Fundamental solutions of homogeneous fully nonlinear elliptic equations
aa r X i v : . [ m a t h . A P ] O c t FUNDAMENTAL SOLUTIONS OF HOMOGENEOUSFULLY NONLINEAR ELLIPTIC EQUATIONS
SCOTT N. ARMSTRONG, BOYAN SIRAKOV, AND CHARLES K. SMART
Abstract.
We prove the existence of two fundamental solutions Φ and ˜Φ ofthe PDE F ( D Φ) = 0 in R n \ { } for any positively homogeneous, uniformly elliptic operator F . Correspond-ing to F are two unique scaling exponents α ∗ , ˜ α ∗ > − F ( D u ) = 0,which is bounded on one side. A Liouville-type result demonstrates that thetwo fundamental solutions are the unique nontrivial solutions of F ( D u ) = 0in R n \{ } which are bounded on one side in both a neighborhood of the originas well as at infinity. Finally, we show that the sign of each scaling exponent isrelated to the recurrence or transience of a stochastic process for a two-playerdifferential game. Introduction and main results
The following fundamental result was proved by M. Bˆocher in 1903.
Theorem 1 (Bˆocher, [4]) . Denote B r = { x ∈ R n : | x | < r } , and assume n ≥ .(i) Suppose u ∈ C ( B \ { } ) is harmonic and bounded below or above in B \ { } .Then either u can be extended to a harmonic function in B , or there exist constants a = 0 and C > such that a Φ − C ≤ u ≤ a Φ + C in B / \ { } , where Φ( x ) = (cid:26) | x | − n if n > − log | x | if n = 2 . Hence, by the linearity of the Laplacian, u − a Φ can be extended to a harmonicfunction in B .(ii) Suppose u is harmonic and bounded below or above in R n \ B , n ≥ . Then u ( x ) → a as | x | → ∞ , for some a ∈ R . The function Φ which appears in this theorem is known as the fundamentalsolution for the Laplacian. Bˆocher’s result easily implies the following extendedLiouville theorem.
Theorem 2 (Bˆocher, [4]) . The set of all harmonic in R n \ { } functions thatare bounded from above or from below in a neighbourhood of zero as well as in aneighbourhood of infinity is in the form { a Φ + b | a, b ∈ R } . Date : October 23, 2018.2000
Mathematics Subject Classification.
Primary 35A08, 35J60, 91A15, 49N70, 35P30.
Key words and phrases. fully nonlinear elliptic equation, Bellman-Isaacs equation, fundamen-tal solution, isolated singularity, Liouville theorem, stochastic differential game.
In this article we construct fundamental solutions of the fully nonlinear equation(1.1) F ( D u ) = 0 , and extend Theorem 1 to solutions of (1.1), under the assumption that F is auniformly elliptic and positively homogeneous operator. Here equation (1.1) isunderstood in the viscosity sense (c.f. [9, 7]). We recall that, in general, the bestregularity available for a solution u of equation (1.1) is u ∈ C ,γ loc , for some constant γ > F (see [7, 29, 22]).Before proceeding to the precise statements of our results, let us give some ad-ditional context. An extension of Theorem 1 to some linear equations appearedalready in [4], while a thorough study of fundamental solutions and isolated sin-gularities of linear equations, in view of more modern theories, was performed byGilbarg and Serrin [13]. Later, in a sequence of papers, Serrin [26, 27, 28] produceda deep study of singular solutions of general quasilinear divergence-form equations − div A ( x, u, Du ) = B ( x, u, Du ) , p -harmonic functions being the model case. Werefer to [30, 23] for more developments and references on solutions of quasilinearequations. We also refer to [21] for more on the existence of fundamental solutionsof linear and quasilinear equations.In recent years, there have been a number of studies of singular solutions ofthe fully nonlinear equation (1.1), in the particular case when F is a rotationallyinvariant operator, that is, F ( D u ) depends only on the eigenvalues of D u . Thework most closely related to ours is the one by Labutin [17], who gave, amongother things, a partial extension of Bocher’s theorem to solutions of Pucci extremalequations. Below we discuss in more detail that paper and the additional hypothesesit involved. We also note that in the last several years there has been a great amountof interest of singular solutions of conformally invariant fully nonlinear equations(we refer to [19, 18, 5] and the references in these works).The essence of all results on isolated singularities is that if a function fails to bea solution at an isolated point and is bounded on one side in a neighbourhood ofthis point, then u behaves like a fundamental solution of the elliptic operator nearthe isolated point. In the literature the term fundamental solution usually refersto a solution in R n (or in some domain of interest) except at zero, which goes toinfinity at the origin and is bounded away from it. We are going to use this termalso for solutions in R n \ { } that have the inverse behavior, that is, are boundedon bounded sets and tend to infinity at infinity. For instance, the fundamentalsolution for the p -Laplace equation is Φ p ( x ) = | x | ( p − n ) / ( p − if p = n . In [26, 27]only the case p < n was considered, but as remarked for instance in [14], similarasymptotics as in Theorem 1(i) hold if p > n .Let us now state our main results. We consider an arbitrary Isaacs operator ,that is, a nonlinear map F from the set S n of n -by- n real symmetric matrices into R , with the following two properties.