 0, d 3 ŵ ds 3 (x 0 ) = 0
as t increases from 0 to 4. The numerical solution ( ŵ, λ) we are interested in corresponds to the case t = 4. The five boundary conditions are due to the fact that we are solving a fourth order equation with an unknown parameter λ.

2) Based on ŵ, λ we construct a C 3 function w which is constant for s ≤ x 0 and piecewise polynomial for x 0 ≤ s ≤ 0. More precisely, we first divide the interval [x 0 , 0] in smaller intervals of length h. Then we generate a cubic spline approximation g f l with floating point coefficients of d 4 ŵ ds 4 . From g f l we generate a piecewise cubic polynomial g ra which uses rational coefficients and we integrate it 4 times to obtain w, where the constants of integration are such that d j w ds j (x 0 ) = 0, 1 ≤ j ≤ 3 and w(x 0 ) is a rational approximation of log(8(N -2)(N -4)/λ). Thus w is a piecewise polynomial function that in each interval is of degree 7 with rational coefficients, and which is globally C 3 . We also let λ be a rational approximation of λ. With these choices note that Lw + 8(N -2)(N -4) -λe w is a small constant (not necessarily zero) for s ≤ x 0 .

3) The condition (44) and (45) we need to check for u are equivalent to the following inequalities for w

Lw + 8(N -2)(N -4) -(λ + ε 0 )e w ≤ 0, -∞ < s < 0 (57) Lw + 8(N -2)(N -4) -(λ -ε 0 )e w ≥ 0, -∞ < s < 0. ( 58
)
Using a program in Maple we verify that w satisfies (57) and (58). This is done evaluating a second order Taylor approximation of Lw +8(N -2)(N -4)-(λ+ε)e w at sufficiently close mesh points. All arithmetic computations are done with rational numbers, thus obtaining exact results. The exponential function is approximated by a Taylor polynomial of degree 14 and the difference with the real value is controlled.

More precisely, we write

f (s) = Lw + 8(N -2)(N -4) -(λ + ε 0 )e w , f (s) = Lw + 8(N -2)(N -4) -(λ + ε 0 )T (w),
where T is the Taylor polynomial of order 14 of the exponential function around 0. Applying Taylor's formula to f at y j , we have for s ∈ [y j , y j+h ],

f (s) ≤ f (y j ) + |f ′ (y j )|h + 1 2 M h 2 ≤ f (y j ) + | f ′ (y j )|h + 1 2 M h 2 + |f (y j ) -f (y j )| + |f ′ (y j ) -f ′ (y j )|h ≤ f (y j ) + | f ′ (y j )|h + 1 2 M h 2 + E 1 + E 2 h,
where

M is a bound for |f ′′ | in [y j , y j + h] E 1 is such that (λ + ε 0 )|e w -T (w)| ≤ E 1 in [y j , y j + h] E 2 is such that (λ + ε 0 )|(e w -T ′ (w))w ′ | ≤ E 2 in [y j , y j + h].
So, inequality (57) will be verified on each interval [y j , y j + h] where w is a polynomial as soon as

f (y j ) + | f ′ (y j )|h + 1 2 M h 2 + E 1 + E 2 h ≤ 0. (59)
When more accuracy is desired, instead of (59) one can verify that

f (x i ) + | f ′ (x i )| h m + 1 2 M ( h m ) 2 + E 1 + E 2 h m ≤ 0,
where (x i ) i=1...m+1 are m + 1 equally spaced points in [y j , y j + h].

We obtain exact values for the upper bounds M, E 1 , E 2 as follows. First note that f ′′ = Lw ′′ -(λ+ ε 0 )e w ((w ′ ) 2 + w ′′ ). On [y j , y j + h], we have w(s) = 7 i=0 a i (s-y j ) i and we estimate |w(s)| ≤ 7 i=0 |a i |h i for s ∈ [y j , y j + h]. Similarly,

d ℓ w ds ℓ (s) ≤ 7 i=ℓ i(i -1) . . . (i -ℓ + 1)|a i |h i-ℓ for all s ∈ [y j , y j + h] (60)
The exponential is estimated by e w ≤ e 1 ≤ 3, since our numerical data satisfies the rough bounds -3/2 ≤ w ≤ 1. Using this information and (60) yields a rational upper bound M . E 1 is estimated using Taylor's formula : 15 15! .

E 1 = (λ + ε 0 ) (3/2)
Similarily, E 2 = (λ + ε 0 ) (3/2) 14 14! B 1 where B 1 is the right hand side of (60) when ℓ = 1.

4)

We show that the operator ∆ 2 -βe u where u(r) = w(log r) -4 log r, satisfies condition (48) for some β ≥ β 0 where β 0 is given by (49). In dimension N ≥ 13 the operator ∆ 2 -βe u has indeed a positive eigenfunction in H 2 0 (B) with finite eigenvalue if β is not too large. The reason is that near the origin

βe u = c |x| 4 ,
where c is a number close to 8(N -2)(N -4)β/λ. If β is not too large compared to λ then c < N 2 (N -4) 2 /16 and hence, using (6), ∆ 2 -βe u is coercive in H 2 0 (B r0 ) (this holds under even weaker conditions, see [START_REF] Dávila | Comparison results for PDEś with a singular potential[END_REF]). It follows that there exists a first eigenfunction ϕ 1 ∈ H 2 0 (B) for the operator ∆ 2 -βe u with a finite first eigenvalue µ 1 , that is

