Rouba Borghol

Laurent Véron

Jim Serrin

Boundary singularities of solutions of N -harmonic equations with absorption *

We study the boundary behaviour of solutions u of -∆N u + |u| q-1 u = 0 in a bounded smooth domain Ω ⊂ R N subject to the boundary condition u = 0 except at one point, in the range q > N -1. We prove that if q ≥ 2N -1 such a u is identically zero, while, if N -1 < q < 2N -1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.

Introduction

Let Ω be a domain is R N (N ≥ 2) with a C 2 compact boundary ∂Ω. Let g be a continous real valued function and a ∈ ∂Ω. This paper deals with the study of solutions u ∈ C 1 ( Ω \ {a}) of the problem

   -div |Du| N -2 Du + g(u) = 0 in Ω u = 0 on ∂Ω \ {a}, (1.1)
and we shall be more specifically interested in the case when g has a power growth at infinity. When N = 2, this problem falls into the scope of the boundary singularity problem for semilinear elliptic equations. The study of the N -dimensional problem

-∆u + g(u) = 0 in Ω u = 0 on ∂Ω \ {a}, (1.2)
has been initiated by Gmira and Véron in [START_REF] Gmira | Boundary singularities of solutions of some nonlinear elliptic equations[END_REF]. Among the subjects under consideration were the question of removability of isolated boundary singularities and, in the case such singularities do exist, their precise description. This seminal article was at the origin of a long series of further works by Dynkin, Kuznetsov, Le Gall, Marcus and Véron in the framework of the trace theory and, later on, the fine trace theory in the case where g(r) = r |r| q-1 , q > 1. One of the main reasons for such a large impact consists of the observation of the existence of a critical exponent q = q * = (N + 1)/(N -1). If q ≥ q * any solution of

-∆u + |u| q-1 u = 0 in Ω u = 0 on ∂Ω \ {a}, (1.3)
is identically zero, while if 1 < q < q * it appears that there exist two possible behaviours of singular solutions near a, the solutions with weak singularities and the ones with the strong singular behaviour. Later on, these two types of singular solutions played a fundamental role in the description of the rough trace of positive solutions of (1.3 ).

Although the techniques needed are considerably more refined, it appeared that the description of solutions of (1.1 ) inherits the same structure as for (1.2 ). The first step is to understand the model case problem

   -div |Du| N -2 Du + |u| q-1 u = 0 in Ω u = 0 on ∂Ω \ {a}, (1.4)
To this equation, we associate the homogeneous equation

   -div |Du| N -2 Du = 0 in Ω u = 0 on ∂Ω \ {a}. (1.5)
It is proved in [START_REF] Borghol | Boundary singularties of N -harmonic functions[END_REF] that for any k > 0 there exists a unique solution u = u k of (1.5 ) satisfying

u k (x) = k ρ(x)
|x -a| 2 (1 + •(1)) as x → a, (x -a)/ |x -a| → σ, (1.6) where ρ(x) = dist (x, ∂Ω). When k = 1, this solution plays the role of the Poisson kernel, although neither any weak formulation nor any reasonable trace theory seems to exists, and we shall denote it by V Ω a . The behaviour (1.6 ) (up to a multiplicative constant) corresponds to weak singularity behaviour for (1.1 ), whenever such singularities exist. The first result we prove is the following: Theorem Let N -1 < q < 2N -1 := q c . Then for any k > 0 there exists a unique solution u = u k,a of problem (1.4 ) satisfying

u k,a (x) = k ρ(x)
|x -a| 2 (1 + •(1)) as x → a, (x -a)/ |x -a| → σ.

(1.7)

Furthermore u ∞,a = lim k→∞ exists and is a solution of (1.4 ) which satisfies

lim x → a x-a |x-a| → σ |x -a| N/(q+1-N ) u ∞,a (x) = ω(σ), (1.8)
and ω is the unique positive solution of the following quasilinear equation on the upper hemisphere of the unit sphere S N -1 ,

           -div σ β 2 q ω 2 + |∇ σ ω| 2 (N -2)/2 ∇ σ ω -Λ β 2 q ω 2 + |∇ σ ω| 2 (N -2)/2
ω + |ω| q-1 ω = 0 on S N -1

+ ω = 0 on ∂S N -1 + , (1.9)
where β q = N/(q + 1 -N ) and Λ = (N -1)β 2 q . The proof of the existence of u k,a , as well as its singular behaviour, is settled upon the conformal invariance of the N -harmonic operator and the construction of subsolution of the same equation. Estimate (1.8 ) is proved by scaling method. The role of the critical exponent q c = 2N -1 is enlighted by the following result.

Theorem Let g be a continuous function such that (i) lim inf r→∞ g(r)/r qc > 0 (ii) lim sup r→-∞ g(r)/ |r| qc < 0.

(1.10)

Then any function u ∈ C 1 (Ω \ {a}) solution of (1.1 ) extends as a function ũ ∈ C(Ω).

As in the semilinear case, the occurrence coincides with the case where the blow-up exponent -β q which is natural for equation (1.4 ) coincides with the one of the function V Ω a solution of (1.5 ). Finally we provide the full classification of positive solutions of problem (1.4 ).

Theorem Let N -1 < q < q c and u is any nonnegative solution of (1.4 ), then (i) Either u ≡ 0, (ii) Either there exists k > 0 such that u = u k,a . (iii) Or u = u ∞,a .

In the proof of (iii) the boundary Harnack inequalities that satisfies any positive solution of (1.4 ) (see [START_REF] Bidaut-Véron | Boundary Harnack inequalities and a priori estimates of singular solutions of quasilinear equations[END_REF]) play a fundamental role.

Our paper is organized as follows 1-Introduction 2-Weak and strong boundary singularities 3-The removability result 4-The classification theorem

Weak and strong boundary singularities

The construction of positive solutions of

-div |Du| N -2 Du + |u| q-1 u = 0, ( 2.1)
is settled upon three facts: the existence of solutions to the homogeneous equation

-div |Du| N -2 Du = 0, (2.2)
the conformal invariance of (2.2 ) and an a priori estimate satisfied by any solution of (2.1 ). Throughout this paper C denotes a positive constant which depends only on the structural assumptions corresponding to N , p, q and Ω. The value of the constant may change from one occurrence to another.

Proposition 2.1 Let Ω ⊂ R N be a domain with a compact boundary and a ∈ ∂Ω. Consider real numbers q > p -1 > 0, A > 0 and

B ≥ 0. If u ∈ C(Ω \ {a}) ∩ W 1,p loc (Ω) is a weak solution of -div |Du| p-2 Du + A |u| q-1 u ≤ B in Ω u ≤ 0 on ∂Ω \ {a}, (2.3
)

it satisfies u(x) ≤ λ A |x -a| p 1/(q+1-p) + µB A 1/q ∀x ∈ Ω \ {a}, (2.4)
where λ and µ depends on N , p and q.

Proof. By assumption

Ω |Du| p-2 Du.Dζ + A |u| q-1 uζ dx ≤ B Ω ζdx (2.5) for any ζ ∈ W 1,p (Ω) with compact support, ζ ≥ 0. Let η ∈ C 2 (R) be a nonnegative function such that 0 ≤ η ′ ≤ 1, η ′′ ≥ 0, η = η ′ = η ′′ on (-∞, 0], 0 < η(r) ≤ r on (0, ∞). For ǫ > 0 we set η ǫ (r) = η((r -ǫ) + ). Let ζ ∈ W 1,p (R N \ {0}) with compact support. Inasmuch (η ′ ǫ (u)) p-1 ζ has compact support in Ω and D (η ′ ǫ (u)) p-1 ζ = (η ′ ǫ (u)) p-1 Dζ + (p -1)(η ′ ǫ (u)) p-2 η ′′ ǫ (u)ζDu,
it belongs to W 1,p (Ω) and is an admissible test function for (2.5 ). Thus

Ω |Du| p-2 Du.D (η ′ ǫ (u)) p-1 ζ + A |u| q-1 u(η ′ ǫ (u)) p-1 ζ dx ≤ B Ω (η ′ ǫ (u)) p-1 ζdx, and
|Du| p-2 Du.D (η ′ ǫ (u)) p-1 ζ ≥ (η ′ ǫ (u)) p-1 |Du| p-2 Du.Dζ = |Dv ǫ | p-2 Dv ǫ .Dζ,
where we have set v ǫ = η ǫ (u). Furthermore, η can be chosen such that r q (η ′ ǫ (r)) p-1 ≥ η q ǫ (r), for example if we fix η(r) = r 2 /2δ on (0, δ] and η(r) = r -δ/2 on [δ, ∞) for some δ > 0. We extend v ǫ by 0 outside Ω \ {a} and denote by ṽǫ the new function, then ṽǫ

∈ W 1,p loc (R N \ {a}) ∩ C(R N \ {a}) and Ω |Dṽ ǫ | p-2 Dṽ ǫ .Dζ + A |ṽ ǫ | q-1 ṽǫ ζ dx ≤ B Ω ζdx. (2.6)
This means that ṽǫ is a weak subsolution in R N \ {a}. By [18, Lemma 1.3], we derive

ṽǫ (x) ≤ λ A |x -a| p 1/q+1-p + µB A 1/q ∀x ∈ R N \ {a},
for some λ > 0 and µ > 0 depending on N , p and q. Letting successively ǫ → 0 and δ → 0 we obtain (2.3 ).

