Isabeau Birindelli

Françoise Demengel

Ergodic pairs for singular or degenerate fully nonlinear operators

Keywords:

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.

Introduction

In 1989, in a fundamental paper [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF], Lasry and Lions study solutions of -∆u + |∇u| q + λu = f (x) in Ω that blow up on the boundary of Ω. Here q > 1 and Ω is a C 2 bounded domain in R N . Among other things, they prove that in the subquadratic case q ≤ 2 there exists a unique constant c Ω , called ergodic constant, and there exists a unique, up to a constant, solution of

-∆ϕ + |∇ϕ| q -c Ω = f (x) in Ω, ϕ = +∞ on ∂Ω.
The couple (ϕ, c Ω ) is referred to as an ergodic pair. It is well known that for q = 2, -c Ω is just the principal eigenvalue of (-∆ + f )(•). Since [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF], a huge and interesting literature generated, also in connection with the stochastic interpretation of the problem. Interestingly, while the concept of principal eigenvalue has been extended to fully nonlinear operators of different types (see e.g. [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF], [START_REF] Busca | Nonlinear eigenvalues and bifurcation problems for Pucci's operators[END_REF]), the notion of ergodic constant has not been much investigated in fully nonlinear settings. The scope of this paper is to give a thoroughly picture of the ergodic pairs and the related blowing up solutions and solutions with Dirichlet boundary condition for approximating equations.

We now detail the main theorems. In the whole paper Ω denotes a C 2 bounded domain of R N ; S denotes the space of symmetric matrices in R N . We consider a uniformly elliptic homogenous operator F , i.e. a continuous function F : S → R satisfying: there exist 0 < a < A such that for all M, N ∈ S, with N ≥ 0, and for all t > 0, a tr(N ) ≤ F (M + N ) -F (M ) ≤ A tr(N ), F (tM ) = tF (M ).

(1.1)

We will always consider the differential operator G(∇u, D 2 u) = -|∇u| α F (D 2 u) + |∇u| β with α > -1 and α + 1 < β ≤ α + 2. This reduces to the Lasry Lions case for α = 0 and a = A = 1.

Theorem 1.1. Suppose that f is bounded and locally Lipschitz continuous in Ω, and that F satisfies (1.1). Consider the Dirichlet problems

-|∇u| α F (D 2 u) + |∇u| β = f in Ω u = 0 on ∂Ω, (1.2)
and, for λ > 0,

-|∇u| α F (D 2 u) + |∇u| β + λ|u| α u = f in Ω u = 0 on ∂Ω. (1.3)
The following alternative holds.

1. Suppose that there exists a bounded sub solution of (1.2). Then the solution u λ of (1.3) satisfies: (u λ ) is bounded and uniformly converging up to a sequence λ n → 0 to a solution of (1.2).

2. Suppose that there is no solution for the Dirichlet problem (1.2). Then, (u λ ) satisfies, up to a sequence λ n → 0 and locally uniformly in Ω, (a) u λ → -∞;

(b) there exists a constant c Ω ≥ 0 such that λ|u λ | α u λ → -c Ω ;

(c) c Ω is an ergodic constant and v λ = u λ +|u λ | ∞ converges to a solution of the ergodic problem

-|∇v| α F (D 2 v) + |∇v| β = f + c Ω in Ω v = +∞ on ∂Ω (1.4)
whose minimum is zero.

The standard notion of viscosity solution fails when the operator is singular, i.e., in this paper, when α < 0, hence we will consider viscosity solutions as defined in [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF]. When α = 0 and F is the laplacian, Theorem 1.1 has been firstly pointed out by Porretta in [START_REF] Porretta | The ergodic limit for a viscous Hamilton-Jacobi equation with Dirichlet conditions[END_REF]. An analogous result for p-laplacian operators has been proved by Leonori and Porretta in [20]. We emphasize that it yields the existence of an ergodic constant, and a sign property, in the case when the Dirichlet problem (1.2) does not have any solution.

In order to show the existence of ergodic pairs when problem (1.2) does admit solutions, we need first to prove the existence of solutions blowing up on the boundary for λ > 0. This is achieved in our Theorem 4.1, where we basically reproduce the construction given in [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF] of explosive sub and super solutions. As a consequence, we will deduce the existence of an ergodic constant in Theorem 6.2. However, in order to establish further properties of the ergodic constant (in particular, its uniqueness), we need to prove new and refined boundary estimates for explosive solutions. These will be obtained under the additional regularity condition

F (∇d(x) ⊗ ∇d(x)) is C 2 in a neighborhood of ∂Ω , (1.5)
where d(x) denotes the distance function from ∂Ω. We observe that (1.5) is certainly satisfied if the domain Ω is of class C 3 and the operator F is C 2 , but there can be also cases with non smooth F satisfying (1.5). For instance, for all operators F (M ) which depend only on the eigenvalues of M , such as Pucci's operators, F (∇d(x) ⊗ ∇d(x)) is a constant function as long as |∇d(x)| = 1. Under condition (1.5), we will show that the ergodic constant c Ω is unique and that it shares some properties with the principal eigenvalue of the operator -|∇u| α F (D 2 u). Let us recall that, when F is the Laplace operator and β = α + 2, then the ergodic constant appearing in problem (1.4) coincides with the opposite of the principal eigenvalue of the operator -|∇u| α ∆u + f |u| α u, meaning that if v is a solution of (1.4), then w = e -v is a positive solution of

-|∇w| α ∆w + f |w| α w = -c Ω |w| α w in Ω w = 0 on ∂Ω
This suggests a deep connection between ergodic constants and principal eigenvalues. We recall at this purpose that a Faber-Krahn inequality for the ergodic constant of the laplacian has been recently proved by Ferone et al. in [START_REF] Ferone | ?????[END_REF].

Even for β = α + 2 and F fully nonlinear, we prove that the ergodic constant can be characterized by an inf-formula analogous to the one which defines the principal eigenvalues for fully nonlinear operators. Following [START_REF] Barles | On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations[END_REF], [START_REF] Porretta | The ergodic limit for a viscous Hamilton-Jacobi equation with Dirichlet conditions[END_REF], we define

µ = inf{µ : ∃ ϕ ∈ C(Ω), -|∇ϕ| α F (D 2 ϕ) + |∇ϕ| β ≤ f + µ} .
Note that µ depends on f and Ω, but if there is no ambiguity we will not precise this dependence. We next state the second main result of the present paper.

Theorem 1.2. Suppose that f is bounded and locally Lipschitz continuous in Ω, and that F satisfies (1.1) and (1.5). Let c Ω be an ergodic constant for problem (1.4); then:

1. c Ω is unique; 2. c Ω = µ ;
3. the map Ω → c Ω is nondecreasing with respect to the domain, and continuous;

4. if either α = 0 or α = 0 and sup Ω f + c Ω < 0, then µ is not achieved. Moreover, if Ω ⊂⊂ Ω, then c Ω < c Ω .
In order to prove these results many questions need to be addressed. Clearly the first one is the existence of solutions for (1.3) when λ > 0, but even though it is fundamental, this is extraneous to the spirit of this note and it can be found in [START_REF] Birindelli | Dirichlet problems for fully nonlinear equations with "subquadratic[END_REF]. The interested reader will see that it is done through a Perron's procedure i.e. constructing sub and super solutions of (1.3) together with a comparison principle and some Lipschitz estimates depending on the L ∞ norm of the solution. 3) that does not depend directly on the L ∞ norm of the solution but only on the norm of the zero order term. In the linear case, these kind of estimates were obtained by Capuzzo Dolcetta, Leoni, Porretta in [START_REF] Capuzzo Dolcetta | Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian[END_REF], and the proof we use is inspired by theirs. In order to extend the result to the present fully nonlinear singular case, we have to address several non trivial technical difficulties, see Section 2. After that, we give the proof of Theorem 1.1 in Section 3.

In Section 4 we focus on existence and estimates for explosive solutions of the approximating λequation, i.e. solutions u of

-|∇u| α F (D 2 u) + |∇u| β + λ|u| α u = f in Ω u = +∞ on ∂Ω.
Here, the function f is assumed to be continuous in Ω, but it is allowed to be unbounded on the boundary, as long as its growth is controlled. This is an important feature that will be needed in the proof of Theorem 1.2. Construction of explosive solutions in the fully nonlinear setting includes the works by Alarcón, Quaas [START_REF] Alarcón | Large viscosity solutions for some fully nonlinear equations[END_REF], by Esteban, Felmer and Quaas [START_REF] Esteban | Super-linear elliptic equation for fully nonlinear operators without growth restrictions for the data[END_REF] and by Demengel, Goubet [START_REF] Demengel | Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations[END_REF], where only suitable zero order terms are considered. Capuzzo Dolcetta, Leoni and Vitolo in [START_REF] Capuzzo Dolcetta | On the inequality F (x, D 2 u) ≥ f (u) + g(u)|Du| q[END_REF][START_REF] Capuzzo Dolcetta | Entire subsolutions of fully nonlinear degenerate elliptic equations[END_REF] construct radial explosive solutions in some degenerate cases. The construction we do in order to give existence and estimates of blowing up solutions slightly differs from the standard proof for linear operators (see Remark 4.3), and we obtain solutions satisfying non constant boundary asymptotics. Moreover, our proof can be carried on for other classes of operators, as e.g. the p-Laplacian or some generalizations such as

F (p, M ) = |p| α (q 1 trM + q 2 M p • p |p| 2 ) ,
with q 1 > 0 and q 1 + q 2 > 0. When p > 2, using the variational form of the p-Laplacian, and its linearity with respect to the Hessian, Leonori and Porretta proved in [START_REF] Leonori | Large solutions and gradient bounds for quasilinear elliptic equations[END_REF] such estimates and existence results. So our result for α < 0 covers the case p < 2 that was not considered there.

In Section 5 we give a comparison theorem for sub and super solutions of equation (1.2), in which zero order terms are lacking. The change of equation that allows to prove the comparison principle of Theorem 5.1 is sort of standard, but the computation which follows is original and ad hoc for our setting. The work of Leonori, Porretta and Riey [START_REF] Leonori | Comparison principles for p-Laplace equations with lower order terms[END_REF] has been a source of inspiration. Finally, in Section 6, after proving the existence of ergodic constants and estimating the ergodic functions near the boundary, we give the proof of Theorem 1.2.

Let us finally remark that we left open the question of uniqueness (up to constants) of the ergodic functions. This is a delicate issue, strongly related with the simplicity of the principal eigenvalue for degenerate/singular operators, which is in general an open problem. We recall that the usual proof for linear operators, see [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF][START_REF] Porretta | The ergodic limit for a viscous Hamilton-Jacobi equation with Dirichlet conditions[END_REF], relies on the strong maximum principle, which does not hold for degenerate operators. Let us also recall that for p-laplacian operators the uniqueness of the ergodic function is obtained in [START_REF] Leonori | Large solutions and gradient bounds for quasilinear elliptic equations[END_REF] for p ≥ 2 and under the condition sup Ω f + c < 0. We believe that the proof given in [START_REF] Leonori | Large solutions and gradient bounds for quasilinear elliptic equations[END_REF] can be extended to the fully nonlinear singular/degenerate setting, provided that one can prove the C 1,γ regularity of the ergodic functions. We demand this study to a future work [START_REF] Birindelli | On the W 2,p and C 1,γ regularity for fully non linear singular or degenerate equations[END_REF].

Notations

• We use d(x) to denote a C 2 positive function in Ω with coincides with the distance function from the boundary in a neighborhood of ∂Ω

• For δ > 0, we set Ω δ = {x ∈ Ω : d(x) > δ}

• We denote by M + , M -the Pucci's operators with ellipticity constants a, A, namely, for all

M ∈ S, M + (M ) = A tr(M + ) -a tr(M -) M -(M ) = a tr(M + ) -A tr(M -)
and we often use that, as a consequence of (1.1), for all M, N ∈ S one has

M -(N ) ≤ F (M + N ) -F (M ) ≤ M + (N )

A priori Lipschitz-type estimates

In the note [START_REF] Birindelli | Dirichlet problems for fully nonlinear equations with "subquadratic[END_REF], we prove the following result Theorem 2.1. Assume that f is bounded and continuous in Ω. Then, for any λ > 0, there exists a unique solution u λ ∈ C(Ω) of (1.3), which is Lipschitz continuous up to the boundary, and satisfies

|u λ | W 1,∞ (Ω) ≤ C(|u λ | ∞ , |f -λ|u λ | α u λ | ∞ , a, A, α, β).
This is obtained, through Perron's method, by constructing sub and super solutions and using the following general comparison principle.