∆ 2 ϕ 1 -βe u ϕ 1 = µ 1 ϕ 1 in B ϕ 1 > 0 in B ϕ 1 ∈ H 2 0 (B). Moreover µ 1 can be characterized as µ 1 = inf ϕ∈C ∞ 0 (B) B (∆ϕ) 2 -βe u ϕ 2
B ϕ 2 and is the smallest number for which a positive eigenfunction in H 2 0 (Ω) exists. Thus to prove that (48) holds it suffices to verify that µ 1 ≥ 0 and for this it is enough to show the existence of a nonnegative ϕ ∈ H

2 0 (B), ϕ ≡ 0 such that        ∆ 2 ϕ -βe u ϕ ≥ 0 in B ϕ = 0 on ∂B ∂ϕ ∂n ≤ 0 on ∂B. (61)
Indeed, multiplication of (61) by ϕ 1 and integration by parts yields

µ 1 B ϕϕ 1 + ∂B ∂ϕ ∂n ∆ϕ 1 ≥ 0.
But ∆ϕ 1 ≥ 0 on ∂B and thus µ 1 ≥ 0. To achieve (61) we again change variables and define φ(s) = ϕ(e s ) -∞ < s ≤ 0.

Then we have to find φ ≥ 0, φ ≡ 0 satisfying

     Lφ -βe w φ ≥ 0 in -∞ < s ≤ 0 φ(0) = 0 φ ′ (0) ≤ 0. (62)
Regarding the behavior as s → -∞, we note that w is constant for -∞ < s < x 0 , and therefore, if

Lφ -βe w φ ≡ 0 -∞ < s ≤ x 0
then φ is a linear combination of exponential functions e -αs where α must be a solution to

α 4 -2(N -4)α 3 + (N 2 -10N + 20)α 2 + 2(N -2)(N -4)α = βe w(x0)
where βe w(x0) is close to 8(N -2)(N -4)β/λ. If N ≥ 13 the polynomial

α 4 -2(N -4)α 3 + (N 2 -10N + 20)α 2 + 2(N -2)(N -4)α -8(N -2)(N -4)
has 4 distinct real roots, while if N ≤ 12 there are 2 real roots and 2 complex conjugate. If N ≥ 13 there is exactly one root in the interval (0, (N -4)/2), 2 roots greater than (N -4)/2 and one negative. We know that ϕ(r) = φ(log r) ∼ r -α is in H 2 , which forces α < (N -4)/2. It follows that for s < x 0 , φ is a combination of e -α0s , e -α1s where α 0 > 0, α 1 < 0 are the 2 roots smaller than α < (N -4)/2. For simplicity, however we will look for φ such that φ(s) = Ce -α0s for s < x 0 where C > 0 is a constant. This restriction will mean that we will not be able to impose φ ′ (0) = 0 at the end. This is not a problem because φ ′ (0) ≤ 0. Notice that we only need the inequality in (62) and hence we need to choose α ∈ (0, N -4/2) such that

α 4 -2(N -4)α 3 + (N 2 -10N + 20)α 2 + 2(N -2)(N -4)α ≥ βe w(x0) .
The precise choice we employed in each dimension is in a summary table at the end of this section.

To find a suitable function φ with the behavior φ(s) = Ce -α for s < x 0 we set φ = ψe -αs and solve the following equation

T α ψ -βe w ψ = f
where the operator T α is given by

T α ψ = d 4 ψ ds 4 + (-4α + 2(N -4)) d 3 ψ ds 3 + (6α 2 -6α(N -4) + N 2 -10N + 20) d 2 w ds 2 + (-4α 3 + 6α 2 (N -4) -2α(N 2 -10N + 20) -2(N -2)(N -4)) dψ ds + (α 4 -2α 3 (N -4) + α 2 (N 2 -10N + 20) + 2α(N -2)(N -4))ψ
and f is some smooth function such that f ≥ 0, f ≡ 0. Actually we choose β > β 0 (where β 0 is given in (49)) find ᾱ satisfying approximately

ᾱ4 -2(N -4)ᾱ 3 + (N 2 -10N + 20)ᾱ 2 + 2(N -2)(N -4)ᾱ = βe w(x0) .
We solve numerically T ᾱ ψβe w ψ = f x 0 < s < 0 ψ(x 0 ) = 1, ψ′′ (x 0 ) = 0, ψ′′′ (x 0 ) = 0 ψ(0) = 0 Using the same strategy as in 2) from the numerical approximation of d 4 ψ ds 4 we compute a piecewise polynomial ψ of degree 7, which is globally C 3 and constant for s ≤ x 0 . The constant ψ(x 0 ) is chosen so that ψ(0) = 0. We use then Maple to verify the following inequalities ψ ≥ 0 x 0 ≤ s ≤ 0 T α ψ -βe w ψ ≥ 0 x 0 ≤ s ≤ 0 ψ ′ (0) ≤ 0 where β 0 < β < β and 0 < α < (N -4)/2 are suitably chosen.

At the URLs: http://www.lamfa.u-picardie.fr/dupaigne/ http://www.ime.unicamp.br/~msm/ we have provided the data of the functions w and ψ defined as piecewise polynomials of degree 7 in [x 0 , 0] with rational coefficients for each dimension in 13 ≤ N ≤ 31. We also give a rational approximation of the constants involved in the corresponding problems.

We use Maple to verify that w and ψ (with suitable extensions) are C 3 global functions and satisfy the corresponding inequalities, using only its capability to operate on arbitrary rational numbers. These operations are exact and are limited only by the memory of the computer and clearly slower than floating point operations. We chose Maple since it is a widely used software, but the reader can check the validity of our results with any other software (see e.g. the open-source solution pari/gp).

