ON THE UNIQUENESS OF BLOW-UP SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS

This paper contains new uniqueness results of the boundary blowup viscosity solutions of second order elliptic equations, generalizing a well known result of Marcus-Veron for the Laplace operator.

In a previous paper [12] the first and the third author proved, together with existence and uniqueness of entire solutions, the existence of boundary blow-up solutions under various assumptions about the dependence of F on x.Our paper was a generalization of Esteban-Quaas-Felmer [11], based on interior estimates which provide the local uniform convergence of approximating solutions.
The issue of uniqueness was considered by Dong-Kim-Safonov in [9], to which we refer for a nice history of the problem.In that paper uniqueness of classical and 772 ANTONIO VITOLO, MARIA E. AMENDOLA AND GIULIO GALISE L p -strong boundary blow-up solutions is proved for semilinear equations Lu = u s , where L is a second order uniformly elliptic operator, in a domain Ω satisfying "the uniform exterior ball condition".The authors notice that a similar result can be obtained when L is replaced by a fully nonlinear operator F of Bellman type.
Here, we consider a different regularity condition on the boundary of domains Ω, called "local graph property", see Definition 2.5 below, and introduced by Marcus-Veron [15] to show the uniqueness of blow-up solutions of equation ∆u = u s .Moreover we investigate the problem in the larger class of viscosity solutions, although our method, which does not use informations about the boundary behavior of solutions, works for F independent of x.In this respect, solutions of Bellman type equations with constant coefficients are in fact classical solutions and therefore are covered by [9], under the uniform exterior ball condition.This is not, generally, the case of Isaacs type equations with constant coefficients, which are instead included in the present paper, see Remark 1 and the examples just below.Let 1 ≤ ϕ(k) → 1 as k → 1 + .In the sequel we need the following additional assumption on F : uniformly with respect to (x, t, ξ, X) ) is an operator (independent of x) satisfying the structure conditions (SC) and ( 6), then problem (1), ( 2) has at most one non-negative solution.
Remark 1. Condition (6) on F is satisfied with ϕ(k) = k α in the case of operators such that F 1 is positively homogeneous of degree α ∈ (0, s], s > 0, i.e.F 1 (kξ, kX) = k α F 1 (ξ, X) for all k > 0 and all (ξ, X) ∈ R n × S n .Observe indeed that ). when t ≥ 0 and k ≥ 1.As one can see in Remark 2 below, f ≤ 0 is a sufficient condition to have non-negative solutions.
By Theorem 1.1 and Remark 1 we have uniqueness of non-negative blow-up solutions for the maximal equation and more generally for Bellman and Isaacs type equations like sup Following Marcus-Veron [16], we also consider more general operators, acting on the convex cone of non-negative continuous functions, which are obtained adding a "positive semilinearity", namely where c is a positive constant, 0 < α < s and F 1 is positively homogeneous of degree β ∈ [α, s], see Remark 1.
Theorem 1.2.Let Ω be a bounded domain of R n of class C gr .Let F be an uniformly elliptic operator satisfying (3) and (5) of type (7), with c ∈ R and has at most one positive solution.
Note that the case c ≤ 0 is already provided by Theorem 1.1.

Preliminaries.
Let Ω be a domain (open connected set) of R n .By S n we denote the set of n × n real symmetric matrices equipped with the usual partial order: and uniformly elliptic if there exist two constants Λ ≥ λ > 0, called ellipticity constants, such that Here P ± λ,Λ are the Pucci's extremal operators, defined by where T r(•) is the trace of a matrix.From (10) it follows the subadditivity, resp.superadditivity, of P + λ,Λ , resp.P − λ,Λ , and the equality P + λ,Λ (X) = −P − λ,Λ (−X).A second order partial differential equation is said to be fully nonlinear uniformly elliptic when the condition (9) holds.We will assume the continuity of the real valued mappings F and f .Definition 2.1.Given a function u : Ω → R, the second order superjet J 2,+ u(x), respectively subjet J 2,− u(x), of u at x ∈ Ω is the convex set (possibly empty) of all pairs (ξ, Definition 2.2.An upper semicontinuous function u : Ω → R is a viscosity subsolution of (11), for short Similarly a lower semicontinuous function u : Ω → R is a viscosity supersolution of (11), for short Finally u ∈ C(Ω) is a viscosity solution of (11), for short F [u] = f or also F = f , if it is simultaneously a viscosity sub and supersolution.
