LIOUVILLE-TYPE RESULTS IN EXTERIOR DOMAINS FOR RADIAL SOLUTIONS OF FULLY NONLINEAR EQUATIONS

. We give necessary and suﬃcient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in R N , and equipped with homogeneous Dirichlet boundary conditions.


Introduction
Let B be any ball in R N and let 0 < λ ≤ Λ. The aim of this paper is to detect the optimal conditions on the exponent p > 1 for the existence of positive radial solutions of the fully nonlinear exterior Dirichlet problem −F(D 2 u) = u p in R N \ B, u = 0 on ∂B, (1.1) where F is either one of the Pucci's extremal operators M ± λ,Λ , defined respectively as Pucci's extremal operators are the prototype of fully nonlinear uniformly elliptic operators. Acting as barriers in the whole class of operators with fixed ellipticity constants λ ≤ Λ, they play a crucial role in the regularity theory for fully nonlinear elliptic equations, see [4]. Moreover, as sup/inf of linear operators, they frequently arise in the equations satisfied by the value function associated with stochastic optimal control problems, see e.g. [11,18], with special application to mathematical finance problems.
We recall that if Λ = λ, both Pucci's operators reduce, up to a multiplicative factor, to the Laplace operator. Thus, they may be considered also as perturbations of the standard Laplacian, and the well-known Lane-Emden-Fowler equation −∆u = u p is included as a very special case of the problems we are considering. In the semilinear case, the Lane-Emden-Fowler equation has been largely studied. The well known existence results, exhibiting the critical Sobolev exponent p * = N +2 N −2 as threshold for the existence of entire solutions or solutions in bounded domains, are intimately related to the (lack of) compactness properties of the Sobolev embeddings. Moreover, the entire solutions existing in the critical case p = p * realize the best constant in the Sobolev inequality, see [19]. In other words, the critical nature of the exponent p * may be largely interpreted in view of structural properties of the functional setting behind the equation.
As soon as Λ > λ, Pucci's operators loose the linear and variational structures. Nevertheless, Lane-Emden-Fowler type equations as (1.1) have been studied and, at least in the radial setting, some critical exponents p * ± acting as thresholds for the existence of entire radial solutions or solutions in balls have been proved to exist, see [10]. In the fully nonlinear radial setting, the exponents p * ± play the same role as the critical Sobolev exponent p * = N +2 N −2 for the Laplacian. Though their appearance is motivated exclusively as threshold for the existence of entire radial solutions or solutions in balls, the recent results of [3], where some weighted energies associated with radial solutions of (1.1) are introduced and proved to be asymptotically preserved by almost critical solutions, suggest that the critical exponents reflect some intrinsic properties of the operators, maybe beyond the radial setting. The Liouville-type results obtained in the present paper may be regarded as a further justification of the critical character of the exponents p * ± , since they are proved to act as thresholds also for the existence of radial solutions to exterior Dirichlet problems. We observe that this result, while well expected for semilinear equations due to the duality between Dirichlet problems in balls and Dirichlet problems in exterior domains (via the Kelvin transform), in the fully non linear case it requires a direct proof. Moreover, the existence of positive solutions for exterior Dirichlet problems is also closely related to the existence or non-existence of sign-changing radial solutions in balls or in the whole space (see also Remark 5.2 in [14]). In particular, in the recent paper [13], the asymptotic behavior of sign changing solutions in balls is studied, and the obtained results strongly rely on the present theorems for exterior Dirichlet problems.
In order to describe the results, let us introduce the dimension like parameters N + : = λ Λ (N − 1) + 1 for M + λ,Λ , which have been proved in some previously studied cases to play a key role in existence results. Let us emphasize that one has alwaysÑ + ≤ N ≤Ñ − , and the equalities hold true if and only if Λ = λ.
The case of entire supersolutions has been considered in [5] , where it has been proved that meaning that, in particular, positive supersolutions never exist ifÑ ± ≤ 2. In the sequel, we will always assume thatÑ ± > 2.
The same threshold has been proved in [2] to be optimal for the existence of solutions in any exterior domain, that is where K ⊂ R N is any nonempty compact set. Let us also mention the results of [1], where more general nonlinearities f (u) replacing u p are considered. In the above results, supersolutions are meant in the viscosity sense and no symmetry property on u is required.