(H1) F is uniformly elliptic and Lipschitz: for some constants 0 < λ ≤ Λ and allreal symmetric matrices M and N , with N nonnegative definite, we have λ trace( N ) ≤ F ( M − N ) − F ( M ) ≤ Λ trace( N ) . (H2) F is positively homogeneous of degree 1: F ( tM ) = tF ( M ) for each t ≥ M. UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 3
We emphasize that (H1) and (H2) will be the only hypotheses on F . In particular,we assume neither that F is convex or concave, nor that F is rotationally invariant.In can be shown that (H1)-(H2) are equivalent to(1.2) F ( D u ) = sup α ∈A inf β ∈B (cid:16) − a α,βij ∂ ij u (cid:17) or F ( D u ) = inf α ∈A sup β ∈B (cid:16) − a α,βij ∂ ij u (cid:17) for index sets A , B and symmetric matrices A α,β = ( a α,βij ), with λI ≤ A α,β ≤ Λ I .In the following theorem we establish the existence and the main properties ofthe fundamental solution. Theorem 3.
There exists a non-constant solution of (1.1) in R n \ { } that isbounded below in B and bounded above in R n \ B . Moreover, the set of all suchsolutions is of the form { a Φ + b | a > , b ∈ R } , where Φ ∈ C ,γ loc ( R n \ { } ) can bechosen to satisfy one of the following homogeneity relations: for all t > x ) = Φ( tx ) + log t or (cid:26) Φ( x ) = t α ∗ Φ( tx ) α ∗ Φ( x ) > in R n \ { } , for some number α ∗ ∈ ( − , ∞ ) \ { } which depends only on F . Definition 1.1.
We call the number α ∗ = α ∗ ( F ) the scaling exponent of F , andwe set α ∗ ( F ) = 0 in the case the first alternative in (1.3) occurs. Definition 1.2.
We call the function Φ whose existence is asserted in Theorem 3,and normalized so that(1.4) min ∂B (sign( α ∗ )Φ) = 1 if α ∗ = 0 , and Z ∂B Φ = 0 if α ∗ = 0 , the upward-pointing fundamental solution of (1.1). Remark 1.3.
By (1.3) and (1.4), the upward-pointing fundamental solution isstrictly decreasing in the radial direction, and we havelim | x |→ Φ( x ) = ∞ if α ∗ ( F ) ≥ , and lim | x |→∞ Φ( x ) = 0 if α ∗ ( F ) > , lim | x |→ Φ( x ) = 0 if α ∗ ( F ) < , and lim | x |→∞ Φ( x ) = −∞ if α ∗ ( F ) ≤ . For any F satisfying (H1)-(H2) we denote the dual operator ˜ F of F by˜ F ( M ) := − F ( − M ) . Notice that the two operators appearing in (1.2) are dual in this sense, as are thePucci extremal operators. By Theorem 3, the operator ˜ F has an upward-pointingfundamental solution which we denote by ˜Φ. It follows that the function − ˜Φ isanother solution of F ( D u ) = 0 in R n \ { } , and we call it the downward-pointingfundamental solution of F .The following result shows Φ and − ˜Φ are the only fundamental solutions of (1.1),and extends Theorem 2 to fully nonlinear operators. Theorem 4.
Suppose that u ∈ C ( R n \ { } ) is a solution of the equation F (cid:0) D u (cid:1) = 0 in R n \ { } . Suppose further that u is bounded from above or below in B \ { } , and that u isbounded from above or below in R n \ B . Then either u ≡ b , or u ≡ a Φ + b , or u ≡ − a ˜Φ + b for some a > , b ∈ R . SCOTT N. ARMSTRONG, BOYAN SIRAKOV, AND CHARLES K. SMART
Prior to this paper, the existence of a fundamental solution of a fully nonlinearequation was known only for certain rotationally invariant operators, when a directcalculation verifies that the function ξ α defined by(1.5) ξ α ( x ) := | x | − α if α > , − log | x | if α = 0 , −| x | − α if α < , satisfies the equation the some α > − P − λ, Λ and P + λ, Λ are ξ λ ( n − / Λ − and ξ Λ( n − /λ − , respectively.It follows from the uniqueness result above that the upward-pointing fundamentalsolution of a rotationally invariant operator must be ξ α for some α > − L , then L Φ is interpretedas the Dirac mass at the origin. For fully nonlinear operators this is not true, asnoticed by Labutin [17] who showed that if λ = Λ, then P + λ, Λ (cid:0) ξ Λ( n − /λ − (cid:1) vanishesnear the origin in a reasonable weak sense.The Liouville-type Theorem 4 is, to our knowledge, the first result of its kindfor fully nonlinear operators, and in particular is new even for the Pucci extremaloperators. Of course, even in the case F ( D u ) = − ∆ u we cannot relax the hy-pothesis that u be bounded on one side in both a neighborhood of the origin and aneighborhood of infinity. This we recall by considering the function u ( x ) = x − x and its Kelvin transform v ( x ) = | x | − n − ( x − x ), both of which are harmonic in R n \ { } . These functions also show that we may not relax the hypotheses on thesolution in our theorems classifying isolated singularities below. Remark 1.4.