The tests were conducted using Maple 9. Remark 6.4. 1) Although we work with λ rational, in the table above we prefer to display a decimal approximation of λ.

Summary of parameters and results

N

2) In the previous table we selected a "large" value of ε 0 in order to have a fast verification with Maple. By requiring more accuracy in the numerical calculations, using a smaller value of ε 0 and using more subintervals to verify the inequalities in the Maple program it is possible to obtain better estimates of λ * . For instance, using formulas (50), we obtained The verification above, however, required to check 1500 subintervals of each of the 4500 intervals of length 0.002, which amounts to substantial computer time.

Proof of Proposition 1.5

Throughout this section, we restrict, as permitted, to the case a = 0. a)Let u denote the extremal solution of (1) with homogeneous Dirichlet boundary condition a = b = 0. We extend u on its maximal interval of existence (0, R).

  Proposition 1.5. a) Fix N ≥ 13 and take any a ∈ R. Assume b ≥ -4. There exists a critical parameter b max > 0, depending only on N , such that the extremal solution u * is singular if and only if b ≤ b max . b) Fix b ≥ -4 and take any a ∈ R. There exists a critical dimension N min ≥ 13, depending only on b, such that the extremal solution u * to (1) is singular if N ≥ N min .

4 .

 4 as ρ → 0. and therefore, with r = 1 ρu ′ (ρ) → -4 as ρ → 0. (31) Proof of Theorem 1.3

Lemma 5 . 1 .

 51 Assume N ≥ 13. Then u * ≤ ū = -4 log |x| in B 1 .

3 •

 3 10 -6 3600 3500 3.1 16 3976.4 1 1 • 10 -5 4100 4000 3.0 17 4568.8 1 2 • 10 -4 4800 4600 3.0 18 5201.1 2 2 • 10 -4 5400 5300 2.7 19 5873.2 2 2 • 10 -4 6100 6000 2.7 20 6585.1 3 7 • 10 -4 7000 6800 2.7 21 7336.7 3 7 • 10 -4 7700 7500 2.6 22 8128.1 4 1 • 10 -3 8600 8400 2.6 23 8959.1 4 1 • 10 -3 9400 9200 2.5 24 9829.8 4 1 • 10 -3 10400 10200 2.5 25 10740.1 4 1 • 10 -3 11400 11200 2.5 26 11690.1 6 2 • 10 -3 12400 12200 2.5 27 12679.7 7 2 • 10 -3 13400 13200 2.4 28 13709.0 7 2 • 10 -3 14500 14300 2.4 29 14777.8 7 2 • 10 -3 15400 15200 2.4 30 15886.2 8 2 • 10 -3 16600 16400 2.4 31 17034.3 10 2 • 10 -3 17600 17500 2.3

  589 0.003 5 • 10 -7 2438.583 2438.595 2550 2510 3.9 14 2911.194 0.003 5 • 10 -7 2911.188 2911.200 3100 3000 3.4

JUAN D ÁVILA, LOUIS DUPAIGNE, IGNACIO GUERRA, AND MARCELO MONTENEGRO

JUAN D ÁVILA, LOUIS DUPAIGNE, IGNACIO GUERRA, AND MARCELO MONTENEGRO

Acknowledgements.

This work has been partly supported by grants FONDECYT 1050725 and FON-DAP, Chile. Part of it was done when J.D. was visiting Laboratoire Amiénois de Mathématique Fondamentale et Appliquée at Faculté de Mathématique et d'Informatique, Amiens, France and when L.D. was visiting Departamento de Ingeniería Matemática at Universidad de Chile, Santiago, Chile. They are grateful for the hospitality and support. I. Guerra was supported by FONDECYT 1061110. M. Montenegro was supported by Project CNPq-CNRS 490093/2004-3. J.D. and L.D. are indebted to J. Coville for useful conversations on some of the topics in the paper.

The authors thank the referee for his careful reading of the manuscript.

Lemma 7.1. R < ∞ and u(r) ∼ log( R -r) -4 for r ∼ R.

Proof. The fact that R < ∞ can be readily deduced from Section 2 of [START_REF] Arioli | Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity[END_REF]. We present an alternative (and more quantitative) argument. We first observe that (63) u ′′ -1 r u ′ > 0 for all r ∈ [1, R).

Integrate indeed (1) over a ball of radius r to conclude that

If r = 1, since u is nonnegative in (0, 1) and u(1) = u ′ (1) = 0, we must have u ′′ (1) ≥ 0. In fact, u ′′ (1) > 0. Otherwise, we would have u ′′ (1) = 0 and u ′′′ (1) > 0 by (64), contradicting u > 0 in (0, 1). So, we may define

and we just need to prove that R = R. Assume this is not the case, then u ′′ (R)

This contradicts (64) and we have just proved (63). In particular, we see that u is convex increasing on (1, R).

Since u is radial, (1) reduces to (65)

Multiply by u ′ :

which we rewrite as

By (63), it follows that for r ∈ [1, R),

Integrating, we obtain for some constant A

Using this information in (66), dropping nonpositive terms and integrating, we obtain for some constant B,

which after multiplication by u ′ and integration provides positive constants c, C such that (u ′ ) 4 ≥ c(e u -Au 2 -Bu -C). At this point, we observe that since u is convex and increasing, u converges to +∞ as r approaches R. Hence, for r close enough to R and for c > 0 perhaps smaller,

. By Gronwall's lemma, R is finite and

It remains to prove that u ≥ -4 log( R -r) -C. This time, we rewrite (1) as

We multiply by r N -1 (∆u) ′ and obtain :

Hence, for r close to R and C perhaps larger, ((∆u) ′ ) 2 ≤ Ce u ∆u and so √ ∆u(∆u) ′ ≤ Ce u/2 ∆u ≤ C ′ e u/2 u ′′ ≤ C ′ (e u/2 u ′ ) ′ , where we have used (63). Integrate to conclude that

Solving for ∆u and multiplying by (u ′ ) 1/3 , we obtain in particular that

Integrating again, it follows that (u ′ ) 4/3 ≤ Ce u/3 , i.e.