When Ω is smooth we have a sharper estimate Proposition 2.2 Let Ω ⊂ R N be a bounded domain with C 2 boundary and a ∈ ∂Ω. Let

q ≥ p -1 > 1 and a > 0. If u ∈ C(Ω \ {a}) ∩ W 1,p loc (Ω) is a weak solution of (2.
3 ) with B = 0, there exists C > 0 depending on Ω, p and q such that

u(x) ≤ Cρ(x) A |x -a| q+1 1/(q+1-p) ∀x ∈ Ω \ {a}, (2.7)
where ρ(x) = dist (x, ∂Ω.

Proof. By translation we can assume that a = 0. For ǫ > 0 let v ǫ be the solution of

-div |Dv ǫ | p-2 Dv ǫ + A |v| q-1 ǫ v ǫ = 0, in Ω ǫ = Ω \ B ǫ v ǫ = u + on ∂Ω ǫ . (2.8)
By [START_REF] Vàzquez | Removable singularities of some strongly nonlinear elliptic equations[END_REF]Lemma 1.3] as in the proof of Proposition 2.1 and the maximum principle, there holds

u + (x) ≤ v ǫ (x) ≤ λ A(|x| -ǫ) p 1/(q+1-p) ∀x ∈ Ω ǫ . Consequently ǫ ≤ ǫ ′ =⇒ v ǫ ≥ v ǫ ′ .
Letting ǫ → 0 and using the previous inequalities and the classical regularity results for solutions of quasilinear equations [START_REF] Libermann | Boundary regularity for solutions of degenerate elliptic equations[END_REF] we conclude that v ǫ converges, as ǫ → 0, to some v which is a nonnegative solution of -div |Dv| p-2 Dv + Av q = 0, in Ω v = 0 on ∂Ω \ {0}.

(2.9) and dominate u. Further, if ℓ > 0 the function v ℓ defined by v ℓ (y) = ℓ p/(q+1-p) v(ℓy) is a solution of (2.9 ) with Ω replaced by

Ω ℓ = ℓ -1 Ω. Let x ∈ Ω \ {0} and ℓ = |x|. Since 0 ≤ v ℓ (y) ≤ λ A(|y|) p 1/(q+1-p) ∀y ∈ Ω ℓ , and max Dv ℓ (y) : y ∈ Ω ℓ ∩ B 3/2 \ B 2/3 ≤ M max v ℓ (z) : z ∈ Ω ℓ ∩ B 2 \ B 1/2 ,
where M is uniformly bounded because the curvature of ∂Ω ℓ is bounded, we obtain that Dv ℓ (y) is uniformly bounded by some constant C on Ω ℓ ∩ B 3/2 \ B 2/3 . Because Dv ℓ (y) = ℓ (q+1)/(q+1-p) Dv(ℓy), it follows that

|Dv(x)| ≤ C A 1/q+1-p |x| (q+1)/(q+1-p)
By the mean value Theorem, and using the fact that v vanishes on ∂Ω \ {0}, we derive

v(x) ≤ Cρ(x) A 1/q+1-p |x| (q+1)/(q+1-p) ,
which implies (2.7 ). .

The construction of solutions of the quasilinear equations (2.1 ) with prescribed isolated singularity on the boundary of a general C 2 bounded domain Ω is settled upon similar constructions when the domain is either a half space, or a ball.

Proposition 2.3 Assume N -1 < q < 2N -1 and let H = R N + = {x = (x 1 , ..., x N ) : x N > 0)} and k > 0.
Then there exists a unique positive solution u = u H k ∈ C 1 (H \ {0}) of (2.1 ) in H which vanishes on ∂H \ {0} and satisfies ,

u(x) = k x N |x| 2 (1 + •(1)) as x → 0. (2.10)
Proof. Since the function x → kx N |x| -2 is N -harmonic in H and vanishes on ∂H \ {0}, it is a supersolution of (2.1 ). We write spherical coordinates in R N under the form

x = (r, σ) ∈ [0, ∞) × S N -1 = (r, sin φ σ ′ , cos φ) : σ ′ ∈ S N -2 , φ ∈ [0, π] ,
(2.11)

then Du = u r i + 1 r ∇ σ u,
where i = x/ |x|, ∇ σ denotes the covariant gradient on S N -1 , and equation (2.1 ) takes the form

-r 1-N r N -1 u 2 r + r -2 |∇ σ u| 2 (N -2)/2 u r r -r -2 div σ . u 2 r + r -2 |∇ σ u| 2 (N -2)/2 ∇ σ u + |u| q-1 u = 0.
(2.12)

Next ∇ σ u = -u φ e + 1 sin φ ∇ σ ′ u
where e is derived from x/ |x| by the rotation with angle π/2 in the plane 0, x, N (N being the North pole), and ∇ σ ′ is the covariant gradient on S N -2 and (see [START_REF] Borghol | Boundary singularties of N -harmonic functions[END_REF])

div σ .   u 2 r + |∇ σ u| 2 r 2 (N -2)/2 ∇ σ u   = 1 sin N -2 φ   sin N -2 φ u 2 r + u 2 φ r 2 + |∇ σ ′ u| 2 r 2 sin 2 φ (N -2)/2 u φ   φ + 1 sin 2 φ div σ ′   u 2 r + u 2 φ r 2 + |∇ σ ′ u| 2 r 2 sin 2 φ (N -2)/2 ∇ σ ′ u   .
(2.13)

If u depends only on r and φ, (2.1 ) takes the form

-r 1-N r N -1 u 2 r + r -2 u 2 φ (N -2)/2 u r r -r -2 sin 2-N φ sin N -2 φ u 2 r + r -2 u 2 φ (N -2)/2 u φ φ + |u| q-1 u = 0. (2.14)
Step 1 We look for a local subsolution w under the form

w(r, σ) = k(1 -r α )r -1 cos φ r > 0 , φ ∈ [0, π/2].
where α > 0 is to be determined. Then

w r = -kr -2 (1 + (α -1)r α ) cos φ and w φ = -kr -1 (1 -r α ) sin φ w 2 r + r -2 w 2 φ := P = k 2 r -4 1 + 2(α cos 2 φ -1)r α + r 2α ((α 2 -2α) cos 2 φ + 1) w rr = kr -3 (2 -(α -1)(α -2)r α ) cos φ and w φφ = -kr -1 (1 -r α ) cos φ P r = -2k 2 r -5 2 + (4 -α)(α cos 2 φ -1)r α + (2 -α)((α 2 -2α) cos 2 φ + 1)r 2α P φ = -k 2 αr α-4 [2 + (α -2)r α ] sin 2φ, P r w r + r -2 P φ w φ = 2k 3 r -7 2 + (5α -6 + (2α -α 2 ) cos 2 φ)r α + O(r 2α )) cos φ, (N -1)r -1 w r + w rr + (N -2)r -2 cot φ w φ + r -2 w φφ = kr -3 [4 -2N ) + (2 -α)(N + α -2)r α ] cos φ. Since -div |Dw| N -2 Dw + w q = Lw = -P (N -2)/2 (N -1)r -1 w r + w rr + (N -2)r -2 cot φ w φ + r -2 w φφ - N -2 2 P (N -4)/2 P r w r + r -2 P φ w φ + w q ,
and w q = k q (1 -r α ) q r -q cos q φ = k q (1 -qr α + O(r 2α ))r -q cos q φ, a straightforward computation leads to

Lw = k p-1 α 3 -2N + (2 + α)(N -2) cos 2 φ + O(r α ) P (N -4)/2 r α-7 cos φ + k q (1 -qr α + O(r 2α ))r -q cos q φ = k p-1 α 3 -2N + (2 + α)(N -2) cos 2 φ r -(2N -1)+α cos φ + k q r -q cos q φ -qk q r -q+α cos q φ + O(r -(2N -1)+2α cos φ) + O(r -q+2α cos φ).
(2.15)