Informally, the scaling exponents α ∗ ( F ) and α ∗ ( ˜ F ) characterize theintrinsic internal scalings of the operator F , and we think of each scaling exponentas a kind of principal eigenvalue of a certain elliptic equation on the unit sphere.Indeed, several of the ideas we employ in our proof of Theorem 3 are related to theprincipal eigenvalue theory for fully nonlinear operators developed in [20, 3, 24, 1].As we will see, α ∗ ( F ) is given by α ∗ ( F ) = sup { α ∈ ( − , ∞ ) \ { } : there exists an ( − α )–homogeneoussupersolution of F ( D v ) ≥ αv > R n \ { } (cid:9) , and satisfies − < λ Λ ( n − − ≤ α ∗ ( F ) ≤ Λ λ ( n − − . Of course, if F is a linear operator, then α ∗ ( F ) = n −
2. In Section 4 we discussmore properties of the scaling exponents. Furthermore, in Section 6 we will seethat α ∗ ( F ) has an interesting stochastic interpretation. Corresponding to the Isaacsoperator is a diffusion process controlled by two competing players. The sign α ∗ ( F )indicates whether the first player can force the diffusion to return to the origin, orwhether the second player can force the process out to infinity (almost surely). Inthe case α ∗ ( F ) = 0, the diffusion is recurrent (that is, it returns infinity manytimes to every neighborhood of the origin almost surely) but returns to the originwith zero probability. This generalizes the well-known fact that Brownian motionis recurrent in dimensions n = 1 ,
2, and transient in dimensions n ≥ UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 5
The principal difficulty in the proof of Theorem 3 is establishing the existence of afundamental solution. We first define α ∗ ( F ) according to the formula in Remark 1.4,and show that F satisfies a maximum principle with respect to ( − α )–homogeneousfunctions, for α < α ∗ ( F ). The rest of our argument is quite different for the cases α ∗ ( F ) > α ∗ ( F ) ≤
0. In the first case, we use a construction based oncomparison principle and the Perron method, while in the second case we appeal toan abstract topological fixed point theorem, which helps us to build approximatefundamental solutions. As we will see, this difference is due to the fact that thecomparison principle is reversed on the space of ( − α )-homogeneous functions, for α <
0, while the regular comparison principle holds for α > F ( D u ) = 0 that are bounded on one side in aneighborhood of the singularity. For brevity we introduce the following notation: u ∼ v if u ( x ) v ( x ) → a as x → , u ≈ v if av − C ≤ u ≤ av + C in B / \ { } , for some a, C >
0, and similarly if in these formulas 0 is replaced by ∞ and B / \{ } is replaced by R n \ B . When we write u ∼ v (resp. u ≈ v ; u ∼ ∞ v ; u ≈ ∞ v ), it is tobe understood that u, v → x → u, v → x → ∞ ; u, v −→ | x |→∞ u, v −→ | x |→∞ −∞ ). Theorem 5.
Suppose u ∈ C ( B \ { } ) is a viscosity solution of the equation (1.6) F ( D u ) = 0 in B \ { } . If u is bounded above or below in a neighborhood of the origin, then precisely oneof the following five alternatives holds. (i) the singularity is removable, that is, u can be defined at the origin so that u ∈ C ( B ) and F ( D u ) = 0 in B ; (ii) α ∗ ( F ) ≥ , and u ≈ Φ ; (iii) α ∗ ( ˜ F ) ≥ , and u ≈ − ˜Φ ; (iv) α ∗ ( F ) < , u can be defined at the origin so that ( u ( x ) − u (0)) ∼ Φ( x ) ; (v) α ∗ ( ˜ F ) < , u can be defined at the origin so that ( u ( x ) − u (0)) ∼ − ˜Φ( x ) . This theorem generalizes a result of Labutin [17], who proved (i)-(iii) above underthe supplementary assumptions that F is rotationally invariant and there exist(fundamental) solutions u and v of F ( D u ) = 0 in R n \ { } such that u ( x ) → ∞ and v ( x ) → −∞ as | x | →
0. In light of our results, this latter assumption isequivalent to α ∗ ( F ) ≥ α ∗ ( ˜ F ) ≥
0. Alternatives (iv) and (v) in Theorem 5are new even for the Pucci extremal operators.Our next result is an analogue of Theorem 5 for solutions of F ( D u ) = 0 in R n \ B near infinity. Since we do not have a Kelvin transform available, this is notsimply a corollary of Theorem 5, although the arguments are very similar. Theorem 6.