It then follows easily that (for r close to R)

Proof of Proposition 1.5 a). Given N ≥ 13, let b max denote the supremum of all parameters b ≥ -4 such that the corresponding extremal solution is singular. We first observe that b max > 0.

In fact, it follows from Sections 5 and 6 that the extremal solution u associated to parameters a = b = 0 is strictly stable :

Extend u as before on its maximal interval of existence (0, R). Choosing R ∈ (1, R) close to 1, we deduce that (67) still holds on the ball B R . In particular, letting v(x) = u(Rx) -u(R) for x ∈ B, we conclude that v is a singular stable solution of (1) with a = 0 and b = Ru ′ (R) > 0. By Proposition 1.7, we conclude that b max > 0. We now prove that b max < ∞.

Assume this is not the case and let u n denote the (singular) extremal solution associated to b n , where b n ր ∞. We first observe that there exists ρ n ∈ (0, 1) such that u ′ n (ρ n ) = 0. Otherwise, u n would remain monotone increasing on (0, 1), hence bounded above by u n (1) = 0. It would then follow from (1) and elliptic regularity that u n is bounded. Let v n (x) = u n (ρ n x) -u n (ρ n ) for x ∈ B and observe that v n solves (1) with a = b = 0 and some λ = λ n . Clearly v n is stable and singular. By Proposition 1.7, v n coincides with u, the extremal solution of ( 1) with a = b = 0. By standard ODE theory, v n = u on (0, R). In addition,

Letting n → ∞, we conclude that u is stable on B R. This clearly contradicts Lemma 7.1.

We have just proved that b max is finite. It remains to prove that u * is singular when -4 ≤ b ≤ b max . We begin with the case b = b max . Choose a sequence (b n ) converging to b max and such that the corresponding extremal solution u n is singular. Using the same notation as above, we find a sequence ρ n ∈ (0, 1) such that 1

Taking subsequences if necessary and passing to the limit as n → ∞, we obtain for some ρ ∈ (0, 1)

If N is sufficiently large, depending on b, we have λ 0 < N 2 (N -4) 2 /16. Then by [START_REF] Boggio | Sulle funzioni di Freen d'ordine m[END_REF] u is a stable subsolution of (1) with λ = λ 0 . As in Lemma 5.2 this implies λ * ≤ λ 0 .

Thus for large enough N we have λ * e u * ≤ r -4 8(N -2)(N -4)e (4+b)/2 < r -4 N 2 (N -4) 2 /16. This and [START_REF] Boggio | Sulle funzioni di Freen d'ordine m[END_REF]


Terence Hébert

Julie Perroy
email: julie.perroy@igf.cnrs.fr

Yan Chastagnier

Enora Moutin

Anne-Laure Hemonnot

Image Processing for Bioluminescence Resonance Energy Transfer Measurement-BRET-Analyzer

Keywords: open source software, automatic image analysis, ratiometric measurements, bioluminescence resonance energy transfer

A growing number of tools now allow live recordings of various signaling pathways and protein-protein interaction dynamics in time and space by ratiometric measurements, such as Bioluminescence Resonance Energy Transfer (BRET) Imaging. Accurate and reproducible analysis of ratiometric measurements has thus become mandatory to interpret quantitative imaging. In order to fulfill this necessity, we have developed an open source toolset for Fiji-BRET-Analyzer-allowing a systematic analysis, from image processing to ratio quantification. We share this open source solution and a step-by-step tutorial at https://github.com/ychastagnier/BRET-Analyzer. This toolset proposes (1) image background subtraction, (2) image alignment over time, (3) a composite thresholding method of the image used as the denominator of the ratio to refine the precise limits of the sample, (4) pixel by pixel division of the images and efficient distribution of the ratio intensity on a pseudocolor scale, and (5) quantification of the ratio mean intensity and standard variation among pixels in chosen areas. In addition to systematize the analysis process, we show that the BRET-Analyzer allows proper reconstitution and quantification of the ratiometric image in time and space, even from heterogeneous subcellular volumes. Indeed, analyzing twice the same images, we demonstrate that compared to standard analysis BRET-Analyzer precisely define the luminescent specimen limits, enlightening proficient strengths from small and big ensembles over time. For example, we followed and quantified, in live, scaffold proteins interaction dynamics in neuronal sub-cellular compartments including dendritic spines, for half an hour. In conclusion, BRET-Analyzer provides a complete, versatile and efficient toolset for automated reproducible and meaningful image ratio analysis.