It is evident that a classical solution of (11), i.e. a C 2 (Ω) function satisfying pointwise the equation, is also a viscosity solution.Conversely a twice differentiable viscosity solution is a classical one.We refer to [2], [5], [13], [14] for major details on viscosity solutions of nonlinear equations.
Proof.By regularity results for convex operators (see [1] and Sections 6.2, 8.1 in [2]) we have w ∈ C 2,a , with 0 < a < 1, so w is a classical solution.
Let (ξ, X) In the sequel p 0 ∈ ( n 2 , n) will be the exponent of Escauriaza [10] (see also Crandall-Swiech [7]) in order that the Alexandroff-Bakelman-Pucci Maximum Principle holds true with p > p 0 in the form (GMP) for solutions u ∈ W 2,p loc (Ω) ∩ C(Ω) of the maximal equation where d = diam(Ω) < +∞ and C a positive constant depending on n, λ, Λ, p, γd.This result can be generalized to viscosity solutions, see Swiech [20], Lemma 1.4.We will use the following Generalized Comparison Principle (GCP), which is deduced by the Maximum Principle of [12], Lemma 3.2.
Lemma 2.4.Let Ω be a domain of R n and F be an uniformly elliptic operator satifying (SC) and independent of x.Suppose that u and v are continuous solutions, resp., of F [u] ≥ f and F [v] ≤ g in viscosity sense, where f, g ∈ C(Ω) ∩ L p loc (R n ) for some p > p 0 .Then for any y ∈ Ω and any ball B R centered at y we have where C 0 = C 0 (n, Λ, s, δ) and Proof.Since F is independent of x, by means of the Jensen's approximations, we may use the structure conditions (SC) just as for smooth functions, see e.g.[4]- [8], to deduce that w = (u − v) + is a viscosity subsolution of From this, reasoning as in Lemma 3.2 of [12] and using GMP (12), for any ball B r centered at y of radius r < R we get from which (13) follows, letting r → 0 + .
Remark 2. If f ≥ g, letting R → ∞, from Lemma 2.4 we obtain the Comparison Principle (CP): Note also that Ω is possibly unbounded in Lemma 2.4.Nonetheless no assumption is made on the growth of u and v at infinity.Definition 2.5.(Marcus-Veron) A domain Ω satisfies the local graph property at P ∈ ∂Ω if there exist a neighborhood Q P and a function ψ ∈ C(R n−1 ) such that in a coordinate system y ≡ (y , y n ) obtained by rotation from x ≡ (x , x n ).
Remark 3. We may assume that Q P is a spherical cylinder centered at P , of radius ρ > 0 and finite height 2σ > 0, as well as |ψ(y )| < σ in Q P so that ) Here x = Ry + x(P ) for an orthogonal matrix R (i.e.R −1 = R T ).As in [15], the class of domains satisfying the local graph property at every P ∈ ∂Ω will be denoted by C gr .
3. Uniqueness of blow-up solutions.Let Q P be a spherical cylinder centered at P as in (16).We start recalling that a non-negative viscosity solution w P ≡ w ∈ C(Q P ) of the boundary blow-up problem 18) is provided by Theorem 1.6 of [12] and by [1], Cor.1.3, w ∈ C 2,a (Q P ) for a ∈ (0, 1).
The main tool to show the uniqueness will be the comparison principle (13).
Let Ω be a domain of R n satisfying the local graph property at x P ∈ ∂Ω, and Q P the cylinder of Remark 3. Assume that satisfies the structure conditions (SC) and the comparison principle (15) holds true with Q P ∩ Ω in place of Ω.If there exists a viscosity subsolution u ∈ C(Q P ∩ Ω) of ( 1) such that then the problem Remark 4. Following [15], by condition (21) we mean v(x) → +∞ as dist(x, A) → 0 for every A ⊂⊂ Γ 1 in the relative topology.
Proof.Following [15], with the notations of ( 16) consider an approximation from below of Θ ≡ Q P ∩ Ω = {x ∈ R n : |y | < ρ, −σ < y n < ψ(y )} , where x = Ry +x(P ) and R −1 = R T , assuming ψ > 0, as we may, using a monotone increasing sequence of smooth positive functions ψ j → ψ as j → ∞.Correspondingly, let [5] we can find a continuous viscosity solution of the problem Here we are using the same boundary conditions of [15], Theorem 2.2.Then by construction for any fixed j ∈ N the sequence (v j,k ) k∈N is increasing, with respect to k ∈ N, on ∂Θ j and so, by the comparison principle, is also increasing in Θ j .On the other side, from Proposition 3.3 in [12] we have uniform boundedness in compact sets K of Θ j , say sup By using Hölder estimates (see Caffarelli-Cabré [2] and Sirakov [19]), Ascoli-Arzelá theorem and stability results for viscosity solutions (see Proposition 4.11 in [2], Theorem 3.8 in [3]), we deduce that is a solution of (20) in Θ j .