By assumption q < 2N -1. If we choose α < min{2N -1 -q, 1/(N -2)}, there exists R ∈ (0, 1] such that Lw ≤ 0 on H ∩ B R .
Step 2 Next we construct a solution u R in B R ∩ H which vanishes on ∂B R ∩ H and on ∂H \ {0} and satisfies for any δ > 0. In order to improve this inequality, we perform a scaling: for r > 0, we set u r (x) = ru(rx). Then u r satisfies

lim r→0 ru R (r, σ) cos φ = k. (2.16) Let ℓ R = k(1 -R α )R -1 . Inasmuch w -ℓ R is a subsolution, for any ǫ > 0 we can construct a nonnegative solution u ǫ of (2.1 ) in H ∩ (B R \ B ǫ ) which vanishes on H ∩ ∂B R and on ∂H ∩ (B R \ B ǫ ) and takes the value kǫ -2 x N on H ∩ ∂B ǫ . By comparison (w(x) -ℓ R ) + ≤ u ǫ (x) ≤ kx N |x| -2 . (2.17) Furthermore, ǫ → u ǫ is increasing. Set u = u R = lim ǫ→0 u ǫ , then u is a solution of (2.1 ) in H ∩ B R which vanishes on ∂B R ∩ H
-div |Du r | N -2 Du r + r 2N -1-q (u r ) q = 0 (2.18)
in H ∩ B R/r where there holds

k(x N |x| -2 (1 -r α |x| α ) -ℓ R ) + ≤ u r (x) ≤ kx N |x| -2 . (2.19)
Since u r is uniformly bounded for 1/2 ≤ |x| ≤ 2, it follows from regularity theory [START_REF] Libermann | Boundary regularity for solutions of degenerate elliptic equations[END_REF] that it is also bounded in the C 1,α -topology of 2/3 ≤ |x| ≤ 3/2. Using Ascoli's theorem and the fact that u r (x) converges to kx N |x| -2 pointwise and locally uniformly, it follows that Du r (x) =

r 2 Du(rx) converges uniformly in {x ∈ H : 2/3 ≤ |x| ≤ 3/2} to -2kx N |x| -4 x + k |x| -2 e N
which is the gradient of x → kx n |x| -2 . Using the expression of Du in spherical coordinates we obtain

r 2 u r i -ru φ e + r sin φ ∇ σ ′ u → -2kσ N i + ke N uniformly on S N -1 + as r → 0, where σ N = σ, e N . Inasmuch i, e and ∇ σ ′ u are orthogonal, the component of e N is sin φ, thus ru φ (r, σ ′ , φ) → -k sin φ as r → 0. (2.20) Since u(r, σ ′ , φ) = φ π/2 u φ (r, σ ′ , θ) dθ, (2.21)
the previous convergence estimate establishes (2.16 ).

Step 3 Construction of the solution in H. Let η be the truncation function introduced in the proof of Proposition 2.1, and η ǫ (r) = η((r -ǫ) + ). Then the function u R,ǫ defined by

u R,ǫ = η ǫ • u R in H ∩ B R
u R,ǫ ≤ u δ ≤ kx N |x| -2 .
When δ → 0, u δ decreases and converges to some u which satisfies (2.1 ) and the previous inequality. Letting successively ǫ → 0 and η(r) → r + we obtain that u satisfies

ǔR (x) ≤ u(x) ≤ kx N |x| -2 in H, (2.22)
where ǔ is the extension of u by zero outside B R . The proof of (2.10 ) is the same as in Step 2.

Step 4 Uniqueness. Let u and û be two solutions of (2.1 ) satisfying (2.10 ) and ǫ > 0. Then

u ǫ = (1 + ǫ)u + ǫ is a super solution which is positive of ∂H \ {0}.
Inasmuch it dominates û both in a neighborhood of 0 and in a neighborhood of infinity, it dominates û in H. Letting ǫ → 0 yields to u ≥ û. Similarly û ≥ u.

Proposition 2.4 Assume N -1 < q < 2N -1 and let B = B 1 (0), a ∈ ∂B and k > 0.
Then there exists a unique function u = u B k,a ∈ C 1 (B \ {a}) which vanishes on ∂B \ {a} and satisfies (2.1 ) in B and

u(x) = k 1 -|x| |x -a| 2 (1 + •(1)) as x → a. (2.23)
Proof. With a change of coordinates, we can assume that B has center m = (0, ..., 0, -1/2) and a is the origin of coordinates. We denote by ω the point (0, ..., 0, -1) and by I ω the inversion with center ω and power 1. By this involutive transformation, the half space

H = {x ∈ R N : x N > 0} is transformed into the ball B * = {x ∈ R N : |x| 2 + x N < 0}. Thus the function x → P k (x) = -k(|x| 2 + x N )/2 |x|
2 is N -harmonic and positive in B * , vanishes on ∂B * \ {0} and is singular at 0. Let v k be the solution of (2.1 ) in H satisfying (2.10 ), and

u k = v k • I ω . Then u k ∈ C(B * \ {0}) satisfies    -div |Du k | N -2 Du k + |x -ω| -2N u q k = 0 in B * u k = 0 on ∂B * \ {0}.
(2.24) Furthermore u k ≤ P k and

P k (x) = k 1/4 -|x -m| 2 2 |x| 2 = k 1/2 -|x -m| 2 |x| 2 (1 + •(1)) = u k (x)(1 + •(1)) (2.25) as x → 0. Inasmuch |x -ω| ≤ 1, u k is a subsolution of (2.1 ) in B * .
For ǫ > 0 we construct a solution v ǫ of (2.1 ) in B * \ B ǫ (0) with boundary value P k . By the maximum principle

u k ≤ v ǫ ≤ P k in B * \ B ǫ (0)
. Since the sequence {v ǫ } is monotone, we obtain that there exists a solution lim ǫ→0 v ǫ := u ∈ C 1 (B * \ {0}) of (2.1 ) in B * which satisfies

u k (x) ≤ u(x) ≤ P k (x) in B * , (2.26)
and

u(x) = k 1/2 -|x -m| 2 |x| 2 (1 + •(1)).
(2.27)

We change the variables in setting x ′ N = x N + 1/2 and x ′ i = x i (i = 1, ..., N -1). We define u ′ (x ′ ) = u(x) and denote by a the point (0, ..., 0, 1). Clearly u ′ satisfies (2.1 ) in B 1/2 , vanishes on ∂B 1/2 \ {a} and

u ′ (x) = k 1/2 -|x| 2 |x -a/2| 2 (1 + •(1)) as x → a/2.
(2.28)

By the transformation ℓ → ℓ p/(q+1-p) u ′ k (ℓx), where ℓ = 1/2, we obtain a solution u k,a of (2.1 ) in B which verifies (2.30)

u k,a (x) = 2 N/(q+1-N ) k 1 -|x| |x -a| 2 (1 + •(1)) as x → a. ( 2
Proof. Uniqueness follows from (2.30 ) by the same method as in Proposition 2.3 and Proposition 2.4. Actually, it will be proved in Theorem 2.7. For existence we perform the inversion I 1 0 with center 0 and power 1. It transforms the function u B k,a constructed in the previous proposition into a function v ∈ C 1 (G \ {a}) which vanishes on ∂B \ {a} and satisfies (2.30 ). Furthermore v is solution of

-div |Dv| N -2 Dv + |x| -2N |v| q-1 v = 0 (2.31) in G. Since |x| > 1, v is a super solution for (2.1 ) in G.
With no loss of generality we can assume that a = (0, ...0, 1) and let u H 1 k,a be the solution of (2.1 ) in

H 1 = {x = (x 1 , ..., x N : x N > 1)} satisfying (2.10 ) already constructed in Proposition 2.3. Then υ ǫ = η(u H 1 k,a
) is a subsolution in G (where η ǫ has been defined in the proof of Proposition 2.1). By the same approximation as in the previous proposition, we construct an increasing sequence {u ǫ } (ǫ > 0) of solutions of (2.1 ) in G \ B ǫ (a) which vanishes on ∂G \ B ǫ (a), takes the value v on G ∩ ∂B ǫ (a) and verifies υ ǫ ≤ u ǫ ≤ v in G \ B ǫ (a). Letting ǫ → 0, we obtain the existence of a solution u * in G which satisfies

ũH 1 k,a ≤ u * ≤ v in G (2.32)
where we denote by ũH 1 k,a the extension of u H 1 k,a by zero in H 1 c . We conclude that (2.30 ) holds in H 1 . In order to extend this convergence to whole G, we proceed as in the proof of Proposition 2.3, with a minor modification due to the geometry. We put the origin of coordinates at a, takes the same spherical coordinates and obtain again that

r 2 u * r i -ru * φ e + r sin φ ∇ σ ′ u * → -2kσ N i + ke uniformly on S N -1 + as r → 0.
Therefore (2.20 ) holds for any φ ∈ [0, π/2]. For r > 0, the angle φ ranges from ψ(r) = cos -1 (-r/2) to 0 (here is the difference with the half-space case) and |x| 2 ∇u(x) remains bounded in this domain, by the regularity theory for quasilinear elliptic equations. Since

u * (r, σ ′ , φ) = φ ψ(r) u * φ (r, σ ′ , θ) dθ, (2.33)
we derive, as in the proof of Proposition 2.3,

lim r→0 u * (r, σ ′ , φ) = k cos φ uniformly on [0, π/2]. (2.34)
The proof that (2.30 ) holds is a particular case of Theorem 2.7.