INTRODUCTION

A perpetual association and dissociation between proteins drives specific cellular signaling in time and space. Bioluminescence Resonance Energy Transfer (BRET) imaging is a sensitive technology to highlight the spatio-temporal dynamics of protein-protein interaction and understand their functions in intact living cells [START_REF] Coulon | Subcellular imaging of dynamic protein interactions by bioluminescence resonance energy transfer[END_REF][START_REF] Perroy | Subcellular dynamic imaging of protein-protein interactions in live cells by bioluminescence resonance energy transfer[END_REF][START_REF] Goyet | Fast and high resolution single-cell BRET imaging[END_REF][START_REF] Faklaris | Fluorescent-based strategies to investigate G protein-coupled receptors: evolution of the techniques to a better understanding[END_REF]. Briefly, the principle of the method stands on an energy transfer between a bioluminescent donor and a compatible fluorescent acceptor [START_REF] Xu | A bioluminescence resonance energy transfer (BRET) system: application to interacting circadian clock proteins[END_REF][START_REF] Angers | Detection of beta 2-adrenergic receptor dimerization in living cells using bioluminescence resonance energy transfer (BRET)[END_REF]. By catalytic oxidation of its substrate, the bioluminescent donor emits light (donor image, D, Figure 1). Upon molecular proximity (<75Å), the BRET compatible acceptor is excited by a non-radiative transfer of energy and in turn emits light at its characteristic wavelength (acceptor image, A, Figure 1). Pixel by pixel division of the light emitted by the acceptor over the light emitted by the donor (A/D) gives rise to the BRET image, a ratiometric measurement expressed as pseudo-colors, allowing live quantification of the interaction between proteins tagged with BRET compatible entities, in subcellular domains [START_REF] Goyet | Fast and high resolution single-cell BRET imaging[END_REF]. The advantage of BRET over other RET methodologies precisely comes from the absence of light to initiate the energy transfer in BRET. Thereby, BRET circumvents many drawbacks linked to light excitation (such as auto-fluorescence of cells, direct excitation of the acceptor fluorophore by the donor exciting light, or photobleaching of fluorophores) giving rise to an excellent signal over noise ratio.

A ratiometric measurement provides the huge benefit to normalize the output signal to the intensity of its stimulus. Hence, any efficient activation of a system can virtually be reported, regardless of the stimulus intensity. This is particularly relevant to put emphasis on effective forces of small ensembles. However, this benefit is counterbalanced by the fact that a nearbackground stimulus will give rise to an aberrant ratiometric measure resulting from the fraction's near-zero denominator. Consequently, the specific signal is lost in a high-intensity background. Therefore, delimitation of the linear detection range of the recording system and proper determination of the stimulus intensity threshold is mandatory to perform ratiometric measurements. In the present work, we have developed a systematic image processing to obtain relevant ratiometric measurements. We applied this automatic processing on BRET images. This homemade toolset is a complete open source solution for Fiji [free software for scientific image analysis, [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF]], available for the scientific community together with a step-by-step tutorial (https://github. com/ychastagnier/BRET-Analyzer).

The BRET-Analyzer toolset includes classical processing of the D and A images by cleaning the background, and pixel by pixel division of the A/D images. We worked mainly on two parameters. First we defined a donor threshold computed as a function of the donor signal intensity in a given area. This step allows defining the frontiers between the sample luminescence per se and the light spread on neighboring pixels. Second we selected and combined different threshold processes independently in each sub-area, depending on the local donor intensity. This is particularly relevant when the recorded cell displays heterogeneous subcellular volumes. For example in neurons, a cytosolic signal from the donor accumulated in the soma will be much brighter than the donor light recorded in thin neuritic processes. A combination of donor threshold processing allows proper reconstitution of the ratiometric image. It is fundamental to note that getting rid of low-level donor pixels permits the selection of pixels from the luminescent specimen only, but does not influence the ratiometric measurement per se. We thus provide here a toolset-BRET-Analyzer-that allows (A) Scheme of two BRET donor compatible entities emission spectra: Nanoluciferase (Nluc) is the BRET donor and Venus the BRET acceptor. The light emitted by the donor alone displays an emission pic at 450 nm (blue curve). When Venus is co-expressed and close enough (<75Å) from Nluc, a non-radiative transfer of energy from Nluc excites Venus, which emits light at its characteristic wavelength (pic at 535 nm, green curve). Light collected with specific filters gives rise to donor and acceptor images. (B) Distribution of the A/D ratio pixel values and possible representations of the BRET images: A stack histogram (left panel) displays the distribution of the A/D ratio values obtained from pixel by pixel division of the A image over the D image, suggesting the minimal and maximal values to be set on the pseudo-color scale expressing the A/D ratio image. "Raw," "weighted" and "specific" panels are three possible representations of the A/D ratio image. The "raw" image expresses the ratio as pseudo-colors, without any selection of pixels. Consequently, pixels with a near-zero intensity on the D image display a maximal A/D ratio, compromising the detection of pixels from the specimen. Alternatively, the "weighted" ratio also displays the A/D ratio intensity as pseudocolors, but the brightness of each pixel is proportional to the donor intensity. This representation shades the ratio background around the cell but also the specific signal coming from low protein expression level in sub-cellular area. The BRET-Analyzer toolset allows obtaining the "specific" ratio by an automatic selection of pixels from the specimen only (obtained by composite thresholding of the D image). The "specific" image displays the intensity of the A/D ratio expressed as pseudo-colors, regardless of protein expression levels.

the elimination of non-specific signal, and performs quantitative analysis of long period recordings of protein-protein interactions, regardless of the protein expression level (Figure 1).

STEPWISE PROCEDURES AND ANTICIPATED RESULTS

Raw images of the donor (D) and acceptor (A) BRET entities were obtained as previously described [START_REF] Goyet | Fast and high resolution single-cell BRET imaging[END_REF], from hippocampal neurons (Figure 1). To summarize the process, the images will first have their background subtracted, then be aligned over time. A threshold will be computed to separate signal from background. A image is divided by D image pixel by pixel. Finally, measures are made on the ratiometric images obtained.