In the present paper we are concerned with solutions of the equation satisfying further homogeneous Dirichlet boundary conditions. By elliptic regularity theory, it is not restrictive to consider classical solutions of problem (1.1), that are C 2 functions satisfying pointwise the equation as well as the boundary condition. In its full generality, that is without assuming radial symmetry of the solutions, the problem is completely open, and also in the semilinear case (i.e. when Λ = λ) few results are known for solutions of exterior Dirichlet problems, see e.g. [7] where solutions are constructed as perturbations of radial solutions. Our results, limited to radially symmetric solutions, may hopefully contribute to tackle the general fully nonlinear problem.
In the radial setting, the existence of entire positive solutions has been studied in [10], where it has been proved that there exist two critical exponents p * + and p * − associated with M + λ,Λ and M − λ,Λ respectively, such that Unfortunately, the dependence of the radial critical exponents on the effective dimensions is not explicitly known. The radial critical exponents are proved in [10] to satisfy, when λ < Λ, the strict inequalitiesÑ Note that inequalities in (1.2) become equalities when Λ = λ. Let us emphasize that inequalities (1.2) say that, with respect to the intrinsic dimensions, the exponent p * + is subcritical, whereas p * − is supercritical. Moreover, (1.2) show that the critical Sobolev exponent p * = N +2 N −2 is not preserved even for small perturbations of the operator, since the critical exponents are instantaneously different as soon as Λ > λ. For an integral characterization of p * ± , as well as for sharp estimates on entire critical solutions, we refer to [3].
Clearly, the analysis on the existence of entire positive radial solutions yields, as a by product, the dual result on the existence of positive solutions of Dirichlet problems in balls, namely Note that, in this case, the radial symmetry of the solutions is not a restriction, since, by [6], any positive solution in the ball is radial. The critical exponents p * ± , therefore, give the optimal thresholds for the existence of positive solutions in balls and, as proved in [9], also in domains sufficiently close to balls.
On the other hand, as recently proved in [14], for annular domains (radial) solutions exist for any p > 1. More precisely, in [14] it has been proved that solutions of the initial value problems for the ODEs associated with the equations −M ± λ,Λ (D 2 u) = u p give radial solutions in annular domains provided that they have sufficiently large initial slope. The results of the present paper, in a sense, complement the results of [14], since here we prove, in particular, that a sufficiently large initial velocity is needed for having radial solutions in annuli if and only if p > p * ± . The results of the subsequent sections are summarized in the following theorem. Theorem 1.1. There exist positive radial solutions of problem (1.1) if and only if p > p * ± . Moreover, for any p > p * ± , problem (1.1) has a unique positive radial solution u * satisfying and infinitely many positive radial solutions u satisfying Borrowing the terminology currently used, the solution u * satisfying (1.3) will be referred to as the fast decaying solution. As far as solutions u satisfying (1.4) are concerned, they will be proved to satisfy either lim r→+∞ r 2 p−1 u(r) = c > 0 , in which case they will be called slow decaying solutions, or 0 < lim inf r→+∞ r 2 p−1 u(r) < lim sup r→+∞ r 2 p−1 u(r) < +∞ , in which case they will be named pseudo-slow decaying solutions (see [10]).
In the semilinear case, a proof of Theorem 1.1 can be found in [16]. For the fully nonlinear case, it will be a straightforward consequence of the results proved in the next sections, where we perform a careful analysis of the initial value problem for the ODE associated to radial solutions of problem (1.1). Recalling that the eigenvalues of the Hessian matrix D 2 u of a smooth radial function u = u(r) are nothing but u (r) and u (r) r (with multiplicity at least N − 1), it is not difficult to write the ODE satisfied by a radial solution of problem (1.1). Nevertheless, since the coefficients of the operators M ± λ,Λ depend on the sign of the eigenvalues of the Hessian matrix, we will obtain an ODE with discontinuous coefficients having jumps at the points where the solution u changes its monotonicity and/or concavity. This is a feature of the fully nonlinear problem which makes techniques previously developed for the semilinear case not directly applicable. In particular, the Kelvin transform which reduces a supercritical exterior Dirichlet problem to a subcritical Dirichlet problem in the punctured ball cannot be used.
We will essentially make use of the results of [10] for entire solutions, in particular of the fact that the critical exponents are the only exponents for which the entire solutions are fast decaying. Moreover, as in [10], we will take advantage of the Emden-Fowler trasformed which produces a new variable x(t) satisfying an autonomous equation. Despite the fact that also the coefficients of the equation satisfied by x will have jumps at the points corresponding to the changes of monotonicity and concavity of u, the phase plane analysis of the trajectories associated to the solution x will be repeatedly used.