Theorem 2.2. Suppose that b is Lipschitz continuous in Ω, ζ : R → R is a non decreasing function and f and g are continuous in Ω. Let u be a bounded by above viscosity sub solution of

-|∇u| α F (D 2 u) + b(x)|∇u| β + ζ(u) ≤ g
and let v be a bounded by below viscosity super solution of

-|∇v| α F (D 2 v) + b(x)|∇v| β + ζ(v) ≥ f. If either g ≤ f and ζ is increasing or g < f then, u ≤ v on ∂Ω implies that u ≤ v in Ω.
For the proofs of the above results we refer to [START_REF] Birindelli | Dirichlet problems for fully nonlinear equations with "subquadratic[END_REF]. The rest of this section is devoted to prove a priori Lipschitz estimates for solutions of the equation

-|∇u| α F (D 2 u) + |∇u| β + λ|u| α u = f , (2.1)
that depend on λ|u| α+1 ∞ , but not on |u| ∞ . Our estimates will be a consequence of the following result, in which we denote by B the unit ball centred at the origin in R N . Proposition 2.3. Let F satisfy (1.1) and, for λ ≥ 0, α > -1 and β > α+1, let u and v be respectively a bounded sub solution and a bounded from below super solution of equation (2.1) in B, with f Lipschitz continuous in B. Then, for any positive p ≥ (2+α-β) + β-α-1 , there exists a positive constant M , depending only on p, α, β, a, A, N, f -λ|u| α u ∞ and on the Lipschitz constant of f , such that, for all x, y ∈ B one has

u(x) -v(y) ≤ sup B (u -v) + + M |x -y| (1 -|y|) β+α - β-α-1 1 + |x -y| (1 -|x|) p
Proof. We argue as in the proof of Theorem 3.1 in [START_REF] Capuzzo Dolcetta | Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian[END_REF].

Let us define a "distance" function d which equals 1 -|x| near the boundary and it is extended in B as a C 2 function satisfying, for some constant c 1 > 0,

     d(x) = 1 -|x| if |x| > 1 2 1-|x| 2 ≤ d(x) ≤ 1 -|x| for all x ∈ B |Dd(x)| ≤ 1 , -c 1 I N ≤ D 2 d(x) ≤ 0 for all x ∈ B For ξ = |x-y| d(x) , we consider the function φ(x, y) = k d(y) τ |x -y| (L + ξ p ) + sup B (u -v) + where p > 0 is a fixed exponent satisfying p ≥ (2+α-β) + β-α-1 , τ = β+α - β-α-1
and L, k are suitably large positive constants to be chosen later. The statement is proved if we show that for all (x, y) ∈ B 2 one has u(x) -v(y) ≤ φ(x, y) .

By contradiction, let us assume that u(x) -v(y) -φ(x, y) > 0 somewhere. Then, necessarily the supremum is achieved on a pair (x, y) with x = y and d(x) , d(y) > 0. Using Ishii's Lemma of [START_REF] Ishii | Viscosity solutions of fully nonlinear equations[END_REF], one gets that on such a point (x, y), for all > 0, there exist symmetric matrices X and Y such that

(∇ x φ, X ) ∈ J 2,+ u(x), (-∇ y φ, -Y ) ∈ J 2,-v(y) - 1 + |D 2 φ| I 2N ≤ X O O Y ≤ D 2 φ + (D 2 φ) 2 .
(2.2)

We proceed in the proof by considering separately the cases α ≥ 0 and α < 0.

The case α ≥ 0. Since u is a sub solution and v a super solution, by the positive 1-homogeneity of F we have in this case

-F (|∇ x φ| α X ) + |∇ x φ| β + λ|u| α u(x) ≤ f (x) -F (-|∇ y φ| α Y ) + |∇ y φ| β + λ|v| α v(y) ≥ f (y)
Subtracting the above inequalities and using also that u(x) -v(y) > φ(x, y) ≥ 0, for any t > 0 we can write

t F (|∇ x φ| α X ) -[F ((1 + t)|∇ x φ| α X ) -F (-|∇ y φ| α Y )] ≤ |∇ y φ| β -|∇ x φ| β + f (x) -f (y) ,
and therefore

t |∇ x φ| β ≤ F ((1 + t)|∇ x φ| α X ) -F (-|∇ y φ| α Y ) +|∇ y φ| β -|∇ x φ| β + t (f -λ|u| α u) + + f (x) -f (y) .
By the uniform ellipticity of F , it then follows that

t |∇ x φ| β ≤ M + ((1 + t)|∇ x φ| α X + |∇ y φ| α Y ) +|∇ y φ| β -|∇ x φ| β + t (f -λ|u| α u) + + f (x) -f (y) .
By multiplying the right inequality of (2.2) on the left and on the right by

√ 1 + t|∇ x φ| α/2 I N O O |∇ y φ| α/2 I N
and testing the resulting inequality on vectors of the form (v, v) with v ∈ R N , we further obtain

(1 + t)|∇ x φ| α X + |∇ y φ| α Y ≤ Z α,t + O( ) , with Z α,t = (1 + t)|∇ x φ| α D 2 xx φ + √ 1 + t|∇ x φ| α/2 |∇ y φ| α/2 D 2 xy φ + D 2 xy φ t + |∇ y φ| α D 2 yy φ . (2.3)
Hence, after letting → 0, we get

t |∇ x φ| β ≤ M + (Z α,t ) + |∇ y φ| β -|∇ x φ| β + t (f -λ|u| α u) + + f (x) -f (y) . (2.4)
We now proceed by evaluating the right hand side terms of (2.4).

An explicit computation shows that, setting η = |x-y| d(y) and ζ = x-y |x-y| , one has

∇ x φ(x, y) = k d(y) τ (L + (1 + p)ξ p )ζ -p ξ p+1 ∇d(x) , as well as ∇ y φ(x, y) = - k d(y) τ [(L + (1 + p)ξ p )ζ + τ η (L + ξ p ) ∇d(y)] .
From now on we denote with c possibly different positive constants which depend only on p, N , a, A, α and β. As discussed in [START_REF] Capuzzo Dolcetta | Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian[END_REF], for L > 1 fixed suitably large depending only on p, one has

|∇ x φ| ≥ ck 1 + ξ p+1 d(y) τ (2.5)
and

|∇ x φ| , |∇ y φ| ≤ ck 1 + ξ p+1 d(y) τ +1 . (2.6)
Moreover, we notice that one has also

|∇ y φ| ≥ k d(y) τ [L + (1 + p)ξ p -τ η(L + ξ p )|∇d(y)|] ≥ ck 1 + ξ p d(y) τ if τ η ≤ 1 2 . (2.7)
On the other hand, the second order derivatives of φ may be written as follows

D 2 xx φ = k d(y) τ [L+(1+p)ξ p ] |x-y| B + p(1 + p) ξ p-1 d(x) T -p(1 + p) ξ p d(x) (C + C t ) +p(1 + p) ξ p+1 d(x) ∇d(x) ⊗ ∇d(x) -p ξ p+1 D 2 d(x) D 2 xy φ = -k d(y) τ [L+(1+p)ξ p ] |x-y| B + p(1 + p) ξ p-1 d(x) T -p(1 + p) ξ p d(x) C t + τ [L+(1+p)ξ p ] d(y) E -τ p ξ p+1 d(y) ∇d(x) ⊗ ∇d(y) D 2 yy φ = k d(y) τ [L+(1+p)ξ p ] |x-y| B + p(1 + p) ξ p-1 d(x) T + τ [L+(1+p)ξ p ] d(y) (E + E t ) + τ (τ +1) η(L+ξ p ) d(y) ∇d(y) ⊗ ∇d(y) -τ η (L + ξ p )D 2 d(y) with B = I N -ζ ⊗ ζ , T = ζ ⊗ ζ , C = ζ ⊗ ∇d(x) and E = ζ ⊗ ∇d(y) .
Therefore, the matrix Z α,t defined in (2.3) is given by

Z α,t = k d(y) τ √ 1 + t|∇ x φ| α/2 -|∇ y φ| α/2 2 (L+(1+p)ξ p ) |x-y| B + p(1 + p) ξ p-1 d(x) T - √ 1 + t|∇ x φ| α/2 √ 1 + t|∇ x φ| α/2 -|∇ y φ| α/2 p(1+p)ξ p d(x) (C + C t ) -|∇ y φ| α/2 √ 1 + t|∇ x φ| α/2 -|∇ y φ| α/2 τ (L+(1+p)ξ p ) d(y) (E + E t ) +p(1 + t)|∇ x φ| α (1+p)ξ p+1 d(x) ∇d(x) ⊗ ∇d(x) -ξ p+1 D 2 d(x) + √ 1 + t|∇ x φ| α/2 |∇ y φ| α/2 τ pξ (p+1)
M + (Z α,t ) ≤ c k d(y) τ √ 1 + t|∇ x φ| α/2 -|∇ y φ| α/2 2 1+ξ p |x-y| + √ 1 + t|∇ x φ| α/2 √ 1 + t|∇ x φ| α/2 -|∇ y φ| α/2 ξ p d(x) +|∇ y φ| α/2 √ 1 + t|∇ x φ| α/2 -|∇ y φ| α/2 1+ξ p d(y) +(1 + t)|∇ x φ| α ξ p+1 d(x) + √ 1 + t|∇ x φ| α/2 |∇ y φ| α/2 ξ p+1 d(y) + |∇ y φ| α η 1+ξ p d(y)
.

By observing that

√ 1 + t|∇ x φ| α/2 -|∇ y φ| α/2 ≤ ( √ t + 1 -1)|∇ x φ| α/2 + |∇ x φ| α/2 -|∇ y φ| α/2
and by applying the trivial inequalities

√ 1 + t -1 ≤ t, √ 1 + t √ t + 1 -1 ≤ t, √ 1 + t ≤ 1 + t, after
rearranging terms we then deduce

M + (Z α,t ) ≤ c k d(y) τ t 2 |∇ x φ| α 1+ξ p |x-y| +t |∇ x φ| α ξ p +ξ p+1 d(x) + |∇ x φ| α/2 |∇ y φ| α/2 1+ξ p +ξ p+1 d(y) + ξ p d(x) + |∇ x φ| α/2 -|∇ y φ| α/2 2 1+ξ p |x-y| + |∇ x φ| α/2 -|∇ y φ| α/2 |∇ x φ| α/2 ξ p d(x) + |∇ y φ| α/2 1+ξ p d(y) +|∇ x φ| α ξ p+1 d(x) + |∇ y φ| α (1+ξ p ) η d(y) + |∇ x φ| α/2 |∇ y φ| α/2 ξ p+1 d(y) (2.8)
We now recall that, as proved in [START_REF] Capuzzo Dolcetta | Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian[END_REF], for all q, γ > 0 one has

ξ q d(x) γ ≤ 2 γ 1 + ξ q+γ d(y) γ . (2.9)
Moreover, if α ≥ 2, the mean value theorem, the bounds (2.6), (2.9) and the explicit expression of

∇ x φ + ∇ y φ imply that |∇ x φ| α/2 -|∇ y φ| α/2 ≤ c max |∇ x φ| α/2-1 , |∇ y φ| α/2-1 |∇ x φ + ∇ y φ| ≤ ck α/2 (1 + ξ p+1 ) α/2-1 d(y) α/2(τ +1)-1 ξ p d(x) + 1 + ξ p d(y) |x -y| ≤ c k (1 + ξ p+1 ) d(y) τ +1 α/2 |x -y|
Analogously, if α < 2 but τ η ≤ 1/2, from (2.5), (2.7) and again (2.9) we deduce

|∇ x φ| α/2 -|∇ y φ| α/2 ≤ c k (1 + ξ p ) d(y) τ α/2-1 k (1 + ξ p+1 ) d(y) τ +1 |x -y|
and therefore, since ξ ≤ 1 2τ -1 for η ≤ 1 2τ , we obtain in this case

|∇ x φ| α/2 -|∇ y φ| α/2 ≤ c k (1 + ξ p+1 ) d(y) τ +2/α α/2 |x -y| .
Finally, if α < 2 and τ η > 1/2, that is |x -y| > d(y)/2τ , we have

|∇ x φ| α/2 -|∇ y φ| α/2 ≤ |∇ x φ + ∇ y φ| α/2 ≤ c k(1 + ξ p+1 )|x -y| d(y) τ +1 α/2 ≤ c k(1 + ξ p+1 ) d(y) τ +1 α/2 2τ d(y) 1-α/2 |x -y| = c k (1 + ξ p+1 ) d(y) τ +2/α α/2 |x -y| .
Thus, in all cases we obtain

|∇ x φ| α/2 -|∇ y φ| α/2 ≤ c k (1 + ξ p+1 ) d(y) τ +max{1,2/α} α/2 |x -y| ≤ c |∇ x φ| α/2 d(y) max{α/2,1} |x -y| .
(2.10) By using inequalities (2.6), (2.5), (2.9) and (2.10), from estimate (2.8) it then follows

M + (Z α,t ) ≤ ck|∇ x φ| α d(y) τ t 2 1 + ξ p |x -y| + t 1 + ξ p+2 d(y) α/2+1 + |x -y| 1 + ξ p+2 d(y) α+2 . (2.11) Moreover, since p ≥ 2+α-β β-α-1 and τ ≥ α+2 2(β-α-1)
, by using again (2.5), we further deduce