The analysis starts with the removal of non-specific signals ("clean" button, Supplementary Material 4.1), in D and A images, in 3 simple steps. First, in each image, applying a median filter of radius 1 (3 * 3 pixels) allows to remove outliers. Second, subtracting the median value of a region corresponding to the background removes the camera offset and global light background, assuming they are homogeneous. The homogeneous assumption is based on the fact that BRET doesn't use illumination, unlike fluorescence based imaging. Subtracting a background image is available as an option. This value is measured on each image of the donor and the acceptor, to account for potential fluctuations of the background's mean intensity from an image to another. Third, in case recording is a timelapse, aligning images might be needed. To do so, we make use of the plugin TurboReg (http://bigwww.epfl.ch/thevenaz/ turboreg/ [START_REF] Thevenaz | A pyramid approach to subpixel registration based on intensity[END_REF], a pyramid approach to subpixel registration based on intensity. This alignment can rescue xy drift that may happen during multi-positioning of the microscopy setup to record distant cells on the same sample. It is important that images are aligned through time to make sure the area in which the measures are made always corresponds to the same part of the sample.

Second step is to divide, pixel by pixel, the A image by the D image ("divide" button, Supplementary Material 4.3). Prior to this division, it is necessary to get rid of all pixels that have an intensity level too close to the noise level on the donor image, to keep only specific signal from the sample (Figure 1B, "specific"). Indeed, the computed A/D ratio tends to infinity for each pixel with a near-zero intensity on the D image. If these pixels are not removed, the ratiometric values can range from zero to hundreds and the boundaries of the cells cannot be distinguished (Figure 1B, "raw"). Alternatively, the brightness of the A/D ratio image can be weighted by the expression level of proteins (Figure 1B, "weighted"), which efficiently hides non-specific pixels, but also fades ratiometric values from small protein numbers. This weighting can be attractive for homogenous samples but should be avoided to perform ratio-metric measurements from heterologous volume specimens since it overshadows proficient strengths from small ensembles.

Hence, in order to remove pixels that are outside of luminescent specimen from the resulting A/D ratio image, we applied available threshold methods to the D image (Figure 2). The simplest approach, applying a static threshold level of a given value across all images (Figure 2B) presented two main problems. First, since the D signal decreases with time, part of the specific D signal falls below the threshold during image time series when the threshold was determined on the first image. Conversely, lowering the threshold resulted in keeping the noise on the first images of the image time series. We circumvented this problem by computing a specific threshold for each individual image of the image time series. We selected a region of interest (which should contain about half pixels to keep and half to remove). For each time point, we computed the median of that area as the threshold to be applied on the D image (Figure 2C).

Alternatively, we used an automatic threshold built in Fiji. Across all the available automatic thresholding methods, Otsu [which searches for the threshold value that minimizes the intra-class variance [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF]] gave the best results at separating the high signal from the rest (Figure 2D).

However, a second problem was related to the fact that, for a given time point, the D image displayed strong intensity differences among neuronal subareas (Figure 2A). Therefore, applying a homogenous constant threshold on the D image did not allow suppressing pixels adjacent to the neuronal cell body (which, even so located out of the neuron, still displayed a relatively high light intensity due to the D light spread), while keeping the low but specific D light from neuritic processes (Figures 2B-D). To solve this spatial problem, one possibility was to crop different regions of interest ("crop" button, Supplementary Material 4.2) and compute a specific threshold for each crop. But the resulting BRET image of the neuron was segmented in several areas. Furthermore this process increased analysis time to quantify the BRET images (step below), as it required a repeated exploration on each crop. We thus favored a second analysis process to determine the threshold of the D image in space. Automatic local thresholds built in Fiji have the advantage to compute the threshold for each pixel based on the surrounding pixels. Hence values that are locally high are selected, allowing inhomogeneous signal across the image. For example Phansalkar threshold [START_REF] Phansalskar | Adaptive local thresholding for detection of nuclei in diversity stained cytology images[END_REF], provided good results at removing noise at the edges of cells making clear delimitation, but kept a good part of the noise far from the cells (Figure 2E).

We thus designed our own composite thresholding method (Chastagnier threshold, Figure 2F). The whole process consisted in drawing an accurate mask around the neuron to keep only pixels of the D image arising from the luminescent specimen. To create this mask (Figure 3, gray rectangle), we first subtracted a blurred version of the D image to itself in order to increase the contrast, and convert it to binary values using Li's automatic threshold method [START_REF] Li | An iterative algorithm for minimum cross entropy thresholding[END_REF]. When the blur effect is decreased, noise areas start to show up, while when blur effect is increased, areas of interest show up thicker than they really are. To get rid of the defects of both low and high blur effect, we used both and combined them with a logical AND in order to keep only pixels that are in both images. As "holes" appeared in the lowest intensity area of two juxtaposed regions of high intensity areas, we combined it using a logical OR with the original image on which Otsu Threshold was applied (to keep only high signal). Logical OR is inclusive, so it keeps pixels that are in an image, in the other, or in both. We thus obtained a binary image with values 0 and 255. Dividing it by 255 gave 0 for pixels we want to remove and 1 for pixels to keep. This is the mask (Figure 3, Chastagnier Threshold). Multiplying it with the original D image removed irrelevant pixels (set them to 0) and did not affect the value for the rest of the D image (Figure 3, Thresholded donor). The Chastagnier's threshold thus unequivocally refined the limits of the luminescent sample in space and time. For user convenience, we nevertheless included the conventional A composite process using Gaussian blur subtraction and Li's thresholding method identifies pixels with locally high intensity. In parallel, Otsu's thresholding method preserves pixels with globally high intensity. A logical OR combines the two processes to obtain the final mask. This mask (red image) with values of 0 for black pixels and 1 for white pixels multiplies the original D image to remove irrelevant pixels without affecting the donor light intensity of pixels from the fluorescent specimen (Thresholded donor). Bottom line shows the thresholded donor, the acceptor and resulting A/D ratio images.

alternative thresholding methods herein tested, as an option in the BRET-Analyzer toolset.