A particularly delicate step in the proof of Theorem 1.1 will be the proof of the non existence of solutions in the critical cases p = p * ± , as well as of the uniqueness of the fast decaying solutions, which will be obtained by using different arguments for M + λ,Λ and M − λ,Λ . For M − λ,Λ , we heavily exploit the fact that p * − >Ñ − +2 N−−2 and the proof relies on some properties of the solutions of supercritical semilinear problems. For M + λ,Λ , for which p * + <Ñ + +2 N+−2 , a different proof will be obtained as an application of Gauss-Green Theorem in the phase plane.
The paper is organized as follows: in Section 2 we recall some basic properties of radial solutions of problem (1.1) and their Emden-Fowler transform, whereas the existence of positive radial solutions for p supercritical is proved in Section 3. Section 4 will be then devoted to the proof of nonexistence of nontrivial solutions when p is strictly subcritical, while the critical cases p = p * + and p = p * − are addressed respectively in Section 5 and Section 6. Finally, in Section 7, we complete the proof of Theorem 1.1 by showing that problem (1.1) has a unique fast decaying solution and infinitely many slow or pseudo-slow decaying solutions, according to the initial slope.

Radial solutions of problem (1.1) and the Emden-Fowler transform
In order to study the existence of solutions for problem (1.1), we can assume, without loss of generality, that B is the unit ball of R N centered at the origin, by the invariance of the equation with respect to translations and to the scalingũ(x) := γ For any α > 0, let us introduce the initial value problem A direct computation shows that u is a radial solution of problem (1.1) with F = M + λ,Λ if and only if u satisfies (2.1) in (1, +∞) for some α > 0.
Analogously, a radial solution of (1.1) with F = M − λ,Λ is nothing but a global solution of For the rest of the section, let us focus on the case F = M + λ,Λ . The same observations can be made for F = M − λ,Λ , by just exchanging Λ with λ. Let us denote by u α the unique positive maximal solution of (2.1), defined on a maximal interval [1, ρ α ), with ρ α ≤ +∞. Some general properties of the function u α have been established in [14]. In particular (see [14, Lemma 2.1]), it is known that, for any α > 0, there exists a unique τ α ∈ (1, ρ α ) such that u α (r) > 0 for r ∈ [1, τ α ) and u α (r) < 0 for r ∈ (τ α , ρ α ). Moreover, if ρ α < +∞ one has u α (ρ α ) = 0, whereas if ρ α = +∞, then lim r→+∞ u α (r) = 0 (see [14,Corollary 2.3]). We remark that if ρ α < +∞, then u α is a radial solution of the Dirichlet problem where A 1,ρα is the annular domain with radii 1 < ρ α , and u α (ρ α ) < 0 by the Hopf boundary maximum principle. On the contrary, if ρ α = +∞, then u α is a positive solution in the exterior domain R N \ B 1 and, by the results of [2], this necessarily implies that p >Ñ + N+−2 . We further notice that the function u α , which satisfies u α (1) < 0, cannot be globally (or even definitely) concave in (1, ρ α ), since otherwise ρ α < +∞ and We will denote by σ α ∈ (τ α , ρ α ) the first zero of u α . We will show below that σ α actually is the first point where the function u α changes its concavity. A useful tool for studying the problem is the so called Emden-Fowler transform (see [12,10]), which reduces the initial problem (2.1) to an autonomous equation. Setting 3) a direct computation shows that .
Hence, the new unknown x satisfies , with coefficients defined respectively as Associated with a solution x of the above problem, one can consider the trajectory γ(t) = (x(t), x (t)) in the phase plane. Thus, the coefficients of the autonomous equation satisfied by x are piecewise constant and have jumps whenever the trajectory γ crosses either the half-line . We will denote by R + and R − the open regions of the right half-plane lying respectively above and below the curve C.
In particular, the trajectory γ α (t) = (x α (t), x α (t)) associated with the transform x α of the solution u α is defined on the interval [0, log ρ α ), it lays in the right half plane for t ∈ (0, log ρ α ) and it satisfies the initial condition γ α (0) = (0, α). We remind the reader that the function x α is increasing as long as the trajectory γ α lays in the first quadrant, as well as x α decreases when γ α is in the fourth quadrant. If ρ α < +∞, then the trajectory γ α reaches the x −axes at the finite time t = log ρ α , with γ α (log ρ α ) = 0, ρ p+1 p−1 α u α (ρ α ) and u α (ρ α ) < 0. The typical behavior of a trajectory γ α having ρ α < +∞ is depicted in the figure above.