M + (Z α,t ) ≤ ck|∇ x φ| α t 2 1 + ξ p d(y) τ |x -y| + t |∇ x φ| β-α k β-α + |x -y|(1 + ξ p+2 ) d(y) τ +α+2 .
Using the above inequality jointly with (2.4) yields

t |∇ x φ| β-α ≤ ck t 2 1 + ξ p d(y) τ |x -y| + t |∇ x φ| β-α k β-α + |x -y|(1 + ξ p+2 ) d(y) τ +α+2 +|∇ x φ| -α |∇ y φ| β -|∇ x φ| β + t |∇ x φ| -α (f -λ|u| α u) + +|∇ x φ| -α (f (x) -f (y)) ,
and therefore, being β > α + 1, for k sufficiently large one has

t 2 |∇ x φ| β-α -t 2 ck(1+ξ p ) d(y) τ |x-y| ≤ ck|x -y|(1 + ξ p+2 ) d(y) τ +α+2 + |∇ x φ| -α |∇ y φ| β -|∇ x φ| β +t |∇ x φ| -α (f -λ|u| α u) + + |∇ x φ| -α (f (x) -f (y)) .
We now choose t > 0 in order to maximize the left hand side, namely

t = |∇ x φ| β-α d(y) τ |x -y| 4ck(1 + ξ p ) .
We then obtain

|∇ x φ| 2(β-α) ≤ c k 2 (1 + ξ 2(p+1) ) d(y) 2τ +α+2 + k|∇ x φ| -α (1 + ξ p ) |∇ y φ| β -|∇ x φ| β |x -y|d(y) τ +|∇ x φ| β-2α (f -λ|u| α u) + + k|∇ x φ| -α (1 + ξ p ) (f (x) -f (y)) |x -y|d(y) τ .
Moreover, arguing as for (2.10) in the case α ≥ 2, we also have

|∇ y φ| β -|∇ x φ| β ≤ ck β |x -y| (1 + ξ p+1 ) β d(y) (τ +1)β , so that |∇ x φ| 2(β-α) ≤ C k 2 (1 + ξ (p+1) ) 2 d(y) 2τ +α+2 + |∇ x φ| -α k β+1 (1 + ξ p+1 ) β+1 d(y) τ (β+1)+β +|∇ x φ| β-2α + |∇ x φ| -α k(1 + ξ p ) d(y) τ ,
for some constant C > 0 depending now also on (f -λ|u| α u) + ∞ and on the Lipschitz constant of f . By inequality (2.5) it then follows

|∇ x φ| 2(β-α) ≤ C |∇ x φ| 2 d(y) α+2 + |∇ x φ| β-α+1 d(y) β + |∇ x φ| β-2α + |∇ x φ| 1-α ≤ C |∇ x φ| 2+ α+2 τ k α+2 τ + |∇ x φ| β-α+1+ β τ k β τ + |∇ x φ| β-2α , Recalling that α > 0, β > α + 1 and τ = β β-α-1 , we see that 2(β -α) = β -α + 1 + β τ > 2 + α+2 τ .
Hence, from the last inequality and from Young's inequality, we obtain that

|∇ x φ| ≤ C
which gives a contradiction to (2.5) for k large enough.

The case α < 0. As proved in [START_REF] Birindelli | C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations[END_REF], if α < 0 a sub solution u and super solution v of equation (2.1) satisfy respectively in the viscosity sense

   -F (D 2 u) + |∇u| β-α + λ|∇u| -α |u| α u ≤ |∇u| -α f -F (D 2 v) + |∇v| β-α + λ|∇v| -α |v| α v ≥ |∇v| -α f From (2.2) in this case it then follows that    -F (X ) + |∇ x φ| β-α + λ|∇ x φ| -α |u| α u(x) ≤ |∇ x φ| -α f (x) -F (-Y ) + |∇ y φ| β-α + λ|∇ y φ| -α |v| α v(y) ≥ |∇ y φ| -α f (y)
and, arguing as in the previous case, we now obtain for any t > 0

t |∇ x φ| β-α ≤ M + (Z 0,t ) + |∇ y φ| β-α -|∇ x φ| β-α + t |∇ x φ| -α (f -λ|u| α u) -(f -λ|u| α u) (|∇ y φ| -α -|∇ x φ| -α ) + |∇ y φ| -α (f (x) -f (y)) ,
where Z 0,t is defined by (2.3) (with α = 0). By applying inequalities (2.11) (with α = 0), (2.5), (2.6), (2.10), in the present case, taking into account that β -α > 1 and 0 < -α < 1, we deduce that

|∇ x φ| 2(β-α) ≤ C |∇ x φ| 2 d(y) 2 + |∇ x φ| β-α+1 d(y) β-α + |∇ x φ| β-2α ,
for some constant C > 0 depending on p, α, β, a, A, N, f -λ|u| α u ∞ and on the Lipschitz constant of f . Since now τ = β-α β-α-1 , we reach a contradiction for k sufficiently large as before.

As in [START_REF] Capuzzo Dolcetta | Hölder's estimates for degenerate elliptic equations with coercive Hamiltonian[END_REF], the above Proposition and a scaling argument for solutions of equation (2.1) give the following result.

Theorem 2.4. Let F satisfy (1.1) and, for λ ≥ 0, α > -1 and β > α + 1, let u be a continuous solution of equation (2.1) in Ω ⊂ R N , with f Lipschitz continuous in Ω. Then, u is locally Lipschitz continuous in Ω and there exists a positive constant M , depending only on α, β, a, A, N, f -λ|u| α u ∞ and on the Lipschitz constant of f , such that at any differentiability point x ∈ Ω one has

|∇u(x)| ≤ M dist ∂Ω (x) 1 β-α-1 .
3 Proof of Theorem 1.1.

By using the Lipschitz estimates obtained in the previous section, we can now prove Theorem 1.1.

Proof of Theorem 1.1. Let u λ be a solution of (1.3). We begin by giving a bound that will be useful in the whole proof. Observe that u + λ is a sub solution of

-|∇u + λ | α F (D 2 u + λ ) ≤ |f | ∞ ;
from known estimates, see [START_REF] Birindelli | Demengel First eigenvalue and Maximum principle for fully nonlinear singular operators[END_REF], this implies that

|u + λ | ∞ ≤ c 1 |f | 1 1+α ∞ . (3.1)
Let us consider first the case when there exists a sub solution ϕ of (1.2). Then, ϕ -|ϕ| ∞ is a sub solution of equation (1.3), and by the comparison principle we deduce u λ ≥ ϕ -|ϕ| ∞ . Thus, in this case (u λ ) is uniformly bounded in Ω. The Lipschitz estimates in Theorem 2.1 then yield that u λ is uniformly converging up to a sequence to a Lipschitz solution of problem (1.2).

We now treat the second case, i.e. we suppose that (1.2) has no solutions. In particular |u λ | ∞ diverges, since otherwise we could extract from (u λ ) a subsequence converging to a solution of (1.2).

On the other hand, since

-|f |∞ λ 1 1+α
is a sub solution of (1.3), by the comparison principle we obtain

u - λ ≤ |f |∞ λ 1 1+α
, which, jointly with (3.1), yields

λ|u λ | 1+α ∞ ≤ c 1 |f | ∞ . Hence, there exists (x λ ) ⊂ Ω such that u λ (x λ ) = -|u λ | ∞ → -∞
and there exists a constant c Ω ≥ 0 such that, up to a subsequence, λ|u λ | 1+α ∞ → c Ω . We will show, as in [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF] (see also [START_REF] Leonori | Large solutions and gradient bounds for quasilinear elliptic equations[END_REF] and [START_REF] Porretta | The ergodic limit for a viscous Hamilton-Jacobi equation with Dirichlet conditions[END_REF]), that

v λ = u λ + |u λ | ∞ = u λ -u λ (x λ ) converges up to a subsequence to a function v such that the pair (c Ω , v) solves (1.4). Clearly, v λ satisfies in Ω -|∇v λ | α F (D 2 v λ ) + |∇v λ | β + λ(v λ ) 1+α = f + λ(v α+1 λ -|u λ | α u λ ) ≥ f . Next, we set γ = 2 + α -β β -1 -α ,
and for s, δ o > 0 to be chosen sufficiently small, let us consider the function

φ(x) = σ (d(x) + s) γ - σ (δ o + s) γ if γ > 0 , φ(x) = -σ log(d(x) + s) + σ log(δ o + s) if γ = 0 , (3.2)
where σ = (γ + 1) a

2 1 β-α-1 γ -1 if γ > 0, σ = a 2 if γ = 0.
A direct computation shows that, for d(x) ≤ δ 0 with δ o small enough, in the case γ > 0 one has

-|∇φ| α M -(D 2 φ) + |∇φ| β + λφ 1+α ≤ - a(σγ) α+1 2(d + s) (γ+1)β + A(σγ) α+1 |D 2 d| ∞ (d + s) (γ+1)(α+1) + λσ α+1 (d + s) γ(α+1) ,
and, in the case γ = 0,

-|∇φ| α M -(D 2 φ) + |∇φ| β + λφ 1+α ≤ - σ α+2 (d + s) α+2 + Aσ α+1 |D 2 d| ∞ (d + s) (α+1) + λ (-σ log(d + s)) α+1 .
In both cases, by the ellipticity of F and for δ o and s sufficiently small, we obtain

-|∇φ| α F (D 2 φ) + |∇φ| β + λφ 1+α ≤ -|f | ∞ ≤ f (x) in Ω \ Ω δ0 .
Moreover, one has φ = 0 ≤ v λ on ∂Ω δo and φ ≤ |u λ | ∞ = v λ on ∂Ω for λ sufficiently small in dependence of s. The comparison principle then yields

v λ ≥ φ > 0 in Ω \ Ω δ0 . (3.3) Since v λ (x λ ) = 0, from (3.
3) we deduce that (x λ ) ⊂ Ω δo . The interior Lipschitz estimate of Theorem 2.4 then yields that v λ = u λ -u λ (x λ ) is locally uniformly bounded and locally uniformly Lipschitz continuous. This proves both statement 2a of the theorem and that (v λ ) is locally uniformly converging up to a subsequence to a Lipschitz continuous function v o ≥ 0 in Ω. Moreover, since also (x λ ) converges up to a subsequence to some point x o ∈ Ω δ0 , we obtain v o (x o ) = 0. We observe further that, locally uniformly in Ω, one has

lim λ→0 λ|u λ | α u λ = lim λ→0 λ |u λ | α+1 ∞ |v λ -|u λ | ∞ | α (v λ -|u λ | ∞ ) |u λ | α+1 ∞ = -c Ω .
This yields statement 2b and, letting λ → 0 in the equation satisfied by v λ , also that v o is a viscosity solution of

-|∇v o | α F (D 2 v o ) + |∇v o | β = f + c Ω .
Finally, letting λ → 0 in inequality (3.3), we obtain v o ≥ φ in Ω \ Ω δ0 , which in turn implies, by letting s → 0 and x → ∂Ω, that v o (x) → +∞ as d(x) → 0. This completely proves statement 2c and concludes the proof of the theorem.

Explosive solutions.

In this section we prove the existence of solutions of equation (2.1) blowing up at the boundary, which will be used in the proof of existence of ergodic pairs. In what follows we drop the assumption on the boundedness of the right hand side f , and we consider continuous functions in Ω, possibly unbounded as d(x) → 0. Let us introduce the nonnegative exponent

γ = 2 + α -β β -1 -α , (4.1)
which plays a crucial role in the next results.

Let us start with a first existence result which follows by the same arguments used in [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF] for the linear case, but requires some additional technical care in the construction of explosive sub and supersolutions, due to the possible singularity of the involved operator.

Theorem 4.1. Let β ∈ (α + 1, α + 2], λ > 0 and let F satisfy (1.1). Let further f ∈ C(Ω) be bounded from below and such that lim

d(x)→0 f (x)d(x) β β-1-α = 0 . (4.2)
Then, the infinite boundary value problem

-|∇u| α F (D 2 u) + |∇u| β + λ|u| α u = f in Ω , u = +∞ on ∂Ω , (4.3)
admits solutions, and any its solution u satisfies, for all x ∈ Ω,

c 0 d(x) γ - D 1 λ 1 α+1 ≤ u(x) ≤ C 0 d(x) γ + D 1 λ 1 α+1 if γ > 0 , c 0 | log d(x)| - D 1 λ 1 α+1 ≤ u(x) ≤ C 0 | log d(x)| + D 1 λ 1 α+1 if γ = 0 , (4.4)
for positive constants c 0 , C 0 and D 1 depending only on α, β, a, A, |d| C 2 (Ω) and on f . Proof. We give the proof in the case γ > 0, the reader can easily see the changes to be made when γ = 0. We will get the conclusion by showing that equation (2.1) has a super solution w and a sub solution w s , for any s > 0 sufficiently small, satisfying, for

D = D 1 /λ 1 α+1 , w(x) ≤ C 0 d(x) -γ + D , w s (x) ≥ c 0 (d(x) + s) -γ -D ,
for all x ∈ Ω, with equalities holding in a neighborhood of ∂Ω. Assume for a while that this is proved. Then, for any R > 0, we can consider the solution u R of

-|∇u R | α F (D 2 u R ) + |∇u R | β + λ|u R | α u R = f R in Ω u R = R on ∂Ω ,
with f R = min{f, R}. By Theorem 2.2, u R is monotone increasing with respect to R and satisfies w s ≤ u R ≤ w, provided that R > max ∂Ω w s (x). Moreover, u R is uniformly locally Lipschitz continuous by Theorem 2.1. Thus, u R is locally uniformly convergent as R → +∞ to a solution u of (4.3) such that w 0 ≤ u ≤ w. By definition, u is the so called minimal explosive solution. The maximal explosive solution u is then obtained as the limit for δ → 0 of the minimal explosive solutions in Ω δ . Thus, it follows that for any solution u of problem (4.3) one has

c 0 d(x) -γ -D ≤ u ≤ u ≤ u ≤ C 0 d(x) -γ + D .
Let us now proceed to the construction of w and w s .