In a general domain we have to extend the solution through the boundary. We denote by ρ(x) the signed distance from x → ∂Ω, that is ρ(x) = ρ(x) if x ∈ Ω and ρ(x) = -ρ(x) if x ∈ Ω c . Since ∂Ω is C 2 , there exists β 0 > 0 such that if x ∈ R N verifies -β 0 ≤ ρ(x) ≤ β 0 , there exists a unique ξ x ∈ ∂Ω such that |x -ξ x | = | ρ(x)|. Furthermore, if ν ξx is the outward unit vector to ∂Ω at ξ x , x = ξ x -ρ(x)ν ξx . In particular ξ x -ρ(x)ν ξx and ξ x + ρ(x)ν ξx have the same orthogonal projection ξ x onto ∂Ω.

Let

T β0 (Ω) = {x ∈ R N : -β 0 ≤ ρ(x) ≤ β 0 }, then the mapping Π : [-β 0 , β 0 ] × ∂Ω → T β0 (Ω) defined by Π(ρ, ξ) = ξ -ρν(ξ) is a C 2 diffeomorphism. Moreover DΠ(0, ξ)(1, e) =
e-ν ξ for any e belonging to the tangent space T ξ (∂Ω) to ∂Ω at ξ. If x ∈ T β0 (Ω), we define the reflection of x through ∂Ω by ψ(x) = ξ x + ρ(x)ν xix . Clearly ψ is an involutive diffeomorphism from Ω ∩ T β0 (Ω) to Ω c ∩ T β0 (Ω). Furthermore for any ξ ∈ ∂Ω, Dψ(ξ) = S T ξ (∂Ω) is the symmetry with respect to the tangent space T ξ (∂Ω) to

∂Ω at ξ. If a function v is defined in Ω ∩ T β0 (Ω), we define ṽ in Ω c ∩ T β0 (Ω) by ṽ(x) = v(x) if x ∈ Ω ∩ T β0 (Ω) -v • ψ(x) if x ∈ Ω c ∩ T β0 (Ω). (2.35) Proposition 2.6 Let v ∈ C 1,α (Ω ∩ T β0 (Ω) \ {0}
) be a solution of (2.1 ) in Ω ∩ T β0 (Ω) vanishing on ∂Ω \ {0}. Then ṽ ∈ C 1,α (T β0 (Ω) \ {0}) is solution of a quasilinear equation

- j ∂ ∂x j Ãj (x, Dṽ) + b(x) |ṽ| q-1 ṽ = 0 (2.36)
in T β0 (Ω) \ {0} where the Ãj and b are C 1 functions defined in T β0 (Ω) where they verify

                     (i) Ãj (x, 0) = 0 (ii) i,j ∂ Ãj ∂η i (x, η)ξ i ξ j ≥ Γ |η| p-2 |ξ| 2 (iii) i,j ∂ Ãj ∂η i (x, η) ≤ Γ |η| p-2 (iv) Γ ≥ b(x) ≥ γ (2.37) for all x ∈ T β (Ω) \ {0} for some β ∈ (0, β 0 ], η ∈ R N , ξ ∈ R N and some 0 < γ ≤ Γ.
Proof. The assumptions (2.37 ) implies that weak solutions of (2.36 ) are C 1,α , for some α > 0 [START_REF] Tolksdorff | Regularity for a more general class of quasilinear elliptic equations[END_REF] and satisfy the standard a priori estimates. As it is defined the function ṽ is clearly

C 1 in T β0 (Ω) \ {0}. Writing Dv(x) = -D(ṽ • ψ(x)) = -Dψ(x)(Dṽ(ψ(x))) and x = ψ(x) = ψ -1 (x) Ω∩T β 0 (Ω) |Dv| p-2 Dv.Dζ + |v| q-1 vζ dx = Ω c ∩T β 0 (Ω)
|Dψ(Dṽ)| p-2 Dψ(Dṽ).Dψ(Dζ) + |ṽ| q-1 ṽζ(ψ(x)) |Dψ| dx.

But

Dψ(Dṽ).Dψ(Dζ)

= k i ∂ψ i ∂x k ∂ṽ ∂x i   j ∂ψ j ∂x k ∂ζ ∂x j   = j   i,k ∂ψ i ∂x k ∂ψ j ∂x k ∂ṽ ∂x i   ∂ζ ∂x j .
We set b(x) = |Dψ|,

A j (x, η) = |Dψ| |Dψ(η)| p-2 i k ∂ψ i ∂x k ∂ψ j ∂x k η i , (2.38) and A(x, η) = (A 1 (x, η), ..., A N (x, η)) = |Dψ| |Dψ(η)| p-2 (Dψ) t Dψ(η). ( 2

.39)

For any ξ ∈ ∂Ω, the mapping Dψ ∂Ω (ξ) is the symmetry with respect to the hyperplane T ξ (∂Ω) tangent to ∂Ω at ξ, so |Dψ(ξ)| = 1. Inasmuch Dψ is continuous, a lengthy but standard computation leads to the existence of some β ∈ (0, β 0 ] such that (2.37 ) holds in

T β (Ω) ∩ Ω c .
If we define à (resp. b) to be |η| p-2 η (resp 1) on T β (Ω) ∩ Ω and A (resp. |Dψ|) on T β (Ω) ∩ Ω c , then inequalities (2.37 ) are satisfied in T β (Ω).

Remark. Notice that, similarly to the p-laplacian, the vector field à is positively homogeneous with exponent p -1 with respect to η. Furthermore, if for r > 0 we set Ãr j (x, η) = Ãj (rx, η) , then Ãr j satisfies the same estimates (2.37 ) as A j , uniformly in T r -1 β (r -1 Ω), for 0 < r ≤ 1. Furthermore

lim r→0 A r j (x, η) = |η| p-2 η j ∀η ∈ R N , ∀j = 1, ..., N,
and this limit is uniform on the bounded subsets of R N .

Theorem 2.7 Let Ω be a bounded domain with a C 2 boundary and a ∈ ∂Ω. Assume N -1 < q < 2N -1 and denote by ρ(x) the distance from x to ∂Ω. Then for any k > 0 there exists a unique function u = u k,a ∈ C(Ω \ {a}) which vanishes on ∂Ω \ {a}, is solution of (2.1 ) and satisfies

u k,a (x) = k ρ(x) |x -a| 2 (1 + •(1)) as x → a.
(2.40)

Proof. Uniqueness follows from (2.40 ) by the same technique as in the previous propositions.

For existence let B i R be a ball of radius R such that B i R ⊂ Ω and a ∈ ∂B i R , and let ω i be its center. We denote by U i the solution of (2.1 ) in B i R , which vanishes on ∂B i R \ {a} and satisfies

U i (x) = k R -|x -ω i | |x -a| 2 (1 + •(1)) as x → a. (2.41)
If we set U δ = η δ (U i ), we have already seen that Ǔδ , the extension of U δ by zero outside its support, is a subsolution of (2.1 ) in Ω. Because V Ω a , the N -harmonic function element of C(Ω \ {a}) vanishing on ∂Ω \ {a}, satisfies

V Ω a (x) = ρ(x) |x -a| 2 (1 + •(1)) as x → a, x ∈ B i R , (2.42)
there holds kV Ω a ≥ Ǔδ . If Ω ǫ = Ω \ {B ǫ (a)} (ǫ > 0), we construct a solution u ǫ ∈ C(Ω ǫ ) of (2.1 ) in Ω ǫ , which vanishes on ∂Ω \ B ǫ (a) and takes the value kV Ω a on ∂B ǫ (a) ∩ Ω. By the maximum principle ǫ → u ǫ is increasing and Ǔδ ≤ u ǫ ≤ kV Ω a in Ω ǫ . Letting ǫ → 0 we obtain that u ǫ converges in the C 1 loc -topology of Ω \ {a} to a solution u = u k,a of (2.1 ) in Ω. It follows from the previous inequalities that

Ǔδ (x) ≤ u(x) ≤ kV Ω a (x) ∀x ∈ Ω \ {a}. ( 2

.43)

In order to prove the asymptotic behaviour, we proceed as in Proposition 2.4 with the help of the reflection principle of Proposition 2.6. We fix the origin of coordinates at a = 0 and the normal outward unit vector at a to be -e N . If ũ is the extension of u by reflection through ∂Ω, it satisfies

- j ∂ ∂x j Ãj (x, Dũ) + b(x) |ũ| q-1 ũ = 0 (2.44)
in T β (Ω) \ {0}. For r > 0, set ũr (x) = rũ(rx). Then ũr is solution of