In the following result we collect some properties of the trajectory γ α , partially observed in previous contributions (we refer in particular to [10,3,14]).
Proof. By setting t α = log τ α and s α = log σ α , properties (i) e (ii) immediately follow by the definition of τ α and σ α . In order to prove (iii), let us assume that, for a time s ∈ (0, log ρ α ), one has γ α (s) ∈ C and γ α (t) ∈ R + for t in a left neighborhood of s. This means that for some δ > 0. It then follows that On the other hand, by (2.4) one has Combining the above relationships, since x α (s) > 0, we then obtain and the intersection is tangential if and only if . But a second order analysis at the intersection point, as in the proof of Lemma 3.1 in [10], shows that intersections from above never occur at the point . This proves statement (iii). The same argument, with reversed inequalities, proves claim (iv).
In order to prove (v), let γ α be such that either ρ α < +∞ or γ α (t) → (0, 0) as t → +∞. Since γ α (0) = (0, α), then the trajectory crosses at least once the x-axis. Let us call t 0 ∈ (0, log ρ α ) the time of the first intersection between γ α and the x-axis, that is Hence, in both cases we obtain x α (t 0 ) < 0 and, for t > t 0 , the trajectory enters the fourth quadrant and x α (t) starts to decrease. We then observe that for t > t 0 the trajectory never intersects the x-axis, since otherwise, arguing as above, it would cross it, entering again into the first quadrant, and, in order to avoid self-intersection, it could not reach any point of the x -axes for subsequent times. This proves that γ α intersects exactly once the x-axis. We notice further that, if s α < t 0 , that is if the first intersection point between γ α and C lies in the first quadrant, then, for t ∈ (s α , t 0 ] the trajectory lies in the first quadrant and x α (t) > x α (s α ), so that γ α (t) cannot intersect C by property (iv). Therefore, γ α (t) stays in R − until it reaches the x-axis at the time t 0 , then it enters the fourth quadrant, x α (t) becomes decreasing, and γ α (t) ∈ R − also for t ≥ t 0 . The same argument can be applied in the case s α ≥ t 0 , since then x α (t) < 0 for any t ∈ (s α , log ρ α ) and, by property (iv), γ α cannot intersect C from below. This completely proves the claim.
Let us now prove (vi). The statement is trivial if ρ α < +∞, so we assume ρ α = +∞ and, thus, p >Ñ + N+−2 . From (iv) it follows that, if γ α (t) intersects again the curve C for t > s α , and if we callŝ > s α the first time such that γ α (ŝ) ∈ C and γ α (t) ∈ R − for t ∈ (s α ,ŝ), then necessarily x α (ŝ) > 0. Hence, γ α (t) intersects twice the x-axis for t ∈ [0,ŝ] and therefore, for all t ∈ [0, +∞), one has It remains to prove the boundedness of the trajectories γ α (t) satisfying γ α (t) ∈ R − for all t > s α . In this case it is convenient to look back at the function u α , which is convex and decreasing in (σ α , +∞). By (2.1), u α satisfies which can be written as Integrating between σ α and any r > σ α , and using the decreasing monotonicity of u α , we get By integrating once again, we deduce Hence, for r ≥σ α := (2.7) Using (2.7) into (2.6), we also have By integrating in [σ α , r] and recalling that p >Ñ + N+−2 , we get Therefore, for r ≥σ α , using again that p >Ñ + N+−2 , we obtain the estimate Also statement (viii) classically follows from the perturbed linear systems theory, λ 1 being the negative eigenvalue of the linear system obtained by linearizing around zero system (2.4). For a detailed proof, we refer e.g. to Lemma 3.3 in [17].
Let us finally prove (ix). We first claim that if a periodic orbit γ of system (2.4) is the ω-limit set of a trajectory γ α , then γ intersects the curve C. Indeed, the trajectory of such a periodic orbit, if any, is a closed curve γ contained in the right half-plane and winding around the equilibrium point (b + Λ) 1/(p−1) , 0 . If, by contradiction, γ does not intersect C, then γ(t) lies entirely in the region R − . This implies that γ is the trajectory associated with a positive periodic solution x(t) of the single equation Hence, the energy function when evaluated along γ(t), satisfies In particular, ifã + = 0, i.e. if p =Ñ + +2 N+−2 , then the function E(γ(t)) is strictly monotone and periodic: a contradiction. On the other hand, ifã + = 0, i.e. if p =Ñ + +2 N+−2 , then E(γ(t)) is constant, that is γ is a level line of the function E contained in R − . Every level line of the function E contained in R − actually is a periodic orbit of system (2.4), and, since γ is contained in R − , by continuity there exists a larger level line of the function E still contained in R − and enclosing γ. This means that γ cannot be approached by any of the trajectories γ α , again a contradiction to the assumptions.