The A image was then divided by the thresholded D image and displayed with a 16 pseudo-colors look up table, ranging from cold to hot colors (Supplementary Material 4.3.5). The minimal and maximal A/D ratio values between which the colors will be distributed have to be selected. The stack histogram displayed between the 5th and 995th permilles (without taking into account pixels with value 0) can be used to help the user choose appropriate values (Figure 1B). Once the range is set, it should be kept the same for every image, so they can be visually compared, before doing the in depth analysis. The resulting BRET image is a quantitative measurement of the energy transfer intensity reporting either the efficiency of interaction between two proteins tagged with BRET compatible entities (intermolecular BRET) or conformational changes of a molecule tagged with D and A entities (intramolecular BRET). This analysis displays BRET intensity regardless of the level of protein expression. Hence, even small numbers of complexes are measured and visualized on the BRET image. This provides an important benefit compared to ratiometric analysis in which the signal is expressed as discontinuous pseudo-color scale representing two different parameters in one, namely A/D ratio and expression level (the brightness of the image being weighted by the protein expression level, e.g., by the D and/or A image intensity). This is potentially confusing and may lead to arbitrary judgments about the extent of BRET. Indeed, this kind of analysis, presented on Figure 1 as "weighted" A/D ratio allows to visually exclude the noise but the evidenced BRET signals come from proteins with high expression level only, neglecting the functional importance of high protein interactions of small ensembles. The weighted representation is nevertheless included as an option in the BRET-Analyzer (Supplementary Material 4.5 Param Tool, Display weighted images?).

Final step is to extract the BRET values out of the ratiometric image, in regions of interest, which can be chosen on a single image (Figure 4), or pooled from multiple images (Supplementary Material 4.4). All tools can be used to define the area of interest (rectangle, polygon, freehand. . . ), at user convenience. It is not necessary to precisely draw the limits of the samples with the tool, because all pixels with a null value are being automatically ignored by the measures and excluded FIGURE 5 | Ratiometric measurements in heterogeneous subcellular volumes using BRET-Analyzer. Hippocampal neurons expressing mGlu5 and Homer proteins fused to BRET compatible entities were recorded as previously described [START_REF] Moutin | Dynamic remodeling of scaffold interactions in dendritic spines controls synaptic excitability[END_REF][START_REF] Guo | Elevated CaMKIIalpha and hyperphosphorylation of homer mediate circuit dysfunction in a Fragile X syndrome mouse model[END_REF] for 30 min. Images were analyzed using conventional (A) and /or BRET-Analyzer protocols (A,B). from the analysis of the BRET quantification in the area. The mean BRET intensity provides a quantification of the number of protein complexes in a given area and the standard deviation of the BRET intensity between pixels reports homogeneity or clusterization of the BRET signal.

In addition to the automation of analysis process, the main strength of BRET-Analyzer resides in the potency to quantify luminescent ratio regardless of protein expression levels. This is particularly relevant when working on heterogeneous subcellular volumes such as neurons. Indeed, a neuron contains a large range of volumes from the neuronal cell body (around 500 µm 3 ) to small dendritic spines (spine head volumes ranging from 0.01 µm 3 to 0.8 µm 3 ). The first challenge, as discussed before is thus to find an adequate composite threshold allowing to perform ratiometric measurements in all volumes at the same time. Moreover for the smallest compartments, measuring protein-protein interactions from few proteins is difficult and live dynamics over time are even more challenging. To illustrate the benefits from the herein developed analysis toolset, we analyzed twice the same images either with the conventional analysis (including a static threshold level of D images, described in Figure 2B) or with BRET-Analyzer (using Chastagnier's threshold, Figure 2F). We thus measured in neurons the interaction between a glutamate receptor, mGlu5, and its cognate scaffold, Homer, together involved in neuronal synaptic transmission [START_REF] Moutin | Dynamic remodeling of scaffold interactions in dendritic spines controls synaptic excitability[END_REF][START_REF] Guo | Elevated CaMKIIalpha and hyperphosphorylation of homer mediate circuit dysfunction in a Fragile X syndrome mouse model[END_REF]. As shown in Figure 5A, the interaction between mGlu5 and Homer was accurately recorded over time in the soma and the quantification indiscernible whatever was the analysis protocol. However, the advantage of using BRET-Analyzer became obvious for small processes, which from the beginning of the experiment were barely identifiable using the classical analysis protocol and totally lost few minutes after. BRET-Analyzer enables long time recording of BRET signals in small subcellular volumes, allowing for the first time live and stable measurements of mGlu5-Homer interaction in individual spines for half an hour. By opposition to previous studies [START_REF] Moutin | Dynamic remodeling of scaffold interactions in dendritic spines controls synaptic excitability[END_REF][START_REF] Guo | Elevated CaMKIIalpha and hyperphosphorylation of homer mediate circuit dysfunction in a Fragile X syndrome mouse model[END_REF], instead of assessing synaptic activity-induced changes in mGlu5-Homer interactions in spines from two distinct cell populations (stimulated or not), we could here follow BRET changes using BRET-Analyzer in the same spines before and during KCl-induced neuronal depolarization. Thus, neuronal depolarization disrupted mGlu5-Homer interaction (Figure 5B). We recorded a 15 and 16% drop of BRET signal in soma and spines, respectively. This BRET decrease was measured as soon as 2 min following KCl 55 mM perfusion and stable for 20 min. Hence, BRET-Analyzer enables image time series of ratiometric measurements even in small cellular compartments.