- j ∂ ∂x j Ãr j (x, Dũ r ) + r 2N -1-q b(rx) |ũ r | q-1 ũr = 0 (2.45) in T βr -1 (Ω r ) \ {0}
, where Ω r := r -1 Ω. By [START_REF] Borghol | Boundary singularties of N -harmonic functions[END_REF]Th 2.4] there exists C > 0 such that

kV Ω 0 (x) ≤ Ck ρ(x) |x| 2 .
Furthermore, for any

x ∈ T β (Ω) \ {0}, ρ(x) := dist (x, Ω) = ρ(ψ(x)) (we recall that ψ(x)
is the symmetric of x with respect to ∂Ω as it is defined in Proposition 2.6), and c |x| ≤ |ψ(x)| ≤ c -1 |x| for some c > 0, the same relations holds if T β (Ω) is replaced by T βr -1 (Ω r ) and ρ(x) by ρ r (x

) := dist (x, Ω r ). Since Ω is C 2 , lim r→0 ρ(rx) rρ r (x) = 1
uniformly on bounded subsets of R N . Consequently

|ũ r | (x) ≤ Ckr -1 ρ(rx) |x| 2 = Ck ρ r (x) |x| 2 (1 + •(1)).
For 0 < a < b fixed and r ≤ r 0 (for some

r 0 ∈ (0, 1]) the spherical shall Γ a,b = {x ∈ R N : a ≤ |x| ≤ b} is included into T βr -1 (Ω r )
. By the classical regularity theory for quasilinear equations [START_REF] Tolksdorff | Regularity for a more general class of quasilinear elliptic equations[END_REF] and Proposition 2.6, there holds x N = h(x ′ )}, and Dh(0) = 0 (this expresses the fact that ∂H = T 0 (∂Ω)). If we define Ψ by

Dũ r C α (Γ 2/3,3/2 ) ≤ C r ũr L ∞ (Γ 1/2,2 ) , ( 2
Ψ(x) = (x ′ , x N -h(x ′ )) ∀x ∈ V ǫ0 .
then det(DΨ) = 1 and DΨ(0) = I. Up to replacing ǫ 0 by a smaller quantity, Ψ is a Clearly the mapping k → u k,a is increasing. As u k satisfies the estimates (2.7 ) and (2.30 ), u k,a converges in the C 1 loc (Ω \ {a})-topology, as k → ∞, to some u ∞,a , solution of (2.1 ) in Ω, vanishes on ∂Ω \ {a} and satisfies

C 2 diffeomorphism from V ǫ0 into a neighborhood V ′ of 0 such that V ǫ0 ∩ ∂Ω) = Θ ǫ0 . Because dist (Ψ(x), ∂H) = x N -h(x ′ ), dist (Ψ(x), ∂H) = ρ(x)(1 + •( 1
lim x→a |x -a| 2 u ∞,a (x) ρ(x) = ∞.
(2.50)

In order to describe the precise behaviour of u ∞,a , we have to introduce separable solutions of (2.1 ) in R N \ {0}: if we look for solutions u under the form u(r, σ) = r β ω(σ), then β = -β q = -N/(q + 1 -N ) and ω satisfies

-div σ β 2 q ω 2 + |∇ σ ω| 2 (N -2)/2 ∇ σ ω -Λ β 2 q ω 2 + |∇ σ ω| 2 (N -2)/2 ω + |ω| q-1 ω = 0
(2.51) on S N -1 where Λ = (N -1)β 2 q . We shall denote by S q the set of (always C 1,α ) solutions of (2.51 ). If u is a separable solution of (2.1 ) in H which vanishes on ∂H \ {0}, the function ω is a solution of (2.51 ) in S N -1 + which vanishes on ∂S N -1 + = S N -2 . We shall denote by S * q the set of such functions and by S * q + the subset of positive solutions. We recall some simple facts Proposition 2.8 (i) For any q > N -1, S q contains at least the three constant functions 0 and ±((N -1)β N q ) 1/(q+1-N ) . (ii) For any q ≥ 2N -1, S * q = {0}. (iii) For any q ∈ (N -1, 2N -1), S * q + contains a unique element.

Proof. Assertion (i) is evident since Λ > 0. Assertion (ii), as well as the existence part of assertion (iii), can be found in [START_REF] Huentutripay | A dynamical system approach to the construction of singular solutions of some degenerate elliptic equations[END_REF] or [START_REF] Véron | Singular p-harmonic functions and related quasilinear equations on manifolds[END_REF]. Furthermore any ω ∈ S * q + is positive in S N -1 + and verifies ω φ < 0 by Hopf boundary lemma as the outward normal derivative on ∂S N -1 + is ∂ /∂φ. We can construct a minimal element in S * q + in the following way: If we denote by u H k the unique solution of (2.1 ) in H which satisfies (2.10 ) and set

T r (u H k )(x) = r βq u H k (rx) for r > 0, then T r (u H k ) is a solution of (2.1 ) in H which satisfies T r (u H k ) = r (2N -1-q)/(q+1-N ) k x N |x| 2 (1 + •(1)) as x → 0.
in R N , where q > p -1 > 0, p not necessarily equal to N or to 2, under the form u(r, σ) = r β ω(σ), then β = β p,q = -p/(q + 1 -p) and ω is a solution of

-div σ β 2 p,q ω 2 + |∇ σ ω| 2 (p-2)/2 ∇ σ ω -Λ(p, q) β 2 p,q ω 2 + |∇ σ ω| 2 (p-2)/2 ω+|w| q-1 ω = 0
(2.53) on S N -1 where Λ(p, q) = β p-1 p,q (qβ p,q -p). If we look for separable solutions in H which vanishes on ∂H \ {0} the solution ω of (2.53 ) is subject to the boundary condition ω = 0 on ∂S N -1 + = S N -2 . A fairly exhaustive theory of existence is developped in [START_REF] Véron | Singular p-harmonic functions and related quasilinear equations on manifolds[END_REF], [START_REF] Huentutripay | A dynamical system approach to the construction of singular solutions of some degenerate elliptic equations[END_REF]. The existence of non-trivial solution of (2.53 ) is insured as soon Λ(p, q) > 0, or equivalently q < N (p -1)/(N -p) if p < N , and no condition if p ≥ N . If q ≥ N (p -1)/(N -p) no solution exists, up to the trivial one. This is linked to the removability result proved by Vàzquez and Véron [START_REF] Vàzquez | Removable singularities of some strongly nonlinear elliptic equations[END_REF]. The existence of non trivial solutions of the same equation in

S N -1 + vanishing on ∂S N -1
+ is much more complicated. However it is proved in [START_REF] Véron | Singular p-harmonic functions and related quasilinear equations on manifolds[END_REF], [START_REF] Huentutripay | A dynamical system approach to the construction of singular solutions of some degenerate elliptic equations[END_REF] that there exists a critical exponent q c > p -1 such that, if q ≥ q c no non-trivial solution exists while if p -1 < q < q c there exist a unique positive solution in S N -1

+ vanishing on ∂S N -1 + .
The uniqueness proof in the previous proposition is valid.

The next result characterizes the solution of (2.1 ) with a strong singularity on the boundary. In order to express the result, we assume that the outward normal unit vector to ∂Ω at a is -e N . Theorem 2.9 Let Ω be a bounded domain with a C 2 boundary and a ∈ ∂Ω. Assume 0 < p -1 < q < 2N -1. Then there exists a unique function u ∈ C 1 (Ω \ {a}) which vanishes on ∂Ω \ {a}, is solution of (2.1 ) in Ω and satisfies

lim x→a |x -a| 2 u(x) ρ(x) = ∞. (2.54) Furthermore lim x → a (x -a)/ |x -a| → σ |x -a| βq u(x) = ω(σ), (2.55)
locally uniformly on S N -1

+ . Finally u = u ∞,a = lim k→∞ u k,a .
Proof. We already know that u ∞,a satisfies (2.54 ). By translation we fix the origin 0 of coordinates at the point a and we assume that -e N is the outward unit vector to ∂Ω at 0. If G is any C 2 domain in R N to the boundary of which 0 belongs, we denote by u G k the solution of (2.1 ) in G, which vanishes on ∂G \ {0} and verifies

u G k = k ρ G (x) |x| 2 (1 + •(1)) as x → 0, (2.56)
where ρ G (x) = dist (x, G). When there is no ambiguity,

u Ω k = u k . By the maximum principle G ⊂ G ′ implies u G k ≤ u G ′ k in G.
(i) u B k (x) ≤ u k (x) ≤ u B ′c k (x) ≤ u B ′ k b ′ + x -b ′ |x -b ′ | 2 = u B k S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ B (ii) u k (x) ≤ u B ′c k (x) ≤ u B k S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ Ω,
(2.57) and similarly

(i) u B k (x) ≤ u H k (x) ≤ u B ′c k (x) ≤ u B k S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ B (ii) u H k (x) ≤ u B ′c k (x) ≤ u B k S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ H, (2.58)
Letting k → ∞, we obtain

(i) u B ∞ (x) ≤ u ∞ (x) ≤ u B ′c ∞ (x) ≤ u B ∞ S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ B (ii) u ∞ (x) ≤ u B ′c ∞ (x) ≤ u B ∞ S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ Ω, (2.59)
as well as