Let δ > 0 be so small that in the set Ω \ Ω 2δ = {d(x) < 2δ} the function d satisfies |∇d| = 1. For x ∈ Ω \ Ω 2δ , let us consider the function

ϕ(x) = C 0 d(x) -γ , with C 0 = γ -1 (2A(γ + 1)) 1 β-α-1
. By a standard computation and assumption (4.6) on f , it is easy to see that

-|∇ϕ| α M + (D 2 ϕ) + |∇ϕ| β ≥ f
for δ small enough. Hence, for D > 0 and ϕ 1 (x) = ϕ(x) + D we obtain

-|∇ϕ 1 | α F (D 2 ϕ 1 ) + |∇ϕ 1 | β + λϕ α+1 1 ≥ f in Ω \ Ω 2δ .
Next, for x ∈ {δ ≤ d(x) ≤ 2δ}, we consider the function

ϕ 2 (x) = C 0 δ γ e 1 d(x)-2δ + 1 δ + D , which satisfies |∇ϕ 2 | α |F (D 2 ϕ 2 )| + |∇ϕ 2 | β ≤ K 1 in Ω δ \ Ω 2δ ,
for a positive constant K 1 depending only on N, a, A, α and β. Thus, if D is chosen satisfying

D ≥ |f | L ∞ (Ω δ ) +K1 λ 1 α+1
, we obtain

-|∇ϕ 2 | α F (D 2 ϕ 2 ) + |∇ϕ 2 | β + λϕ α+1 2 ≥ f (x) in Ω δ \ Ω 2δ .
We then conclude that the function

w(x) =        C 0 d(x) -γ + +D for d < δ δ -γ C 0 e 1 d(x)-2δ + 1 δ + D for δ ≤ d ≤ 2δ D for d > 2δ
is the required super solution in Ω. Indeed, in the set Ω δ , w is of class C 2 and it is a super solution by the properties of ϕ 2 and by the fact that locally constant functions satisfy |∇u| α F (D 2 u) = 0. On the other hand, w is a super solution in Ω \ Ω 2δ by the properties of ϕ 1 and ϕ 2 and the fact that ϕ 2 (x) < ϕ 1 (x) for d(x) > δ.

As far as the sub solution is concerned, for s > 0,

c 0 = γ -1 (γ+1)a 2 1 β-α-1 and x ∈ Ω \ Ω 2δ , let us consider the function ϕ s (x) = c 0 (d(x) + s) -γ .

Symmetric computations as above give that

-|∇ϕ s | α F (D 2 ϕ s ) + |∇ϕ s | β + λ|ϕ s | α ϕ s ≤ -|∇ϕ s | α M -(D 2 ϕ s ) + |∇ϕ s | β + λ|ϕ s | α ϕ s ≤ f (x) ,
for δ and s sufficiently small, since f is bounded from below.

Moreover, for

D ≥ c 0 δ -γ + |f -|∞ λ 1 α+1 , the constant function c 0 (δ + s) -γ -D is also a sub solution in Ω.
Therefore, the function

w s (x) = ϕ s (x) -D in Ω \ Ω δ c 0 (δ + s) -γ -D in Ω δ
is the wanted sub solution.

In order to obtain refined boundary estimates for the explosive solutions, and to show uniqueness of solution in some cases, we need to assume the extra regularity condition (1.5) involving both the operator F and the domain Ω.

Under assumption (1.5), we denote by C(x) a non negative function of class C 2 in Ω satisfying in a neighborhood of ∂Ω C(x) = ((γ + 1)F (∇d(x) ⊗ ∇d(x)))

1 β-α-1 γ -1 if γ > 0, C(x) = F (∇d(x) ⊗ ∇d(x)) if γ = 0.
lim d(x)→0 f (x)d(x) β β-1-α -γ0 = 0 , (4.6)
for some γ 0 ≥ 0. Then, any solution u of (4.3) satisfies: for any ν > 0 and for any 0 ≤ γ 1 ≤ γ 0 , with γ 1 < 1, and γ 1 < γ when γ > 0, there exists D = D1 λ 1/(α+1) , with D 1 > 0 depending on ν, γ 1 , α, β, a, A, |d| C 2 (Ω) , |C| C 2 (Ω) and on f , such that, for all x ∈ Ω,

C(x) d(x) γ - ν d(x) γ-γ1 -D ≤ u(x) ≤ C(x) d(x) γ + ν d(x) γ-γ1 + D if γ > 0 , | log d(x)| (C(x) -νd(x) γ1 ) -D ≤ u(x) ≤ | log d(x)| (C(x) + νd(x) γ1 ) + D if γ = 0 . (4.7)
Furthermore, the solution u is unique

• for α ≥ 0 and any β,

• for α < 0 and any β > 1-α-α 2 1-α , provided that f satisfies (4.6) with γ 0 > -α γ. Proof. As in the previous proof, we detail only the case γ > 0. 1. Refined boundary estimates. By arguing as in the proof of Theorem 4.1, we get the conclusion by showing that, for every ν > 0 and for any 0 ≤ γ 1 ≤ γ 0 , with γ 1 < min{1, γ}, there exist D = D1 λ 1/α+1 > 0, a super solution w and a sub solution w s , for s > 0 sufficiently small, satisfying

w s (x) ≥ C(x)(d + s) -γ -ν(d + s) -γ+γ1 -D , w(x) ≤ C(x)d -γ + νd -γ+γ1 + D ,
for any x ∈ Ω, with equalities holding in a neighborhood of ∂Ω.

Let δ > 0 be so small that in the set Ω \ Ω 2δ = {d(x) < 2δ} the function d satisfies |∇d| = 1 and C satisfies (4.5). For x ∈ Ω \ Ω 2δ , let us consider the function

ϕ(x) = C(x)d(x) -γ + νd(x) γ1-γ . One has ∇ϕ(x) = -γ C(x)d -γ-1 1 + ν (γ -γ 1 ) γ C(x) d γ1 ∇d + d -γ ∇C and D 2 ϕ(x) = γ (γ + 1)C(x)d -γ-2 1 + ν (γ-γ1)(γ-γ1+1) γ (γ+1)C(x) d γ1 ∇d ⊗ ∇d -γ C(x)d -γ-1 1 + ν (γ-γ1) γ C(x) d γ1 D 2 d -γ d -γ-1 (∇d ⊗ ∇C + ∇C ⊗ ∇d) + d -γ D 2 C .
By ellipticity of F and by definition of C(x) it then follows

F (D 2 ϕ) ≤ γ (γ+1)C(x) d γ+2 1 + ν (γ-γ1)(γ-γ1+1) γ (γ+1)C(x) d γ1 F (∇d ⊗ ∇d) + K1 d γ+1 = (γ C(x)) β-α d γ+2 1 + ν (γ-γ1)(γ-γ1+1) γ (γ+1)C(x) d γ1 + K1 d γ+1 , for a constant K 1 > 0 depending on ν, α, β, γ 1 , a, A, |D 2 d| ∞ and |C| C 2 (Ω) .
In what follows we denote by K i , i = 1, 2 . . ., different constants depending on these quantities. Hence, we obtain

-|∇ϕ| α F (D 2 ϕ) + |∇ϕ| β ≥ |∇ϕ| α -(γ C(x)) β-α d γ+2 1 + ν (γ-γ1)(γ-γ1+1) γ (γ+1)C(x) d γ1 -K1 d γ+1 + |∇ϕ| β-α ≥ |∇ϕ| α -(γ C(x)) β-α d γ+2 1 + ν (γ-γ1)(γ-γ1+1) γ (γ+1)C(x) d γ1 -K1 d γ+1 + (γ C(x)) β-α d (γ+1)(β-α) 1 + ν (γ-γ1) γ C(x) d γ1 β-α - K2 d (γ+1)(β-α)-1 ≥ |∇ϕ| α -(γ C(x)) β-α d γ+2 1 + ν (γ-γ1)(γ-γ1+1) γ (γ+1)C(x) d γ1 -K1 d γ+1 + (γ C(x)) β-α d (γ+1)(β-α) 1 + ν (β-α)(γ-γ1) γ C(x) d γ1 - K2 d (γ+1)(β-α)-1
Recalling that γ = α+2-β β-α-1 , we finally deduce

-|∇ϕ| α F (D 2 ϕ) + |∇ϕ| β ≥ |∇ϕ| α ν (γ-γ1)(1+γ1) (γ+1) (γ C(x)) β-α-1 d γ+2-γ 1 -K3 d γ+1 = |(γ C(x)+ν(γ-γ1)d γ 1 )∇d-d ∇C| α d β β-α-1 -γ 1 ν (γ-γ1)(1+γ1) (γ+1) (γ C(x)) β-α-1 -K 3 d 1-γ1
Since γ 1 ≤ γ 0 , by assumption (4.6) on f the last inequality implies that, for δ sufficiently small,

-|∇ϕ| α F (D 2 ϕ) + |∇ϕ| β ≥ f (x) in Ω \ Ω 2δ ,
and therefore also that

-|∇ϕ| α F (D 2 ϕ) + |∇ϕ| β + λϕ α+1 ≥ f (x) in Ω \ Ω 2δ .
Clearly, the same inequality holds also for ϕ 1 (x) = ϕ(x) + D, for any D > 0. The function ϕ 1 can then be extended to the whole of Ω as in the proof of Theorem 4.1, and this construction yields the supersolution w.

As far as the sub solution is concerned, for s > 0, ν > 0 and x ∈ Ω \ Ω 2δ , let us consider the function

ϕ s (x) = C(x)(d(x) + s) -γ -ν(d(x) + s) -γ+γ1 .
Analogous computations as above give that

-|∇ϕ s | α F (D 2 ϕ s ) + |∇ϕ s | β + λ|ϕ s | α ϕ s ≤ |∇ϕ s | α -(γ C(x)) β-α-1 (d+s) γ+2-γ 1 ν (γ-γ1)(1+γ1) γ+1 + K5 (d+s) γ+1 + λ A α+1 (d+s) γ (α+1) ≤ f (x) ,
for δ and s sufficiently small, since f is bounded from below. The function ϕ s (x) -D, for suitable large D > 0, can then be constantly extended in Ω in order to give the wanted subsolution w s .

Uniqueness.

We prove that u = u.

Remark that, by estimates (4.7), for any θ < 1 and for any c ∈ R, there exists δ such that θu(x) -c ≤ u(x) for d(x) ≤ δ.

The case α ≥ 0. Observe that, for all t ∈ R and c > 0, one has

|t -c| α (t -c) -|t| α t ≤ -2 -α c α+1 .
From this, we deduce that

-|∇(θu -c)| α F D 2 (θu -c) + |∇(θu -c)| β + λ|θu -c| α (θu -c) ≤ θ α+1 f (x) -λ2 -α c α+1 ,
and the choice

c = c θ = 2 α (1 -θ α+1 )|f -| ∞ λ 1 α+1 then yields -|∇(θu -c)| α F D 2 (θu -c) + |∇(θu -c)| β + λ|θu -c| α (θu -c) ≤ θ α+1 f (x) -(1 -θ α+1 ) f - ∞ ≤ f (x)
. By applying Theorem 2.2, it then follows that θu -c θ ≤ u in Ω, and letting θ → 1 we obtain the uniqueness of the explosive solution in the case α ≥ 0.

The case α < 0. In this case we use the inequality

|t -c| α (t -c) -|t| α t ≤ -2 α (α + 1)K α c ,
which holds true for all 0 < c < K and t ∈ R such that |t| ≤ K.