MATERIALS Source Code

The data were analyzed using a toolset made for Fiji (https:// fiji.sc) [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF]. We here provide a stepby-step tutorial as Supplementary Material. The toolset and documentation details of how to use it are also publicly available at (https://github.com/ychastagnier/BRET-Analyzer), and will be regularly updated. Moreover, the tools can be freely downloaded, modified and improved to fit future research needs.

TurboReg, the plugin used to align stacks of images can be downloaded at (http://bigwww.epfl.ch/thevenaz/turboreg/) [START_REF] Thevenaz | A pyramid approach to subpixel registration based on intensity[END_REF].

Example Luminescent Images

The donor and acceptor fluorescent images used here to exemplify the use of BRET-Analyzer corresponds to an hippocampal neuron from primary cell culture transfected with an intramolecular BRET-based sensor for ERK activity [YEN [START_REF] Goyet | Fast and high resolution single-cell BRET imaging[END_REF]], except in Figure 5, where mGlu5-NanoLuc and Venus-Homer intermolecular BRET was recorded. Raw images are available at https://github.com/ychastagnier/BRET-Analyzer.

FIGURE 1 |

 1 FIGURE 1 | Principle of the BRET-Analyzer toolset. The BRET-Analyzer toolset provides automatic analyses of donor (D) and acceptor (A) images (A) to generate a quantitative analysis of the specific A/D ratio image (B).

FIGURE 2 |

 2 FIGURE 2 | Methods to define a donor threshold in space and time. Applying a threshold to the donor image before division allows masking irrelevant pixels (background around the luminescent specimen) in the A/D ratio image. Various threshold methods (B-F) can be applied to the donor image (A). Each column represents a different time point. (A) Intensity of the donor for each time point. (B) Mask using a constant threshold all over the image time series. (C,D) Mask using a global threshold computed for each time point (median threshold of the area outlined in red, (C) or Otsu thresholding method, (D). (E,F) Mask using a local threshold computed for each time point, Phansalkar (E) or Chastagnier threshold (F), described in Figure 3. Note that computing a local threshold for each image over the time series allows to precisely define the luminescent specimen limits, all along the experiment.

FIGURE 3 |

 3 FIGURE 3 | Principle of the Chastagnier threshold. The Chastagnier threshold computed on the donor image (Gray rectangle) creates the mask (red image) to be applied on the donor image (thresholded donor) prior to A/D images division. Left column shows the donor, acceptor and A/D ratio images without mask. The gray rectangle is a diagram representation of the process used for Chastagnier thresholding method. A composite process using Gaussian blur subtraction and Li's thresholding method identifies pixels with locally high intensity. In parallel, Otsu's thresholding method preserves pixels with globally high intensity. A logical OR combines the two processes to obtain the final mask. This mask (red image) with values of 0 for black pixels and 1 for white pixels multiplies the original D image to remove irrelevant pixels without affecting the donor light intensity of pixels from the fluorescent specimen (Thresholded donor). Bottom line shows the thresholded donor, the acceptor and resulting A/D ratio images.

FIGURE 4 |

 4 FIGURE 4 | Quantitative analysis of the A/D ratio image. (A) Regions of interest are drawn on the ratiometric image. (B,C) The mean intensity of pixels within areas of interest quantifies BRET signals over time (B) whereas the standard deviation of A/D ratio between pixels (C) provides a measurement of the homogeneity of the BRET signal within areas and the evolution as a function of time.

  FIGURE 5 | hour. By opposition, the conventional analysis failed to quantify the BRET in spines over time. (B) BRET-Analyzer enables live, BRET image time series of mGlu5-Homer interaction in neurons. We quantified the mean BRET intensity in single areas drawn on soma and dendritic spines over time, before and during neuronal depolarization (KCl 55 mM). KCl 55 mM decreases BRET signals in soma and spines. BRET-Analyzer enables precise measurements of ratiometric signals over time. Each point of the curves is the mean ± SEM of BRET intensity recorded in 3 areas.

Frontiers in Computational Neuroscience | www.frontiersin.org

January 2018 | Volume 11 | Article 118

ACKNOWLEDGMENTS

We are grateful to Vincent Compan, Laurent Fagni, Elise Goyet, and Vincent Ollendorff for helpful reading of the manuscript. This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (JP, grant agreement No. 646788), the Agence Nationale de la Recherche (JP, ANR-13-JSV4-0005-01) and the Région Languedoc-Roussillon (JP, Chercheur d'Avenir 146090).

AUTHOR CONTRIBUTIONS

YC designed the Fiji BRET-Analyzer plugin, computed all steps of the automated image analysis protocol and wrote the manuscript and step by step tutorial. EM and A-LH performed BRET imaging experiments. JP conceived, designed, led the project, supervised the analysis strategy and wrote the manuscript.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fncom. 2017.00118/full#supplementary-material

Conflict of Interest Statement:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Chastagnier, Moutin, Hemonnot and Perroy. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.