(i) u B ∞ (x) ≤ |x| -βq ω x |x| ≤ u B ′c ∞ (x) ≤ u B ∞ S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ B (ii) |x| -βq ω x |x| ≤ u B ′c ∞ (x) ≤ u B ∞ S b ′ + x -b ′ |x -b ′ | 2 ∀x ∈ H.
(2.60) From (2.60 )-(i) and the fact that b ′ = -b, we also derive

|x| -βq ω(x/ |x|) ≤ u B ′c ∞ (x) ≤ S x + b |x + b| 2 -b -βq ω   S x + b -|x + b| 2 b S x + b -|x + b| 2 b   . (2.61) But S x + b |x + b| 2 -b = |x| |x + b| = |x| (1 + •( 1
T -1 (x) -βq ω T -1 (x) |T -1 (x)| ≤ u B ∞ (x) ≤ |x| -βq ω x |x| . (2.64) Furthermore T -1 (x) = x 1 |x -b| 2 , ..., x N -1 |x -b| 2 , 1 -x N |x -b| 2 -1 = x -|x| 2 e N |x -b| 2 .
Then uniformly on any subset of Ω such that |x| 2 = •(x N ) near 0. In order to obtain the precise behaviour (2.55 ), we proceed and in the proof of Theorem 2.7. We extend u by reflection through ∂Ω near 0 and denote by ũ the extended function defined in T β (Ω). For r ∈ (0, 1] we define w r := T r (ũ)(x) = r βq ũ(rx).

T -1 (x) = b + x -b |x -b| 2 = |x| |x -b| , and
T -1 (x) -βq ω T -1 (x) |T -1 (x)| = |x| -βq |x
Then w r satisfies

- j ∂ ∂x j Ãr j (x, Dw r ) + b(rx) |w r | q-1 w r = 0 (2.67)
in T βr -1 (Ω r ). Since w r is uniformly bounded on Γ 1/2,2 (by Proposition 2.1 applied to u and -u) and the definition of the refected function), Dw r (u) is bounded in C α (Γ 2/3,3/2 ). By Ascoli's theorem w r converges in the C 1 (Γ 2/3,3/2 )-topology to x → |x| -βq ω(x/ |x|), where ω is defined from ω by reflection through the equator ∂S N -1

+

. In ordre to get rid of the boundary, we use again the C 2 diffeomorphism Ψ which sends B ǫ0 onto itself and verifies Ψ(B ǫ0 ∩ ∂Ω) = B ǫ0 ∩ ∂H. We set x = Ψ -1 (y) and ũ(x) = u * (y).

The removability result

In this section Ω is a C 2 domain of R N and a ∈ ∂Ω. The next result extends the removability result of Gmira-Véron [START_REF] Gmira | Boundary singularities of solutions of some nonlinear elliptic equations[END_REF] dealing with semilinear equations. where q c := 2N -1 and let u ∈ C 1 (Ω \ {a}) be a solution of

-div |Du| N -2 Du + g(u) = 0 in Ω (3.2)
which coincides with some φ ∈ C 1 (∂Ω) on ∂Ω \ {a}. Then u extends to Ω as a continuous function.

Proof. Without any loss of generality, we can assume that Ω is bounded, a = 0 and -e N is the outward normal vector to ∂Ω at 0. We denote by V Ω 0 the solution of (2.2 ) in Ω which vanishes on ∂Ω \ {0} and satisfies

V Ω 0 (x) = ρ(x) |x| 2 (1 + •(1)) as x → 0.
Let M be the supremum of |φ| on ∂Ω and M = max{M, (B/A) 1/q }. By assumption there exists A > 0 and B ≥ 0, depending only on g, such that

-div |Du| N -2 Du + Au qc ≤ B in {x ∈ Ω : u(x) > 0}. (3.3) If v = u -M , then v ≤ 0 on ∂Ω \ {0} and -div |Dv| N -2 Dv + Av qc ≤ 0 in {x ∈ Ω : v(x) > 0}, (3.4)
Using the same functions η ǫ as in the proof of Proposition 2.1 we deduce that η ǫ (v) satisfies the same inequality as v, but on whole Ω. By Proposition 2.2 with q = q c and the expression of

V Ω 0 it follows that v(x) ≤ cV Ω (x) ∀x ∈ Ω, (3.5)
where the constant c depends on A and N . Furthermore, there exists a function

u * ∈ C 1 (Ω \ {0}) such that 0 ≤ v + ≤ u * (x) ≤ cV Ω 0 in Ω, and
-div |Du * | N -2 Du * + Au * qc = 0 in Ω. (3.6)
As in the proof of Theorem 2.9 we extend u * through the boundary into ũ and scale it by setting T r (ũ) := w r (x) = rũ(rx) for r > 0. Inasmuch all the previous a priori estimates apply (compactness), it follows that there exists a subsequence {r n } converging to 0 and a function w

∈ C 1 (R N \ {0}) such that w rn → w in the C 1 loc -topology of R N \ {0}, w is a solution of          -div |Dw| N -2 Dw + Aw qc = 0 in R N \ {0} w ≥ 0 in H = {x ∈ R N : x N > 0} w = 0 on ∂H \ {0}. (3.7)
At end, (3.5 ) transforms into

0 ≤ w(x) ≤ c x N |x| 2 ∀x ∈ H. (3.8)
For ǫ > 0 we denote by W ǫ the solution of

         -div |DW ǫ | N -2 DW ǫ + AW qc ǫ = 0 in H \ B ǫ (0) W ǫ = cǫ -2 x N on H ∩ ∂B ǫ (0) W ǫ = 0 on ∂H \ B ǫ (0). (3.9)
By the maximum principle 0 ≤ w(x) ≤ W ǫ (x) ≤ cx N |x| -2 for any ǫ > 0, and by uniqueness, T r (W ǫ )(x) = rW ǫ (rx) = W ǫ/r (x). Furthermore ǫ → W ǫ is increasing. Letting ǫ → 0 we conclude that W ǫ decreases to some W 0 , which is a solution of

         -div |DW 0 | N -2 DW 0 + AW qc 0 = 0 in H W 0 ≥ 0 in H W 0 = 0 on ∂H \ {0}, (3.10)
Proof. We assume that a = 0 with ν 0 = -e N and define k = lim sup Suppose k = 0. It follows from the maximum principle that for any ǫ > 0 there exists a sequence r n → 0 such that 0 ≤ u(x) ≤ ǫV Ω 0 (x) in Ω \ {B rn (0)}. This fact implies the nullity of u. Therefore we assume that k = 0. Assume first that k is finite. Then, for any ǫ > 0, there exists a sequence of points x n converging to 0 such that

lim n→∞ u(x n ) V Ω 0 (x n ) = k (4.

  .29) Because k is arbitrary, (2.23 ) follows. Uniqueness of the solution is obtained as in Proposition 2.3 with u ǫ = (1 + ǫ)u. Proposition 2.5 Assume N -1 < q < 2N -1 and let G = B c , a ∈ ∂B and k > 0. Then there exists a unique function u = u B c k,a ∈ C 1 (G \ {a}) which vanishes on ∂B \ {a} and satisfies (2.1 ) in G and u(x) = k |x| -1 |x -a| 2 (1 + •(1)) as x → a.

r 2

 2 .46) where C r remains bounded because r ≤ 1. By Ascoli's theorem and (2.43 ) ũr (x) converges to kx N |x| -2 in the C 1 (B 3/2 \ B 1/2 )-topology. This implies in particular lim r→0 Dũ(rx) = -2kx N x |x| -4 + k |x| -2 e N . If we take in particular |x| = 1, we derive lim r→0 (rũ(r, σ), r 2 ∇ũ(r, σ)) = (k cos φ, -k sin φe N ), (2.47) uniformly with respect to σ = (sin φ σ ′ , cos φ) ∈ S N -2 × [0, π]. Because ∂Ω is C 2 there exists ǫ 0 > 0 and a C 2 real valued function h defined in Θ ǫ0 := B ǫ0 ∩ ∂H (we recall that ∂H = {x = (x ′ , 0)}) and an open neighborhood V ǫ0 of 0 such that ∂Ω ∩ V ǫ0 = {x = (x ′ , x N :

1 +

 1 )) as x → 0. Thus, if we set x = Ψ -1 (y) and ũ(x) = u * (y),(2.47 ) is equivalent to lim |y|→0 (|y| u * (|y| , σ), |y| 2 ∇u * (|y| , σ)) = (k cos φ, -k sin φ e N ), (2.48) uniformly on S N -1 , thus |y| u * (|y| , σ) = k sin φ (1 + •(1)) as |y| → 0 (2.49) uniformly with respect to σ ∈ S N -, because u * vanishes on B ǫ0 ∩ ∂H \ {0}. This implies (2.40 ).