Let C 1 = sup Ω |u(x)|d(x) γ , which is finite by (4.7). Then, for any δ > 0, one has |u| < C1 δ γ in Ω δ . Therefore, for any 0 < θ < 1 and 0 < c < C1 δ γ , and for x ∈ Ω δ , we have

-|∇(θu -c)| α F D 2 (θu -c) + |∇(θu -c)| β + λ|θu -c| α (θu -c) ≤ θ α+1 f (x) -λ(2C 1 ) α (α + 1)δ -αγ c .
We choose, as before,

c = c θ,δ = |f -| ∞ (1 -θ α+1 ) λ(2C 1 ) α (α + 1)δ -αγ ,
which is admissible for δ sufficiently small, since α > -1. This yields

-|∇(θu -c θ,δ )| α F D 2 (θu -c θ,δ ) + |∇(θu -c θ,δ )| β + λ|θu -c θ,δ | α (θu -c θ,δ ) ≤ f in Ω δ .
On the other hand, by estimates (4.7) with ν = 1, we have

θu -c θ,δ ≤ u on ∂Ω δ provided that δ = δ θ = a(1 -θ) 2(1 + D) 1 γ 1
.

With this choice of δ, we then deduce from Theorem 2.2 that θu -c θ,δ θ ≤ u in Ω δ θ . Finally, we let θ → 1. We observe that, by the restrictions assumed on β and f in the case α < 0, we can choose γ 1 satisfying γ 1 > -αγ. Therefore, c θ,δ θ → 0 as θ → 1, and we conclude that u ≤ u in Ω.

Remark 4.3. Let us put in evidence that estimates (4.7) imply that any solution u of (4.3) satisfies lim

d(x)→0 u(x) d(x) γ C(x) = 1 if γ > 0, lim d(x)→0 u(x) | log d(x)| C(x) = 1 if γ = 0 .
Moreover, if f satisfies (4.6) with γ 0 = 0, then necessarily γ 1 = 0 and (4.7) reduce to

(C(x) -ν)d(x) -γ -D ≤ u(x) ≤ (C(x) + ν)d(x) -γ + D if γ > 0 , (C(x) -ν) | log d(x)| -D ≤ u(x) ≤ (C(x) + ν) | log d(x)| + D if γ = 0 ,
for any ν > 0, with D > 0 depending in particular on ν and λ. The above estimates are the classical ones for explosive solutions firstly obtained in the semilinear case in [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF], where C(x) is a constant function. Also the case γ 0 = 1 has been considered in [START_REF] Lasry | Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with state Constraints[END_REF], and in this case more refined estimates have been obtained. In the nonlinear case, analogous estimates for γ 0 ≥ 1 would require further regularity assumptions on the non constant function C(x). Estimates (4.7) are interesting in the intermediate cases 0 ≤ γ 0 < 1, in which they are new also for linear operators and yield a uniqueness result in the non linear singular case α < 0.

A comparison principle for non linear degenerate/singular equations without zero order terms

This section is devoted to some comparison principle for fully non linear equations without zero order terms. For analogous results concerning non singular operators, see [START_REF] Barles | Existence and comparison results for fully non linear degenerate elliptic equations without zeroth order terms[END_REF], [START_REF] Barles | Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions[END_REF].

Theorem 5.1. Let b be a continuous and bounded function in Ω and, when α = 0, let f be a bounded continuous function such that f ≤ -m < 0. Let u and v be respectively sub and super solution of

-|∇u| α F (D 2 u) + b(x)|∇u| β = f in Ω. (5.1)
If u and v are bounded and at least one of the two is Lipschitz continuous then the comparison principle holds i.e.

u ≤ v on ∂Ω ⇒ u ≤ v in Ω.
Proof. Without loss of generality, we will suppose that u is Lipschitz continuous.

The case α = 0 is quite standard, it is enough to construct strict sub solutions that converge uniformly to u. We choose > 0 such that a > 2Bβ(|∇u| ∞ + 1) β-1 , where B = |b| ∞ . Let u = u + e -x 1 -, with e.g. Ω ⊂ {x 1 > 0}. Then, u is a strict sub solution of (5.1), being

F (D 2 u ) ≥ F (D 2 u) + a e -x 1 ≥ b(x)|∇u | β -f + b(x) |∇u| β -|∇u | β + a e -x 1 ≥ b(x)|∇u | β -f + a 2 e -x 1 .
Furthermore u ≤ u ≤ v on ∂Ω. By Theorem 2.2, we obtain that u ≤ v in Ω. To conclude, let → 0. This computation has been done for a classical solution u, but, with obvious changes, it can be made rigorous for viscosity solutions.

For the case α = 0 and f < 0, we use the change of function u = ϕ(z), v = ϕ(w) with

ϕ(s) = -γ(α + 1) log δ + e -s α+1
.

This function is used in [START_REF] Barles | Existence and comparison results for fully non linear degenerate elliptic equations without zeroth order terms[END_REF], [START_REF] Barles | Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions[END_REF], [START_REF] Barles | Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equation Ann[END_REF], [START_REF] Leonori | Large solutions and gradient bounds for quasilinear elliptic equations[END_REF], [START_REF] Leonori | Comparison principles for p-Laplace equations with lower order terms[END_REF]. We choose δ small enough in order that the range of ϕ covers the ranges of u and v. The constant γ will be chosen small enough depending only on a, α, β, inf Ω (-f ) and |b| ∞ ; in this proof, any constant of this type will be called universal . Observe that ϕ > 0 while ϕ < 0.

In the viscosity sense, z and w are respectively sub and super solution of

-|∇z| α F (D 2 z + ϕ (z) ϕ (z) ∇z ⊗ ∇z) + b(x)ϕ (z) β-α-1 |∇z| β + -f (ϕ (z)) α+1 = 0. (5.2)
We define

H(x, s, p) = -aϕ (s) ϕ (s) |p| 2+α + b(x)ϕ (s) β-α-1 |p| β + -f (x) ϕ (s) α+1 .
The point is to prove that at x, a maximum point of z -w, ∂H(x,s,p) ∂s > 0 for all p. This will be sufficient to get a contradiction. A simple computation gives

ϕ = γe -s α+1 δ + e -s α+1 , ϕ = -γδe -s α+1 (α + 1)(δ + e -s α+1 ) 2 . Hence -ϕ ϕ = δ (α + 1) 2 e -s α+1 (δ + e -s α+1 ) 2 i.e. -ϕ ϕ = - ϕ (α + 1)γ > 0.
Differentiating H with respect to s gives:

∂ s H = a|p| α+2 -ϕ (α + 1)γ + (-f ) -(α + 1)ϕ (ϕ ) α+2 + b(x)|p| β (β -α -1)(ϕ ) β-α-2 ϕ .
Since -ϕ is positive, we need to prove that

K := a|p| α+2 (α + 1)γ + (-f ) α + 1 (ϕ ) α+2 -|b| ∞ |p| β (β -α -1)(ϕ ) β-α-2 > 0.
We start by treating the case β < α + 2.

Observe first that the boundedness of u and v, implies that there exists universal positive constants c o and c 1 such that

c o γ ≤ ϕ ≤ c 1 γ.
Hence, it is easy to see that there exist three positive universal constants C i , i = 1, 2, 3 such that

K > C 1 |p| α+2 γ + C 2 γ α+2 - C 3 |p| β γ α+2-β . We choose γ = min 1, ( C3 C2 ) β , ( C3 C1 ) 1 α+1-β . With this choice of γ, for |p| ≤ 1, C 1 |p| α+2 γ + C 2 γ α+2 - C 3 |p| β γ α+2-β ≥ C 2 γ α+2 - C 3 γ α+2-β > 0; while for |p| ≥ 1, C 1 |p| α+2 γ + C 2 γ α+2 - C 3 |p| β γ α+2-β ≥ (C 1 )|p| α+2 γ - C 3 |p| β γ α+2-β > 0.
If β = α + 2, just take γ < a (α+1)|b|∞ . This gives that for γ small enough depending only on min(-f ) , α, |b| ∞ and β one has, for some universal constant C, ∂ s H(x, s, p) ≥ C > 0.

(5.3)

We now conclude the proof of the comparison principle. We will distinguish the case α > 0 and α < 0. In the first case we introduce ψ j (x, y) = z(x) -w(y)j 2 |x -y| 2 while in the second case we use ψ j (x, y) = z(x) -w(y) -j q |x -y| q where q > α+2 α+1 . We detail the case α > 0. Suppose by contradiction that u > v somewhere in Ω, then z > w somewhere, since ϕ is increasing, while on the boundary z ≤ w. Then the supremum of z -w is positive and it is achieved inside Ω. Hence ψ j reaches a positive maximum in (x j , y j ) ∈ Ω × Ω. By Ishii's lemma [START_REF] Ishii | Viscosity solutions of fully nonlinear equations[END_REF], there exists (X j , Y j ) ∈ S × S such that (p j , X j ) ∈ J 2,+ z(x j ), (p j , -Y j ) ∈ J 2,-w(y j ), with p j = j(x j -y j ) and X j 0 0 Y j ≤ j I -I -I I .

On (x j , y j ), by a continuity argument, for j large enough one has z(x j ) > w(y j ) + sup(z -w) 2 .

Note for later purposes that since z or w are Lipschitz, p j = j(x j -y j ) is bounded. Observe that, the monotonicity of ϕ ϕ implies that

N = p j ⊗ p j ϕ (z(x j )) ϕ (z(x j )) - ϕ (w(y j )) ϕ (w(y j )) ≤ 0.
Using the fact that z and w are respectively sub and super solutions of the equation (5.2), the estimate (5.3) and that H is decreasing in the second variable, one obtains:

0 ≥ -f (x j ) (ϕ ) α+1 (z(x j )) -|p j | α F (X j + ϕ (z(x j )) ϕ (z(x j )) p j ⊗ p j ) + b(x j )|p j | β ϕ (z(x j )) β-α-1 ≥ -f (x j ) (ϕ ) α+1 (z(x j )) -|p j | α F (-Y j + ϕ (w(y j )) ϕ (w(y j )) p j ⊗ p j ) + +a|p j | 2+α ϕ (w(y j )) ϕ (w(y j )) - ϕ (z(x j )) ϕ (z(x j )) + |p j | β b(x j )(ϕ (z(x j )) β-α-1 ≥ f (y j ) -f (x j ) (ϕ (w(y j )) α+1 + (b(x j ) -b(y j ))|p j | β (ϕ (w(y j ))) β-α-1
+H(x j , z(x j ), p j ) -H(x j , w(y j ), p j ) ≥ C(z(x j ) -w(y j )) + o(1) γ α+1 .

Here we have used the continuity of f and b, the boundedness of p j and that ψ(x j , y j ) ≥ sup(ψ(x j , x j ), ψ(y j , y j )).

Passing to the limit one gets a contradiction, since (x j , y j ) converges to (x, x) such that z(x) > w(z). This ends the case α ≥ 0.

In the case α < 0, the proof is similar but we need to make sure that one can choose x j = y j . This can be done proceeding as in [START_REF] Birindelli | Dirichlet problems for fully nonlinear equations with "subquadratic[END_REF].

Ergodic pairs

In this section we consider, for c ∈ R, the equation

-|∇u| α F (D 2 u) + |∇u| β = f + c in Ω. ( 6
.1) Definition 6.1. Suppose that c is some constant (depending on f , Ω, β, α, and F ) such that there exists ϕ ∈ C(Ω), solution of (6.1), such that ϕ → +∞ at ∂Ω. We will say that c is an ergodic constant, ϕ is an ergodic function and (c, ϕ) is an ergodic pair.

We suppose, as usual, that α > -1, β ∈ (α + 1, α + 2] and recall that γ = 2+α-β β-α-1 and C(x) satisfies (4.5). In the following subsections, we prove the existence and show several properties of ergodic pairs.

Existence of ergodic constants and boundary behavior of ergodic functions

Theorem 1.1 provides the existence of a nonnegative ergodic constant under the assumption that problem (1.2) does not have a solution. In the next result we obtain the existence of ergodic constants using approximating explosive solutions.

Theorem 6.2. Let F and f be as in Theorem 4.1, and assume further that f is locally Lipschitz continuous in Ω. Then, there exists an ergodic constant c ∈ R.

Proof. By Theorem 4.1, for λ > 0 there exists a solution U λ of problem (4.3), which satisfies estimates (4.4). It then follows that λ|U λ | α U λ is locally bounded in Ω, uniformly with respect to 0 < λ < 1.

Let us fix an arbitrary point x 0 ∈ Ω. Then, there exists c ∈ R such that, up to a sequence λ n → 0,

λ|U λ (x 0 )| α U λ (x 0 ) → -c .
On the other hand, Theorem 2.4 yields that U λ is locally uniformly Lipschitz continuous. Therefore, for x in a compact subset of Ω, one has

λ ||U λ (x)| α U λ (x) -|U λ (x 0 )| α U λ (x 0 )| ≤ λ|U λ (x) -U λ (x 0 )| α+1 → 0 if α ≤ 0 ,
as well as, using again estimates (4.4),

λ ||U λ (x)| α U λ (x) -|U λ (x 0 )| α U λ (x 0 )| ≤ λ K λ α α+1 |U λ (x) -U λ (x 0 )| → 0 if α > 0 .
It then follows that c does not depend on the choice of x 0 and, up to a sequence and locally uniformly in Ω, one has λ|U λ | α U λ → -c .