Remark 2.2. Positive entire radial solutions of equations
−M ± λ,Λ (D 2 u) = u p in R N have been studied in detail in [10]. In this case the initial value problems for the ODE to be considered are and for the maximal solutions v α , defined on [0, r α ) for some 0 < r α ≤ +∞, one has r α < +∞ if and only if p < p * ± . We recall that, by the homogeneity property of the equation, for any α, α > 0 one has that is changing the initial condition just reflects in scaling the solution. Therefore, the property r α < +∞ or r α = +∞ only depends on p.

Relationship (2.10) implies that
and thus the initial value α for the solution v α (r) does not affect the support of the trajectory Γ α (t), which is then denoted simply by Γ(t). In the phase plane, the trajectory Γ(t) is a curve exiting from the origin, lying always below the line L, and staying above the curve C for t ∈ (−∞, S α ), for some S α < log r α . We recall from [10] that Γ(t) satisfies properties (iii)-(ix) of Lemma 2.1, and fast, slow or pseudo-slow decaying solutions v α are accordingly defined. A crucial result proved in [10] is that, independently of α, v α is a fast decaying solution if and only if p = p * ± .
3. Existence in the supercritical case p > p * ± for M ±

λ,Λ
This section is devoted to the proof of the existence statement in Theorem 1.1. Proof. We write the proof for M + λ,Λ . Unless otherwise said, the proof for M − λ,Λ is obtained just by exchanging λ with Λ.
Therefore, fixing r ∈ (0, l 1 ) and passing to the limit as α → 0 + , we get which gives lim r→0ũ (r) = 0. Thus,ũ in particular satisfies and this implies that it is of class C 2 up to 0. Summing up,ũ is a smooth nontrivial radial solution to Since p > p * + , this is a contradiction to the results of [6,10]. The proof of Step 3 is complete.
On the other hand, by (3.14), we also have and we reach a contradiction in this case as well.
Consider now the compact region D bounded by the closed piecewise smooth curve formed by the trajectories γ(t) for t ≥ t ε , γ α (t) for t ≥ t α,ε and the vertical segment joining γ(t ε ) and γ α (t α,ε ).

Existence and uniqueness of fast decaying solutions
In this section we complete the proof of Theorem 1.1, by showing in particular the existence and the uniqueness of fast decaying solutions of problems (1.1) for p > p * ± . For any α > 0, let u α be the unique maximal solution of either (2.1) or (2.2), defined on the maximal interval [1, ρ α ) with ρ α ≤ +∞. As for the proof of Proposition 6.2, let us set D = {α ∈ (0, +∞) : ρ α < +∞} , and α * = α * (p) = inf D .
Next, in order to prove that u α * is the only fast decaying solution, let us consider separately the cases of M + λ,Λ and M − λ,Λ . Assume first that u α solves (2.1) and, by contradiction, suppose that γ α (t) → (0, 0) as t → +∞ for some α < α * . If p * + < p <Ñ + +2 N+−2 , then we can apply the argument of the proof of Theorem 5.1 to the two trajectories γ α * (t) and γ α (t), reaching a contradiction. On the other hand, if p =Ñ + +2 N+−2 , then the trajectory associated with any fast decaying solution lies for sufficiently large t on the zero level set of the energy function E given by (2.9). Hence, γ α * (t) and γ α (t) definitively coincide, again a contradiction. Finally, if p >Ñ + +2 N+−2 , then we can argue as in the proof of Theorem 6.1 and, thanks to Proposition 6.2 for ν =Ñ + , we obtain a contradiction as well.
For the operator M − λ,Λ , the uniqueness of the fast decaying solution follows by the same proof of Theorem 6.1.
Therefore, in both cases, we obtain that, for α < α * , the trajectories γ α (t) cannot approach the origin as t → +∞. Furthermore, since, for any α, γ α (t) cannot intersect the trajectory γ(t) associated with any entire solution u of −M ± λ,Λ (D 2 u) = u p , and since, for p > p * ± and α < α * , neither u nor u α is a fast decaying solution, it follows that γ(t) and γ α (t) must have the same behavior as t → +∞.
According to Theorem 1.1 of [10], we then obtain the following result.