  By dilation we can assume that there exist two balls of radius 1, B ⊂ Ω and B ′ ⊂ Ω c with respective center b = e N and b ′ = -b with the property that 0 = ∂B∩∂B ′ . It follows from the maximum principle, the fact that u B k (x) = u B ′ k (S(x)) where S is the symmetry with respect to the hyperplane ∂H and Proposition 2.4, Proposition 2.5

Theorem 3 . 1

 31 Let g be a continuous function defined on R which satisfies lim inf r→∞ g(r)/r qc > 0 and lim sup r→-∞ g(r)/|r| qc < 0, (3.1)

16 )

 16 Since u k satisfies (2.40 ) with a = 0, the two previous relations can be replaced by(i) lim n→∞ u(x n ) u k (x n ) = 1 (ii) sup |x|≤rn u(x) u k (x) ≤ 1 + ǫ.

(4. 17 ) 1 + 1 + 1 +,σ β 2 φ 2 + 1 +φ = 0 on ∂S N - 1 +.( 4 2

 1711121142 We denote r n = |x n |, ξ n = x n /r n and define u n = r n u(r n x) and u k n = r n u k (r n x). By the previous arguments combining a priori estimate and regularity theory, there exist a subsequence {r nj } and two nonnegative functions v and v ′ , N -harmonic in H and vanishing on ∂H \ {0}, such that u nj , u k nj converges to (v, v ′ ) in the C 1 loc -topology of H = R N + . Clearly equality (2.40 ) implies that r nj u k (r nj x) converges to kV H 0 (x) (which is defined by kV H 0 (x) := kx N / |x| -2 ) in the same topology. Since v ′ is uniquely determined by its blow-up at 0, this implies v ′ = kV H 0 in H. Furthermore there exists ξ ∈ S N -such that ξ n k → ξ. If ξ ∈ S N -, v(ξ) = v ′ (ξ), while, if ξ ∈ ∂S N -∂v/∂ν(ξ) = ∂ xN v(ξ) = ∂v ′ /∂ν(ξ).In both situation, the tangency conditions of the graphs of v and v ′ and the strong maximum principle implies that v = v ′ = kV H 0 . By estimate (4.17 )-(i) and the convergence properties, it followslim n→∞ u(r n ξ) u k (r n ξ) = 1, uniformly on |ξ| = 1.Consequently, for any δ > 0, there holds,(1 -δ)u k (x) ≤ u(x) ≤ (1 + δ)u k (x) ∀x ∈ Ω \ B rn ,for n large enough, which leads to u k = u. At end we consider the case k = ∞. Writting (2.1 ) under the form-div |Du| N -2 Du + d(x)u N -1 = 0 (4.18)whered(x) = |u| q+1-N (x) ≤ C |x| -N, by (2.4 ), We use the boundary Harnack principle. By [2, Th 2.2] there exists a constant c = c(N, q, Ω) > 0 the specific cases N = 2 or p = N or p = 2. Notice that in that case the function φ satisfies the so-called spherical p-harmonic spectral equation|∇ σ φ| 2 (p-2)/2 ∇ σ φ -λ β 2 φ 2 + |∇ σ φ| 2 (p-2)/2 φ = 0 in S N -.21) where λ = β(β(p -1) + p -N ). If p = 2 then β = N -1, while if N = 2,β is the positive root of the equation 3β to now and due to the lack of conformal invariance, it has not been possible to construct the equivalent of the V Ω a in a general smooth bounded domain Ω, that are positive p-harmonic functions in Ω, vanishing on ∂Ω \ {a} and satisfying lim x → a (x -a)/ |x -a| → σ |x -a| β u(x) = φ(σ). (4.23) However, if Ω = H = R N + the removability and the classification results of Sections 3 and 4 are still valid. The proofs of these theorems are developed in [1].

  and zero outside, is a subsolution of (2.1 ) in H which vanishes on ∂H \ {0} and satisfies(2.16 ). Using the same device as in Step 2, we construct a sequence of solutions u δ (δ > 0) of (2.1 ) in H \ B δ with boundary value kδ -2 x N on ∂B δ ∩ H, zero on ∂H \ B δ and satisfies

  )) as x → 0(remember that |b| = 1). If x = (x 1 , . . . , x N ), |x + b| 2 = |x| 2 + 1 + 2x N and

	Thus (2.61 ) becomes
							
	|x|	-βq ω(x/ |x|) ≤ u B ′c ∞ (x) ≤ |x|	-βq |x + b| βq ω		x + |x| 2 e N |x| 1 + |x| 2 + 2x N	 .	(2.62)
	If we assume |x| x → 0, and If we define T by 2 = •(x N ) then (x + |x| u B ′c ∞ (x) = |x| -βq ω(x/ |x|)(1 + •(1)). 2 e N )/(|x| 1 + |x| 2 + 2x N ) = x(1 + •(1))/ |x| as (2.63) T (x) = S x + b |x + b| 2 -b ,
	then (2.60 )-(i) reads also as
		S x + b -|x + b| 2 b = (x 1 , . . . , x N + |x|	2 ).

Thus T r (u H k ) = u H r (2N -1-q)/(q+1-N ) k . Furthermore, if ω ∈ S * q + , the maximum principle at 0 and at infinity (replacing u ω by u ω + ǫ and letting ǫ → 0) leads to u ω (r, σ) := r -βq ω(σ) > u H k (r, σ) ∀(r, σ) ∈ (0, ∞) × S N -1

+

, ∀k > 0.

Letting k → ∞ implies u ω (r, σ) ≥ u H ∞ (r, σ) and T r (u H ∞ ) = u H ∞ given that 2N -1 -q > 0. Then the function u H ∞ is invariant with respect to the transformation T r . It is therefore selfsimilar, and consequently under the form u H ∞ (r, σ) = r -βq ω(σ). As a result of the previous inequality ω is the minimal element of S * q + . Next we denote δ * = max{δ ≥ 0 : δω ≤ ω} and u ω,δ * = δ * u ω . Notice that δ * ∈ (0, 1] as ω > 0 in S N -1 + and satisfies Hopf boundary lemma on ∂S N -1

, and (i) either there exists σ 0 ∈ S N -1

In case (i), and as Du H

∞ never vanishes in H, it follows from [6, Lemma 1.3] (a variant of the strong comparison principle) that u ω,δ * = u. This implies that u ω,δ * is a solution, δ * = 1 and, consequently ω = ω. In case (ii) we follow the linearization procedure already introduced in [START_REF] Friedman | Singular solutions of some quasilinear elliptic equations[END_REF]. By the mean value theorem

where

is the Hessian of a strictly convex function therefore it is nonnegative and that

, we derive a contradiction with Hopf lemma. Therefore case (ii) cannot occur and ω = ω.

Remark. If we look for separable solutions of

by the standard regularity results, and satisfies 0 ≤ w ≤ W 0 . Finally, W 0 inherits the following scaling invariance property T r (W 0 )(x) = W 0 (x) for any r > 0. Therefore W 0 is a separable solution which endows the following form

where ω is nonnegative on S N -1 + and satisfies

By Proposition 2.8, ω = 0. Thus W 0 = 0 =⇒ w = 0, which implies w r (x) → 0 as r → 0 and equivalently rũ(rx) → 0 in the C 1 loc -topology of R N \ {0}. Consequently Dũ(x) = •(|x| -2 ) as x → 0 and finally u * (x) = •(V Ω 0 (x)) as x → 0. The maximum principle and the positivity of u * yields to u * ≡ 0 and finally u ≤ M in Ω. In the same way u ≥ -M . Because the modulus of continuity of u is uniformly bounded near 0, by the classical regularity theory of degenerate elliptic equations (see [START_REF] Libermann | Boundary regularity for solutions of degenerate elliptic equations[END_REF] for example), u extends as a continuous function in whole Ω.

The classification theorem

The next result extends some of Gmira-Véron's classification theorem [START_REF] Gmira | Boundary singularities of solutions of some nonlinear elliptic equations[END_REF]Sect. 4,[START_REF] Dynkin | Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary[END_REF] obtained in the study of problem (1.3 ). In the above mentioned article, the main idea was to reduce the equation to a infinite dimensional quasi-autonomous evolution system in R + × S N -1 + and to use Lyapounov-energy function. Such an approach cannot be adapted in the quasilinear case. Our method is based upon scaling and uniqueness arguments. Theorem 4.1 Assume N -1 < q < 2N -1, Ω is a bounded domain with a C 2 boundary, a ∈ ∂Ω and -e N is the outward normal unit vector to ∂Ω at a. Let u ∈ C 1 (Ω \ {a}) be a positive function satisfying (2.1 ) in Ω and vanishing on ∂Ω \ {a}. Then the following alternative holds.

(i) Either there exists k > 0 such that

.12)

Furthermore u = u k,a , the unique solution of (2.1 ) defined in Theorem 2.7.

(ii) Or

where ω is the unique positive solution of (2.51 ) on S N -1

for any x and y in Ω such that |x| = |y| be small enough. Since there exists a sequence

Thus u satisfies (2.54 ); Theorem 2.9 and (2.55 ) imply that (4.13 ) holds.