Moreover, the function V λ (x) = U λ (x) -U λ (x 0 ) is locally uniformly bounded, locally uniformly Lipschitz continuous and satisfies

-|∇V λ | α F (D 2 V λ ) + |∇V λ | β = f -λ|U λ | α U λ in Ω .
If V denotes the local uniform limit of V λ for a sequence λ n → 0, then one has

-|∇V | α F (D 2 V ) + |∇V | β = f + c in Ω .
Finally, arguing as in the proof of Theorem 1.1 and using Theorem 2.2, we have that, for some δ 0 > 0 sufficiently small, V λ ≥ φ + min

d(x)=δ0
V λ in Ω \ Ω δ0 , with φ defined in (3.2) for arbitrary s > 0. Letting λ, s → 0 we deduce that V (x) → +∞ as d(x) → 0. This shows that (c, V ) is an ergodic pair and concludes the proof.

We now prove that, under assumption (1.5), ergodic functions satisfy on the boundary the same asymptotic identities as the explosive solutions of (4.3).

  d(y) [∇d(x) ⊗ ∇d(y) + ∇d(y) ⊗ ∇d(x)] +τ |∇ y φ| α (1+τ )η(L+ξ p ) d(y) ∇d(y) ⊗ ∇d(y) -η(L + ξ p )D 2 d(y) , and, recalling that ξ = |x -y|/d(x) and that d < 1 in B, this yields the estimate

(4. 5 ) 4 . 2 .

 542 Theorem Let β ∈ (α + 1, α + 2], λ > 0 and let F satisfy (1.1) and (1.5). Let further f ∈ C(Ω) be bounded from below and such that

Theorem 6 . 3 . 1 where c 1 >

 6311 Let F and f be as in Theorem 4.2. Then, any ergodic function u satisfieslim d(x)→0 u(x) d(x) γ C(x) = 1 if γ > 0, lim d(x)→0 u(x) | log d(x)| C(x) = 1 if γ = 0 . (6.2)Proof. As usual, we consider only the case γ > 0.The computations made in the proof of Theorem 4.2 (for γ 1 = 0) show that, for all ν > 0 and for δ 0 > 0 sufficiently small, the function w ν,δ (x) := C(x)+ν (d(x)-δ) γ satisfies for δ < d(x) < δ 0 -|∇w ν,δ | α F (D 2 w ν,δ ) + |∇w ν,δ | β ≥ c 1 ν(d(x) -δ) -β β-α-0 is a constant depending on α, β, a, A, |D 2 d| ∞ and |C| C 2 (Ω). By assumption (4.6) on f , this implies that-|∇w ν,δ | α F (D 2 w ν,δ ) + |∇w ν,δ | β > f (x) + c = -|∇u| α F (D 2 u) + |∇u| β in Ω δ \ Ω δ0for δ 0 = δ 0 (ν) small enough. Hence, we are in the hypothesis of Theorem 2.2 and we deduce thatu ≤ M ν + w ν,δ in Ω δ \ Ω δ0 , with M ν = sup d(x)=δ0 u(x). Letting δ → 0 we obtain that u ≤ M ν + (C(x) + ν)d(x) -γ in Ω \ Ω δ0 .This in turn implies that limd(x)→0 u(x) α+1 d(x) β β-α-1 -γ0 = 0for all γ 0 such thatγ 0 < β β -α -1 -(α + 1)γ = α + 2.Since α + 2 > 1, we obtain in particular that the function |u| α u satisfies condition (4.6) with γ 0 = 1. Note also that |u| α u is bounded from below in Ω since it is continuous and blows up on the boundary. Finally, we observe that u satisfies-|∇u| α F (D 2 u) + |∇u| β + |u| α u = f + c + |u| α u ,where the right hand side f + c + |u| α u satisfies condition (4.6) with an exponent γ 0 = min{γ 0 (f ), 1}, γ 0 (f ) being the exponent appearing in the condition (4.6) satisfied by f . Hence, by applying Theorem 4.2, we obtain that u satisfies the boundary estimates (4.7) with λ = 1 and the constant D depending also on u itself. Estimates (4.7) in turn imply relations (6.2).

  Theorems 1.1 and 1.2 are obtained by means of several intermediate results, most of which are of independent interest. A first fundamental tool is an interior Lipschitz estimate for solutions of equation (1.

Acknowledgment. Part of this work has been done while the first and third authors were visiting the UMR 80-88, University of Cergy Pontoise, and the second one was visiting Sapienza University of Rome supported by GNAMPA-INDAM.

6.2 Uniqueness and further properties of the ergodic constant: proof of Theorem 1.2.

Throughout this section we assume that f is bounded and locally Lipschitz continuous.

In the introduction we have defined µ ∈ R ∪ {-∞} as

It is easy to see that µ ≤ -inf Ω f . A better upper bound on µ depending on the domain Ω is given by the following result.

Proposition 6.4.

Hence, by its definition, µ ≤ -

We are now ready to give the proof of Theorem 1.2.

Proof of Theorem 1.2. Here we set c Ω = c. Note that the existence of c is given by Theorem 6.2. Proof of 1. Suppose that c and c are two ergodic constants, and let ϕ and ϕ be respectively corresponding ergodic functions. By Theorem 6.3 the ratio of ϕ and ϕ goes to 1 as d(x) → 0; hence, for any θ < 1, the supremum of θϕ -ϕ is achieved in the interior of Ω since θϕ -ϕ blows down to -∞ as d(x) → 0. We observe that

From standard comparison arguments in viscosity solutions theory, see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], it follows that at a maximum point x of θϕ -ϕ one has

Exchanging the roles of c and c we conclude that c = c . Proof of 2. Let µ < c and suppose by contradiction that there exists ϕ ∈ C(Ω) satisfying

Let u be an ergodic function corresponding to c. Clearly, sup Ω (ϕ -u) is attained in Ω. Again by standard viscosity arguments, we obtain that at a maximum point x of ϕ -u one has

which is a contradiction. Hence, we deduce

which implies that µ is finite and µ ≥ c.

On the other hand, by definition of µ , for any µ < µ the problem

does not have solution. Theorem 1.1 then implies that there exists an ergodic constant c f +µ ≥ 0 for the right hand side f + µ. On the other hand, by the uniqueness proved in 1. above, one has c = µ + c f +µ . Hence, c ≥ µ and, therefore, c ≥ µ .

Proof of 3. The nondecreasing monotonicity of c with respect to the domain Ω is an immediate consequence of point 2. above and the definition of µ .

Let us now prove the "continuity" of Ω → c Ω , in the following weak sense. For δ > 0 small, let c δ denote the ergodic constant in Ω δ . Then, c δ is nondecreasing as δ decreases to zero, and c δ ≤ c = c Ω . Let u δ be an ergodic function in Ω δ , x 0 ∈ Ω be a fixed point and let us set v δ = u δ -u δ (x 0 ). By Theorem 2.4, v δ is locally uniformly bounded and locally uniformly Lipschitz continuous in Ω δ . Thus, up to a sequence δ n → 0, v δ converges locally uniformly in Ω to a solution v of the equation with right hand side f + lim δ→0 c δ . Moreover, arguing as in the proof of Theorem 6.2, we have that

, for some constant C 0 > 0 and for δ < d(x) ≤ δ 0 . Letting δ → 0, we get that v(x) → +∞ as d(x) → 0. Hence, v is an ergodic function in Ω and, by point 1., lim δ→0 c δ is the ergodic constant c.

Proof of 4. We prove that the constant µ is not achieved. Suppose by contradiction that there exists

On the other hand, let u be an ergodic function in Ω.

We observe that for all constants M , ϕ + M is still a bounded sub solution, whereas u is a solution satisfying u = +∞ on ∂Ω. Theorem 5.1 applied in a smaller domain Ω δ then yields u ≥ ϕ + M for arbitrarily large M , which clearly is a contradiction. A similar argument proves the strict increasing behavior of the ergodic constant. Let Ω ⊂⊂ Ω and suppose by contradiction that c Ω = c Ω . Let u Ω and u Ω be ergodic functions respectively in Ω and Ω.

For every constant M , both u Ω + M and u Ω satisfy (6.1) in Ω , with u Ω + M bounded and u Ω = +∞ on ∂Ω . Hence, Theorem 5.1 yields the contradiction u Ω ≥ u Ω + M for every M . Remark 6.5. We remark that, thanks to Proposition 6.4, the condition sup Ω f + c < 0 appearing in Theorem 1.2-4. is satisfied in one of the following cases:

-f is constant in Ω;

-the oscillation sup Ω f -inf Ω f of f is suitably small, in dependence of the length of the projections of Ω on the coordinated axes;

-in at least one direction Ω is suitably narrow, in dependence of the oscillation of f in Ω.

As a direct consequence of Theorems 1.1 and 1.2, we can finally establish a connection between the existence of solutions of the Dirichlet problem (1.2) and the sign of the ergodic constant c = c Ω,f . Corollary 6.6. Let F, f be as in Theorem 1.2 and let c denote the ergodic constant in Ω for f . Then:

(i) if c < 0, then problem (1.2) does admit solutions;

(ii) if c > 0, then problem (1.2) does not admit any solution.

Proof. From Theorem 1.1 and from the uniqueness of the ergodic constant c proved in Theorem 1.2-1., we deduce that if there is no solution of problem (1.2) then c ≥ 0, that is statement (i).

On the other hand, if c > 0, then, by Theorem 1.2-2., it follows that there does not exist any function ϕ ∈ C(Ω) satifying -|∇ϕ| α F (D 2 ϕ) + |∇ϕ| β ≤ f in Ω .

In particular, there cannot exist any solution of problem (1.2), that is statement (ii).


Isabeau Birindelli

Françoise Demengel

C 1,γ regularity for singular or degenerate fully nonlinear operators and applications

C 1,γ regularity for singular or degenerate fully nonlinear operators and applications

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni

Introduction

The goal of the present paper is to establish C 1,γ regularity results and to obtain a priori estimates for viscosity solutions of a class of fully nonlinear elliptic equations which may be singular or degenerate at the points where the gradient of the solution vanishes.

Regularity properties of viscosity solutions of fully nonlinear elliptic equations have been studied since a long time, starting with the seminal paper of Caffarelli [START_REF] Caffarelli | Interior a priori estimates for solutions of fully nonlinear equations[END_REF] in 1989, which contains in particular C 1,γ estimates for F (x, D 2 u) = f when f ∈ L p , p > n. His results were extended to L p viscosity solutions of operators F (x, D 2 u, Du) at most linear in the gradient by Swiech in [START_REF] Swiech | p -interior estimates for solutions of fully nonlinear, uniformly elliptic equations[END_REF]. Later, Winter [START_REF] Winter | Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations[END_REF] proved C 1,γ (Ω) estimates in the presence of a regular boundary datum. In the recent preprint [START_REF] Nornberg | C 1,α regularity for fully nonlinear elliptic equations with superlinear growth in the gradient[END_REF], Saller Nornberg proves C 1,γ and W 2,p results for L p viscosity solutions, when F (x, D 2 u, Du) is fully nonlinear, uniformly elliptic and at most quadratic in the gradient.

In [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], the first two authors of the present paper consider singular or degenerate equations of the form

|∇u| α F (D 2 u) = f (x, ∇u) ,
where F is fully nonlinear uniformly elliptic, α > -1, and f has growth at most of order 1 + α in the gradient. Lipschitz regularity results are proved in [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], and C 1,γ regularity up to the boundary in the case α ≤ 0 in [START_REF] Birindelli | Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators[END_REF], for the Dirichlet problem with homogeneous boundary conditions. Later, C 1,γ interior regularity was obtained in the case α > 0 by Imbert and Silvestre [START_REF] Imbert | C 1,α regularity of solutions of degenerate fully non-linear elliptic equations[END_REF], when f does not depend on ∇u. These results have been extended to the case where f depends on the gradient with growth at most α + 1, and to boundary C 1,γ results in the presence of sufficiently regular boundary datum, in [START_REF] Birindelli | C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations[END_REF][START_REF] Birindelli | Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient[END_REF][START_REF] Birindelli | Hölder regularity of the gradient for solutions of fully nonlinear equations with sub linear first order term, Geometric Methods in PDE?s[END_REF].

In this note we prove C 1,γ interior and boundary regularity results when the equation possesses some Hamiltonian "superlinear" but at most "quadratic" in the gradient. More precisely, we consider equations of the form

-|∇u| α F (D 2 u) + b(x)|∇u| β = f (x) , (1.1)
where the coefficient functions b and f are continuous, and the exponents α and β always satisfy α > -1 and α + 1 < β ≤ α + 2. On the second order operator F , we assume it is a continuous function F : S N → R defined on the set S N of N × N symmetric matrices, positively homogeneous of degree one, and satisfying further the uniform ellipticity condition

a tr(N ) ≤ F (M + N ) -F (M ) ≤ A tr(N ) , (1.2)
for any M, N ∈ S N , with N ≥ 0, for given positive constants A ≥ a > 0.