Next consider the sequence (v j,∞ ) j∈N .Since v j+1,k ≤ v j,k on ∂Θ j , we have v j+1,∞ ≤ v j,+∞ on ∂Θ j so that, again by the comparison principle, the sequence (v j,∞ ) j∈N is monotone decreasing in Θ j and, by reasoning as before to show that v j,∞ are solutions, in turn converges locally uniformly to a solution v of (20) in Θ.
It is easy to check that v = 0 on Γ 2 , which is regular enough in order that the boundary condition is satisfied with continuity, see [6].In order to prove (21), let us observe that for all k Since u is bounded on ∂Θ j , then u ≤ v j,∞ on Γ 1j , as well as u ≤ w on Γ 2j , by (18).Moreover from Lemma (2.3) the function v j,∞ + w is a supersolution of (20) in Θ j and hence by the comparison principle u ≤ v j,∞ + w in Θ j .
Passing to the limit as j → ∞ we obtain in Θ, from which condition (21) follows.
Corollary 1. Suppose that the assumptions of Proposition 1 are satisfied for positive functions u = u i ∈ C(Q P ∩ Ω), i = 1, 2. Let Q * P ⊂⊂ Q P be a spherical cylinder centered at P .If F is independent of x, then there exists a positive constant C such that Proof.Since F is independent of x, the comparison principle holds true by Remark 2. Therefore, from (23) we have u 2 ≤ v + w in Q P ∩ Ω, where, up to a rotation, we may suppose the axis of the cylinder Q P parallel to x n , see (17).Since w is bounded in Q * P , we get then with C = sup Q * P w.On the other side, consider v h (x , x n ) = v(x , x n − h) for sufficiently small h > 0: v h is continuous in Q h P ∩ Ω and v h = 0 on Γ h 2 .Here Q h P and Γ h i , i = 1, 2, result from the corresponding sets Q P and Γ i moved up by h along the axis of Q P .Then Since F is independent of x, the function v h satisfies the equation where Therefore, fixing y ∈ Q P ∩ Ω, choosing h > 0 small enough in order that y ∈ Q h P ∩ Ω and applying Lemma 2.4 in For references about maximum principles and related methods see [17] and [18].
Lemma 4.1.Let Ω be a bounded domain of R n and F be an uniformly elliptic operator satisfying (3) and (5) of form (33) with F positively homogeneous of degree β ∈ [α, s] for t > 0 and c ≥ 0. Suppose that u and v are continuous subsolutions and supersolutions, respectively, of F = f in viscosity sense, where f ∈ C(Ω) and f ≤ 0. In addition we assume u, v ∈ C 1 (Ω) and F independent of x.Suppose v > 0 in Ω, then lim sup Remark 5.If we assume at least one of u and v to be C 2 (Ω), then we do not need to assume F independent of x.
Proof of Lemma 4.1.By contradiction, suppose Setting u = e U and v = e V , by straightforward computation, we obtain, in viscosity sense, where we have used the positive homogeneity of F .Let w = U − V .Subtracting (36) from ( 35), as we may in viscosity setting when F is independent of x, using (3) we have We are ready to develop the program of the previous Section to establish an uniqueness result for the blow-up problem (1) & (2) with fully nonlinear uniformly elliptic operators of type (7), i.e.F = F 1 (ξ, X) in (33).
Let ε ∈ (0, 1).By the local graph property and the boundary blow-up condition, for every P ∈ ∂Ω we can find a spherical cylinder Q P as (16) such that (17) holds true and u i ≥ ( c ε ) 1 s−α in Q P ∩ Ω, i = 1, 2, so that by Lemma 4.1 the comparison principle (15) holds true with Q P ∩ Ω in place of Ω.Then 2 , say, in the definition (18) of w, we conclude as in Corollary 1 that for Q * P ⊂⊂ Q P there exists C such that (24) holds true.As in the proof of Theorem 1.1 we find a neighborhood N ε of ∂Ω where (31) holds true and set Ω ε = {x ∈ Ω , u 2 > (1 + ε )u 1 = u 1ε }.We infer that Ω ε = ∅.