The assumption of positivity on u can be weakened if a better a priori estimate is already known. The next result extends [6, Th 1.2] into the framework of boundary singularities. Theorem 4.2 Assume N -1 < q < 2N -1, Ω is a bounded domain with a C 2 boundary, a ∈ ∂Ω and -e N is the outward normal unit vector to ∂Ω at a. Let u ∈ C 1 (Ω \ {a}) be a solution of (2.1 ) in Ω vanishing on ∂Ω \ {a} such that u/V Ω a is bounded in Ω. Then there exists k ∈ R such that u = u k,a .

Proof. The outline of the proof are very similar to the finite case of the previous theorem. We still assume a = 0 and define k by (4.14 ). If k = 0 the maximum principle implies u ≤ 0 and we return to Theorem 4.1 in the case u ≤ 0. If k = 0, k > 0 for example, (4.15 ) and (4.16 ) apply. By the previous scaling method we derive that u n k converges to some function v in the C 1 loc -topology of H = R N + which is N -harmonic in H and vanishes on ∂H \ {0}. Because r n k u k (r n k x) converges to kV H 0 , the tangency condition of v and kV H 0 at some ξ implies that v = kV H 0 . Thus u(x) ≥ 0 for |x| = r n k for n k large enough. This implies that u ≥ 0 in Ω and we are back to Theorem 4.1.

Remark. In the semilinear case of problem (1.3 ), it is proved in [START_REF] Gmira | Boundary singularities of solutions of some nonlinear elliptic equations[END_REF] that any signed solution u which satisfies lim x→a |x -a| N u(x) = 0 has constant sign. The exponent N characterize the minimal changing sign harmonic function vanishing on ∂Ω, with an isolated singularity at a. Changing sign singular N -harmonic functions are constructed in [START_REF] Borghol | Boundary singularties of N -harmonic functions[END_REF]. In particular there exist singular N -harmonic functions w under the form w(r, σ) = r -β2 ω(σ)

where

and ω is defined on S N -1 + = {x ∈ S N -1 : x N > 0}, vanishes on the equator ∂S N -1 + , is positive on S N -1 + ∩ {x : x N -1 > 0} and negative on S N -1 + ∩ {x : x N -1 < 0}. A natural question is therefore wether any signed solution u of (2.1 ) in Ω which vanishes on ∂Ω \ {a} and satisfies lim x→a |x -a| β2 u(x) = 0 has constant sign, and can be henceforth classified through Theorem 4.1.

Final remark. If one replaces the N -harmonic operator by the p-harmonic operator (p > 1) and tries to extend the results of sections 2, 3, 4, several difficulties will appear. Even if the existence of separable singular solutions is known, the precise value of the exponent β > 0 such that (r, σ) → r -β φ(σ) is p-harmonic and positive in H and vanishes on ∂H \ {0} is


J C Fritton

Kawashima

H-W Courtland

Elis

Insulin-like Growth Factor-1 Binding Proteins Alter Osteoclastogenesis

Introduction:

Aging is accompanied by a shift in bone marrow (BM) cell fate and greater osteoclastogenesis [1]. The production of IGF-1 and the binding proteins (BP) that comprise the IGF-1 complexes are also affected by aging [2,3]. To determine the auto/paracrine relationship between these constituents and osteoclastogenesis in BM we evaluated the osteoclastogenic potential of marrow-derived mesenchymal stromal cells (MSCs) from mice with IGF-1 deficiency due to knockdown of IGF-1 production by the liver or knockout of the binding proteins (BP) that stabilize IGF-1.

Methods:

We employed 10-16 week old, male, liver-specific IGF-1 deficient (LID), IGFBP-3 knockout (BP3KO) and acid-labile subunit knockout (ALSKO) mice and assessed BM cell differentiation in vitro and BM gene expression in vivo (n=5 mice per strain). All mouse strains were backcrossed to a C57BL6/J background. IACUC approval was obtained.

BM cells were flushed from femurs immediately after sacrifice and collected in α-MEM culture. Cells were then either cultured or sorted (FACS). For sorting cells were preincubated with rat anti-mouse CD16/CD32 (BD Biosciences) at 4°C to block Fc receptors. Cells were then incubated with fluorescently labeled antibodies: R-Phycoerythrin (PE)-c-Fms/CSF-1R (Santa Cruz Biotechnology) and PE-Cy7conjugated rat anti-mouse CD45R/B220 (BD Biosciences). The cells were washed and then resuspended in staining buffer (Streptavidin-FITC, BD Biosciences). The cells were incubated for 20 min, washed, and resuspended in PBS containing 1% paraformaldehyde. Cell acquisition was performed in a Becton Dickinson FACScan, and a minimum of 10,000 events was acquired for each test. Data was analyzed with FlowJo software (v7.2).

Osteoblastogenesis was performed with MSCs cultured in αMEM supplemented with 10% heat-inactivated FBS with β-glycerolphosphate and ascorbic acid. Cellular alkaline phosphatase activity was determined using a para-nitrophenol phosphate based colorimetric assay with staining from a kit (Sigma-Aldrich's).

Osteoclastogenesis was determined by culturing BM cells in FBSsupplemented αMEM medium. Nonadherent cells were re-plated in αMEM medium (10% FBS with RANKL and M-CSF) in an incubator for 4 -7 d. Cells were then fixed and stained for tartrate-resistant acid phosphatase (TRAP) activity, and TRAP+ cells containing ≥3 nuclei were counted as mature osteoclasts (OC). IGF-1, ALS, or IGFBP-3 were added in separate rescue experiments.

For co-culture experiments, primary osteoblast (OB) cultures were established as above from each mouse strain. Cells were harvested after 14 d and re-seeded in αMEM. Non-adherent cells from Control mice only were co-cultured with each OB culture for 7 d. Co-cultures were then fixed, stained and TRAP+ cells counted.

BM cells were isolated and cultured as above for 18 h. Non-adherent cells were re-plated on BD BioCoat,™ Osteologic™ calcium hydroxyapatite (HA)-coated discs (BD Biosciences) in osteoclastogenic conditions for 7 d as above. On day 8, a pit assay was performed by quantifying the resorption areas on each disc after Von Kossa staining.

Differences between groups were evaluated by 2-way ANOVA with Fisher PLSD post-hoc testing. Results are expressed as means ± SE.

Results:

ALSKO and BP3KO mouse cells exhibited an increase in alkaline phosphatase activity in culture. Despite this indication of enhanced early osteoblastogenesis we found that ALSKO mice had a significant decrease in the number of OCs formed in vitro (Fig. 1) and reduced c-fms+ and B220+ BM cells in vivo (Fig. 2). Likewise, in co-culture, OBs from ALSKO mice failed to induce OC formation (Fig. 3). In contrast, BP3KO mice exhibited an increase in the number of OCs in culture (Fig. 1), and an increase in c-fms+ BM cells in vivo (Fig. 2). The pit assays for OC formation and function on HA confirmed our results for culture on plastic.

The defect in OC formation for primary ALSKO culture was rescued by reintroduction of ALS protein and not by addition of IGF-1 (Fig. 1). Our results indicate that impairment of IGF-1 complex formation in BM alters cell fate by modifying osteoclastogenesis. Importantly, the determination of BM cellular populations that express the mouse colony stimulating factor-1 (M-CSF1) receptor (i.e., c-fms), which is a critical factor for osteoclastogenesis, and B220+ cells, pre-pro-B cells, which express membrane-bound and secreted RANKL, suggest an in vivo role for both ALS and IGFBP-3.

The inability of IGF-1 to rescue osteoclastogenesis in ALSKO combined with the data indicating that the als gene need not be active for OC formation to be rescued with ALS protein suggest a direct role for ALS in determining cell fate in the BM micro-environment.

Our study further reinforces the tenet that there is a strong interaction among hematopoietic and mesenchymal stem cells in BM, and raises the possibility that the IGF-1 complex constituents IGFBP-3 and ALS play a more active role in the assignment of cell fate in BM than previously thought.

References:

[1] Meunier et al., CORR 1971;80:147. [2] Kasukawa et al., Curr Pharm Des 2004;10:2577. [3] Niu and Rosen, Gene 2005;361:38.

Acknowledgements:

Funds provided by NIH (SY, DC, and CJR) & Charles H. Revson Foundation (JCF).

Figure 1 .

 1 Figure1. TRAP staining of OC cultures derived from non-adherent BM cells of mutant mice revealed that both ALS and IGFBP-3 gene inactivation affect OC formation. However, attempts at rescue by addition of IGF-1 only affected BP3KO.

Figure 3 .

 3 Figure 3. TRAP stained co-cultures. Only in experiments including OBs derived from ALSKO was there failure to support osteoclastogenesis of non-adherent cells derived from Control mice.

Figure 2 .

 2 Figure 2. FACS revealed c-fms, a receptor critical for osteoclastogenesis, to be increased and decreased in BP3KO and ALSKO, respectively. ALSKO mice exhibited a significantly lower fraction of B220+ cells which express RANKL.