The considered equations include, as a very special case, the semilinear equation

-∆u + b(x)|Du| β = f (x)
with 1 < β ≤ 2, and it is for this reason that the growth of the first order terms of equations (1.1) is referred to as "superlinear" and "subquadratic". The definition of viscosity solution we adopt is the one firstly introduced in [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], which is equivalent to the usual one in the case α ≥ 0, and, in any cases, allows not to test points where the gradient of the test function is zero, except in the locally constant case.

In the paper [START_REF] Birindelli | Dirichlet problems for fully nonlinear equations with "subquadratic" Hamiltonians[END_REF], we proved local and global Lipschitz regularity results for viscosity solutions of (1.1). In particular, we showed that if u satisfies in the viscosity sense equation (1.1) in a domain Ω ⊆ R N , then, for any pair of bounded subdomains ω ⊂⊂ ω ⊂⊂ Ω, there exists a positive constant M depending on a, A, N, α, β, u

L ∞ (ω ) , f L ∞ (ω ) , b W 1, ∞ (
ω ) and on ω, ω such that |∇u(x)| ≤ M a.e. in ω .

Our main result in the present paper reads as follows.

Theorem 1.1. Suppose that Ω is an open subset in R N , that f ∈ C(Ω), and b ∈ W 1,∞ loc (Ω). Let u ∈ C(Ω) be a viscosity solution of (1.1) in Ω ⊆ R N . Then, there exists 0 < γ ≤ 1 1+α + depending on the data, such that u belongs to C 1,γ loc (Ω) and, moreover, for any pair of subsets ω ⊂⊂ ω ⊂⊂ Ω one has

|∇u| C 0,γ (ω) ≤ C 1 |u| L ∞ (ω ) , |b| W 1,∞ (ω ) , |f | L ∞ (ω ) . (1.3)
Note that some more explicit bound, depending on the Lipschitz norm of u, is given by

|∇u| C 0,γ (ω) ≤ C |u| W 1,∞ (ω ) + |b| 1 1+α ∞ |u| β 1+α W 1,∞ (ω ) + |f | 1 1+α L ∞ (ω )
. Theorem 1.1 complements the case β ≤ α + 1, treated in [START_REF] Birindelli | C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations[END_REF]. We remark that the arguments used in [START_REF] Birindelli | C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations[END_REF] are different and fail in the case β > α + 1. Furthermore, Theorem 1.1 includes the results of [START_REF] Nornberg | C 1,α regularity for fully nonlinear elliptic equations with superlinear growth in the gradient[END_REF] for α = 0. We observe that in [START_REF] Nornberg | C 1,α regularity for fully nonlinear elliptic equations with superlinear growth in the gradient[END_REF] the author uses essentially the ABP estimate of [START_REF] Koike | Maximum principle for fully nonlinear equations via the iterated comparison function method[END_REF] for fully nonlinear elliptic equations with quadratic growth in the gradient, and Caffarelli's iterative method. This method cannot be employed when α = 0. The main ingredients in the proof of Theorem 1.1 are the Lipschitz continuity of solutions, a fixed point argument, the existence and uniqueness result for Dirichlet problems proved in [START_REF] Birindelli | Dirichlet problems for fully nonlinear equations with "subquadratic" Hamiltonians[END_REF], and the C 1,γ estimates of [START_REF] Caffarelli | Interior a priori estimates for solutions of fully nonlinear equations[END_REF] for α = 0, of [START_REF] Imbert | C 1,α regularity of solutions of degenerate fully non-linear elliptic equations[END_REF][START_REF] Birindelli | Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient[END_REF] for α > 0, and the one proved in Proposition 2.1 when α < 0 . As an application of the C 1,γ loc regularity of viscosity solutions of equation (1.1), we prove the uniqueness of the ergodic function associated with the considered operators. Let us recall that, as recently proved in [START_REF] Birindelli | Ergodic pairs for singular or degenerate fully nonlinear operators[END_REF]

, if Ω ⊂ R N is an open, bounded, C 2 domain, if d(x)
denotes the distance function from ∂Ω and if the operator F satisfies the extra regularity assumption

F (∇d(x) ⊗ ∇d(x)) is C 2 in a neighborhood of ∂Ω , (1.4)
then, given a locally Lipschitz continuous datum f , there exists a unique constant c Ω , called the ergodic constant or additive eigenvalue of F , such that the infinite boundary condition problem

-|∇u| α F (D 2 u) + |∇u| β = f + c Ω in Ω u = +∞ on ∂Ω (1.5)
has a solution u ∈ C(Ω), called ergodic function. In Section 3 we will prove the following result.

Theorem 1.2. Let Ω ⊂ R N be a bounded domain of class C 2 and let F satisfy (1.2) and (1.4). Assume further that α > -1, α + 1 < β < α + 2 and that f ∈ C(Ω) is bounded. Then, up to additive constants, problem (1.5) has at most one solution, provided that, when α = 0, sup Ω f < -c Ω and ∂Ω is connected.

2 Proof of Theorem 1.1.

In all this section we set |v| W 1,∞ := |v| ∞ + |∇v| ∞ . Furthermore, we always assume that F satisfies (1.2) and that α > -1.

We begin by proving a C 1,γ interior estimate for the case β = 0, which completes the results in [START_REF] Birindelli | Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators[END_REF]. The proof is very similar to the one employed in [START_REF] Birindelli | Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators[END_REF], but we reproduce it here for the sake of completeness. We denote by C lip some constant such that, by [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], for any u ∈ W 1,∞ (B(0, 1)) and for any g continuous and bounded in B(0, 1), any solution w of

-|∇w| α F (D 2 w) = g in B(0, 1) w = u on ∂B(0, 1) (2.1)
satisfies |w| W 1,∞ (B(0,1)) ≤ C lip (|u| W 1,∞ (B(0,1)) + |g| 1 1+α
L ∞ (B(0,1)) ).

(2.2)

We have the following Proposition 2.1. Under the above assumptions, there exists 0 < γ ≤ 1 1+α + , depending only on N, a, A and α, such that any solution w of (2.1) belongs to C 1,γ loc (B(0, 1)) and, for any r < 1, there exists C r > 0 such that

|w| C 1,γ (B(0,r)) ≤ C r (|u| W 1,∞ (B(0,1)) + |g| 1 1+α L ∞ (B(0,1)) ) (2.3)
Proof. This result is well known in the case α = 0, see [START_REF] Caffarelli | Interior a priori estimates for solutions of fully nonlinear equations[END_REF], while the case α > 0 is proved in [START_REF] Imbert | C 1,α regularity of solutions of degenerate fully non-linear elliptic equations[END_REF]. It remains to consider the case α < 0. Take = 1 4C lip and 0 < δ ≤ 1. Let

K R = {v ∈ C 1 (B(0, 1)) ∩ W 1,∞ (B(0, 1)) : |v| W 1,∞ ≤ R}
where R has to be chosen large enough. Precisely, we fixed such

R satisfying R ≥ |w| W 1,∞ and R ≥ 2 C lip |f | ∞ (1 + R -α ) + R 1+α 2 C lip + |u| W 1,∞ ,
which is possible since both -α and 1 + α are lesser than 1.

We define the map T : v ∈ K R → w δ , where w δ satisfies

-F (D 2 w δ ) = (f -|v| α v + |w| α w)(δ 2 + |∇v| 2 ) -α 2 in B(0, 1) w δ = u on ∂B(0, 1).
The map T is well defined since the right hand side is continuous and bounded. Furthemore, w δ belongs to C 1 (B(0, 1)) and, by (2.2) in the case α = 0, one has

|w δ | W 1,∞ ≤ C lip (f -|v| α v + |w| α w)(δ 2 + |∇v| 2 ) -α 2 ∞ + |u| W 1,∞ ≤ C lip (|f | ∞ + 2 R 1+α 4C lip )(δ -α + R -α ) + |u| W 1,∞ ≤ C lip |f | ∞ (1 + R -α ) + R 2C lip + R 1+α 2C lip + |u| W 1,∞ ≤ R
by the choice of R. Hence, K R is a closed convex set invariant for T . Moreover, T is a compact operator, see [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], so that, by Schauder's theorem, T possesses a fixed point denoted by w δ . We further observe that, by the convexity inequalities

δ -α |w δ | 1+α ∞ ≤ (1 + α) |w δ | ∞ + (-α) δ and |f + |w| α w| ∞ |w δ | -α W 1,∞ ≤ (-α) |w δ | W 1,∞ + (1 + α) α 1+α (|f | ∞ + |w| 1+α ∞ ) 1 1+α , it follows that |w δ | W 1,∞ ≤ 2 C lip α 1+α (|f | ∞ + |w| 1+α ∞ ) 1 1+α + (-α) δ +δ -α |f | ∞ + |w| α+1 ∞ + |u| W 1,∞ .
Moreover, by (2.3) for α = 0, we also deduce that

|w δ | C 1,γ (B(0,r)) ≤ C r |f | ∞ + |w| 1+α ∞ 4 C lip + |w δ | 1+α ∞ 4 C lip (1 + |w δ | -α W 1,∞ ) + |u| W 1,∞ (2.4)
Note that w δ satisfies

-(δ 2 + |∇w δ | 2 ) α 2 F (D 2 w δ ) + |w δ | α w δ = f + |w| α w in B(0, 1) w δ = u on ∂B(0, 1)
Using estimate (2.4), which is uniform with respect to δ ≤ 1, we obtain that as δ → 0, up to a subsequence, w δ converges uniformly and locally in C 1 to a solution w of

-|∇w| α F (D 2 w) + |w| α w = f + |w| α w in B(0, 1) w = u on ∂B(0, 1)
By uniqueness of solutions of this boundary value problem, see [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], one gets that w = w and then, by (2.4), w is C 1,γ (B(0, r)). In order to get the precise estimate (2.3), we observe that, being of class C 1 , w is a solution of

-F (D 2 ϕ) = |∇w| -α f in B(0, 1) ϕ = u on ∂B(0, 1)
and therefore, by ( 2.3) in the case α = 0,

|w| C 1,γ (B(0,r)) ≤ C r (|∇w| -α ∞ |f | ∞ + |u| W 1,∞ ) ≤ C r (|f | 1 1+α ∞ + |w| W 1,∞ + |u| W 1,∞ ) ≤ C r (1 + C lip )(|f | 1 1+α ∞ + |u| W 1,∞ ) .
Thus, for α < 0 we get estimate (2.3) with C r replaced by C r (1 + C lip ).

Next, we recall the Lipschitz estimate proved in [START_REF] Birindelli | Dirichlet problems for fully nonlinear equations with "subquadratic" Hamiltonians[END_REF].

Theorem 2.2. Suppose that F is uniformly elliptic, f is continuous in B(0, 1), b is locally Lipschitz continuous in B(0, 1), α > -1 and β ∈ (0, α + 2]. Let u be a locally bounded viscosity solution of

-|∇u| α F (D 2 u) + b(x)|∇u| β = f (x) in B(0, 1)
Then u is locally Lipschitz continuous in B(0, 1), that is, for any r < r < 1, there exists some constant c depending on the ellipticity constants of F , on r, r and on universal constants, such that

|u| W 1,∞ (B(0,r)) ≤ c(|u| L ∞ (B(0,r )) , |f | L ∞ (B(0,r )) , |b| W 1,∞ (B(0,r )) ) Remark 2.3.
The assumption that b is Lipschitz continuous is needed only in the case β = α + 2. For the case β < α + 2, b bounded is sufficient.

We can now proceed with the Proof of Theorem 1.1. We will use both a truncation method and a fixed point argument. Estimate (2.3) will then enable us to get the conclusion.

With no loss of generality, we assume that u ∈ W 1,∞ (B(0, 1)) is a Lipschitz continuous solution of (1.1) in B(0, 1), and we set M = |u| W 1,∞ (B(0,1)) . In the following, T M denotes the truncation operator at level M , that is

T M (s) = inf{|s|, M } s |s| . Let α = 2 -1-(-α) + 1+α C lip
. For g bounded and continuous, let w be the unique solution, see [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], of the Dirichlet problem

-|∇w| α F (D 2 w) + 1+α α |w| α w = g in B(0, 1) w = u on ∂B(0, 1)
.

By (2.
2), the solution w satisfies

|w| W 1,∞ ≤ C lip |g -1+α α |w| α w| 1 1+α ∞ + |u| W 1,∞ ≤ 2 (-α) + 1+α C lip 1 1+α ∞ + α |w| ∞ + |u| W 1,∞ ≤ |w| W 1,∞ 2 + 2 (-α) + 1+α C lip |g| 1 1+α ∞ + |u| W 1,∞ , hence |w| W 1,∞ ≤ 2 1+( -α) + 1+α C lip |g| 1 1+α ∞ + |u| W 1,∞ .
(2.5)

We set

R = 2 1+ (-α) + 1+α C lip |f | ∞ + 1+α α |u| 1+α ∞ + |b| ∞ M β 1 1+α + |u| W 1,∞ ,
and we define

K R = {v ∈ C 1 (B(0, 1)) ∩ W 1,∞ (B(0, 1)) : |v| W 1,∞ ≤ R} .
We also define the map T : v ∈ K R → w, where w is the unique solution, see [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], of

-|∇w| α F (D 2 w) + 1+α α |w| α w = f + 1+α α |u| α u -b(x)T M (|∇v|) β in B(0, 1) w = u on ∂B(0, 1)
The map T is well defined, since the right hand is continuous and bounded. By (2.5)

w satisfies |w| W 1,∞ ≤ 2 (-α) + 1+α C lip |f | ∞ + 1+α α |u| 1+α ∞ + |b| ∞ M β 1 1+α + |u| W 1,∞
and, therefore,

|w| W 1,∞ ≤ R
by the choice of R. Hence, T sends K R into itself and, by classical uniform estimates (see [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF]), T is a compact operator. Then, by Schauder's fixed point Theorem, T possesses a fixed point w, that is a solution of

-|∇w| α F (D 2 w) + b(x)|T M (∇w)| β + 1+α α |w| w = f + 1+α α |u| α u in B(0, 1) w = u
on ∂B(0, 1) (2.6) Since |∇u| ≤ M , it is easy to see that u satisfies the same equation. By a mere adaptation of the comparison principle in [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF], there is uniqueness of solution for problem (2.6), hence u = w. By (2.3) one gets that u is C 1,γ for the γ allowed by (2.3).

Remark 2.4. Let us observe that, if α ≤ 0 and if the operator F is convex or concave in the Hessian argument, then we can repeat the above proof with the C 1,γ loc norm replaced by the W 2,p loc norm, for any 1 < p < ∞, see [START_REF] Caffarelli | Interior a priori estimates for solutions of fully nonlinear equations[END_REF][START_REF] Caffarelli | on Viscosity solutions of Fully Non linear equations with measurable coefficients[END_REF]. This yields the local a priori estimate, for any ω ⊂⊂ ω ⊂⊂ Ω

u W 2,p (ω) ≤ C p u L ∞ (ω ) , f L ∞ (ω ) , |b| W 1,∞ (ω )
for any viscosity solution u of equation (1.1). Furthermore, suppose that Ω ⊂ R N is a smooth bounded domain, that f ∈ C(Ω), b ∈ W 1,∞ (Ω), and that u is a viscosity solution of (1.1) in Ω, satisfying the boundary condition u = ψ on ∂Ω, with ψ ∈ C 1,γ 0 (∂Ω). By using the up to the boundary estimates of [START_REF] Winter | Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations[END_REF][START_REF] Birindelli | C 1,β regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations[END_REF], we obtain the global regularity bound

u C 1,γ (Ω) ≤ C ψ C 1,γ (∂Ω) , f L ∞ (Ω) , |b| W 1,∞ (Ω)
for some exponent γ ≤ inf γ 0 , 1 1+α + .

Remark 2.5. As in [START_REF] Nornberg | C 1,α regularity for fully nonlinear elliptic equations with superlinear growth in the gradient[END_REF], for α ≤ 0 our results can be extended to the case where F (M ) is replaced by F (p, M ), satisfying the following structural assumptions: there exist positive constants µ, b such that for any (p,

q) ∈ (R N ) 2 , (X, Y ) ∈ (S N ) 2 one has -b|p -q| -µ(|p| + |q|)(|p -q|) + M -(X -Y ) ≤ F (p, X) -F (q, Y ) ≤ M + (X -Y ) + b|p -q| + µ(|p| + |q|)(|p -q|)
where M + and M -denote the Pucci extremal operators. The fixed point argument and the truncation method can be easily adapted to this case.

3 Gradient estimates and proof of Theorem 1.2.

As an application of the regularity results proved in the previous section, we now focus on the ergodic pairs associated to the class of operators we are considering. Precisely, given a function f ∈ C(Ω), let c Ω ∈ R be a constant for which there exist solutions u ∈ C(Ω) of the infinite boundary value problem (1.5).

In [START_REF] Birindelli | Ergodic pairs for singular or degenerate fully nonlinear operators[END_REF] we gave sufficient conditions for the existence of ergodic pairs (c Ω , u) which solve (1.5), and we proved the uniqueness of c Ω in some cases. Here, we are concerned in particular with the uniqueness of u.

As proved in [START_REF] Birindelli | Ergodic pairs for singular or degenerate fully nonlinear operators[END_REF] and recalled in the introduction, the rate of boundary explosion of any ergodic function can be made precise assuming that the operator F satisfies the "boundary" regularity condition

F (∇d(x) ⊗ ∇d(x)) is a C 2 function in a neighborhood of ∂Ω . ( 3
u(x) d(x) χ C(x) = 1 if χ > 0, (3.2)
and lim

d(x)→0 u(x) | log d(x)| C(x) = 1 if χ = 0 , (3.3)
where

χ = 2 + α -β β -1 -α ,
and, for x in a neighborhood of ∂Ω, C(x) = ((χ + 1)F (∇d(x) ⊗ ∇d(x)))

1 β-α-1 χ -1 if χ > 0, C(x) = F (∇d(x) ⊗ ∇d(x)) if χ = 0. (3.4)
Let us now observe that any ergodic function verifies the assumptions of Theorem 1.1, so that it is a C 1,γ loc (Ω) function and satisfies the local a priori bound (1.3). This regularity property, jointly with the asymptotic estimates (3.2), allows us to precise, at least in the case χ > 0, the boundary asymptotic behaviour of its gradient. We obtain the analogous of the result proved in [START_REF] Porretta | The ergodic limit for a viscous Hamilton-Jacobi equation with Dirichlet conditions[END_REF] for Laplace operator and in [START_REF] Leonori | Large solutions and gradient bounds for quasilinear elliptic equations[END_REF] for p-Laplace operator.

Theorem 3.1. Assume that F satisfies (1.2) and (3.1), let f ∈ C(Ω) be bounded and suppose that χ > 0 (i.e. β < α + 2). Then, for any ergodic function u, one has

lim d(x)→0 d(x) χ+1 ∇u(x) • ∇d(x) C(x) = -χ . (3.5)
Proof. Let us consider, for x 0 ∈ ∂Ω fixed and δ > 0, the function

u δ (ζ) = δ χ u(x 0 + δ ζ) , defined for ζ ∈ 1 δ (Ω -x 0 ). By (3.2), one has lim δ→0 u δ (ζ) = C(x 0 ) (∇d(x 0 ) • ζ) χ
locally uniformly with respect to ζ in the halfspace H = {ζ ∈ R N : ζ • ∇d(x 0 ) > 0}, and uniformly with respect to x 0 ∈ ∂Ω. In particular, u δ is locally uniformly bounded in H. Moreover, by direct computation, u δ satisfies the equation

-|∇u δ | α F (D 2 u δ ) + |∇u δ | β = δ β β-α-1 [f (x 0 + δ ζ) + c] in 1 δ (Ω -x 0 ) .
Thus, as a consequence of Theorem 1.1, u δ belongs to C 1,γ loc (H) and verifies estimate (1.3). This implies that u δ is converging in C 1 loc (H), and, therefore,

lim δ→0 ∇u δ (ζ) = -χ C(x 0 ) ∇d(x 0 ) (∇d(x 0 ) • ζ) χ+1
locally uniformly with respect to ζ ∈ H and, again, uniformly with respect to x 0 ∈ ∂Ω.

Hence, we deduce that lim

δ→0 d(x 0 + δ ζ) χ+1 ∇u(x 0 + δ ζ) = -χ C(x 0 ) ∇d(x 0 )
locally uniformly with respect to ζ ∈ H and uniformly with respect to x 0 ∈ ∂Ω. This immediately yields (3.5).

Remark 3.2. In the case χ = 0, one can try to use an analogous argument as above, and to consider the function

u δ (ζ) = u(x 0 + δζ) + C(x 0 ) log(δ) .
By Theorem 4.2 and Theorem 6.3 of [START_REF] Birindelli | Ergodic pairs for singular or degenerate fully nonlinear operators[END_REF], it follows that u δ is uniformly bounded. Moreover, arguing as in the above proof, we obtain that u δ actually converges in C 1 loc (H) to a solution of

-|∇v| α F (D 2 v) + |∇v| 2+α = 0 in H v = +∞ on ∂H
Using the same argument as in Section 4 of [START_REF] Imbert | C 1,α regularity of solutions of degenerate fully non-linear elliptic equations[END_REF], one gets that v satisfies

. 1 )

 1 Here, d(x) denotes the distance function from ∂Ω, and it is of class C 2 in a neighborhood of ∂Ω by the regularity assumption on the domain. Condition (3.1) is certainly satisfied if the domain Ω is of class C 3 and the operator F is C 2 , but there can be also cases with non smooth F satisfying (3.1) in C 2 domains. For instance, when F (M ) depends only on the eigenvalues of M , as in the case of Pucci's operators,F (∇d(x) ⊗ ∇d(x))is a constant function as long as |∇d(x)| = 1. Under assumptions (1.2) and (3.1) on F , and if f ∈ C(Ω) is locally Lipschitz continuous, bounded from below and satisfying lim d(x)→0 f (x)d(x) β β-α-1 = 0 , then ergodic pairs (c Ω , u) exist, and any ergodic function u satisfies lim d(x)→0

-F

  (D 2 v) + |∇v| 2 = 0 in H v = +∞ on ∂HNow, consider first the case when F is a linear operator, that is F (M ) = a tr(M ). Then, defining ϕ = e -v/a , one sees that ϕ is positive and harmonic in H, and it satisfies zero boundary conditions. Hence,ϕ(ζ) = c ∇d(x 0 ) • ζ for some constant c > 0. Coming back to v, one gets that v(ζ) = -a log ∇d(x 0 ) • ζ -a log c, and, by the local C 1 convergence of u δ to v, we conclude that lim δ→0 δ ∇u(x 0 + δ ζ) = -a ∇d(x 0 ) ∇d(x 0 ) • ζ .Observing that in this case a = C(x 0 ), we deduce for linear operators the asymptotic gradient behaviour limd(x)→0 d(x)∇u(x) • ∇d(x) C(x) = -1 ,which is the analogous of (3.5) for χ = 0. For general F , the same result could be obtained as a consequence of the following Liouville type property: if u is a solution in the half spaceH = {x N > 0} of -F (D 2 u) + |∇u| 2 = 0 in {x N > 0} u(x , 0) = +∞,then there exists some constant c such thatu(x) = F (e N ⊗ e N )| log x N | + c .As far as the authors know, this is an open question.of[START_REF] Birindelli | Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient[END_REF], and we get that for any maximum point x ∈ Ω of u -(v + m) there exists a neighborhood V x of x where u ≡ v + m. Denote O δ = {x ∈ Ω 2δ : u(x) = v(x) + m}. Then, we have proved that O δ is non empty and open. On the other hand, by definition, it is closed in Ω 2δ , hence O δ = Ω 2δ . Finally, the comparison principle applied in Ω \ Ω δ yields u = v + m in Ω \ Ω δ , and therefore u = v + m in the whole of Ω.

Acknowledgment.

This work was done while the first and third author were visiting the University of Cergy -Pontoise, and the second one the Sapienza University of Roma, supported by INdAM-GNAMPA.

We are finally in the position to prove the uniqueness, up to additive constants, of the ergodic function.

Proof of Theorem 1.2. By Theorem 3.1, any ergodic function u satisfies the asymptotic gradient boundary behaviour (3.5). Hence, there exists a positive constant

. Suppose now that u and v are two ergodic functions related to the same ergodic constant c Ω . For > 0, consider u = (1 -)u. For further computations, let c α and c β be positive constants such that, for < 1 2 ,

and take

by the choices of δ and . By the boundary asymptotic behavior both of u and v, there exists a neighborhood V of ∂Ω, contained in Ω δ , on which u < v. Since u is a strict subsolution, the comparison principle (see [START_REF] Birindelli | First eigenvalue and maximum principle for fully nonlinear singular operators[END_REF]) can be applied in

On the other hand, the comparison principle for equations having no zero order terms proved in [START_REF] Birindelli | Ergodic pairs for singular or degenerate fully nonlinear operators[END_REF] under the assumption sup(f +c Ω ) < 0 yields that u ≤ v +sup {d=δ} (u-v) in Ω \ Ω δ . Therefore, we obtain that u ≤ v + sup {d=δ} (u -v) in the whole of Ω. We claim that u coincides with v + sup d=δ (u -v) := v + m in Ω. When α = 0, this is a direct consequence of the strong maximum principle, since the supremum of u -(v + m) is zero and is achieved inside Ω. When α = 0, we use the assumption that ∂Ω has only one connected component. This implies that Ω 2δ is connected. We note that, by the choice of δ, ∇u = 0 in Ω δ , since |∇u| ≥ C 0 δ -χ-1 in Ω δ . This allows to apply the strong comparison principle