Suppose u ∈ C ( R n \ B ) is a viscosity solution of the equation (1.7) F ( D u ) = 0 in R n \ ¯ B . If u is bounded above or below in R n \ B , then precisely one of the following fivealternatives holds. SCOTT N. ARMSTRONG, BOYAN SIRAKOV, AND CHARLES K. SMART (i) u ∞ = lim | x |→∞ u ( x ) exists, and min ∂B r u ≤ u ∞ ≤ max ∂B r u , for all r > ; (ii) α ∗ ( F ) > , u ∞ := lim | x |→∞ u ( x ) exists, and ( u ( x ) − u ∞ ) ∼ ∞ Φ( x ) ; (iii) α ∗ ( ˜ F ) > , u ∞ := lim | x |→∞ u ( x ) exists, and ( u ( x ) − u ∞ ) ∼ ∞ − ˜Φ( x ) ; (iv) α ∗ ( F ) ≤ , and u ≈ ∞ Φ ; (v) α ∗ ( ˜ F ) ≤ , and u ≈ ∞ − ˜Φ . We expect the scaling exponents α ∗ ( F ) and α ∗ ( ˜ F ) to govern many propertiesof equations which involve the operator F . In addition to the behavior of thefundamental solutions and isolated singularities of F ( D u ) = 0, we have describedanother such property in Remark 1.4, and we do not doubt many others are to come.For instance, the results and techniques in this paper can be used to generalizeand sharpen several theorems concerning the removability of singularities, criticalexponents and Liouville type results for equations like F ( D u ) ± u p = 0. We referin particular to results obtained by Cutr`ı and Leoni [10, Theorems 3.2 and 4.1],Labutin [16, Theorem 1], Felmer and Quaas [11, Theorem 1.3 and 1.4].This paper is organized as follows. In the next section, we give some preliminarydefinitions and recall some standard results for fully nonlinear equations which weuse in our arguments. In Section 3 we study the scaling number α ∗ ( F ) and constructfundamental solutions of (1.1), establishing the existence part of Theorem 3. Theuniqueness part of Theorem 3 is a consequence of Theorem 4, and is postponedto the end of Section 5. In Section 4 we discuss some examples. We study thesingularities of solutions of (1.1) and prove our main results in Section 5. Finally,in Section 6 we show that the scaling exponent α ∗ ( F ) is related to the behavior ofa certain controlled stochastic process.2. Preliminaries
We begin by introducing some notation. The set of n -by- n real symmetric ma-trices is denoted by S n , and I n is the identity matrix. If M, N ∈ S n , then we write M ≥ N if M − N is nonnegative definite. If x, y ∈ R n , we denote by x ⊗ y thesymmetric matrix with entries ( x i y j + x j y i ). If U is a matrix, then the transposeof U is written U t .For 0 < λ ≤ Λ we define the operators P + λ, Λ ( M ) := sup A ∈ J λ, Λ K [ − trace( AM )] and P − λ, Λ ( M ) := inf A ∈ J λ, Λ K [ − trace( AM )] , for M ∈ S n , and where J λ, Λ K ⊆ S n is the subset of S n consisting of A for which λI n ≤ A ≤ Λ I n . The nonlinear operators P + λ, Λ and P − λ, Λ are called the Puccimaximal and minimal operators , respectively. For ease of notation, we will oftendrop the subscripts and simply write P + and P − . A convenient way to write thePucci extremal operators is(2.1) P + ( M ) = − λ X µ j > µ j − Λ X µ j < µ j and P − ( M ) = − Λ X µ j > µ j − λ X µ j < µ j , where µ , . . . , µ n are the eigenvalues of M .In this article, we require our nonlinear operator F : S n → R to be uniformlyelliptic in the sense that UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 7 (H1) there exist 0 < λ ≤ Λ such that for every
M, N ∈ S n , P − λ, Λ ( M − N ) ≤ F ( M ) − F ( N ) ≤ P + λ, Λ ( M − N ) , and positively homogeneous of order one:(H2) For all M ∈ S n and t ≥ F ( tM ) = tF ( M ).Notice that we have written (H1) in a different but equivalent way from how itappeared in the introduction. We remark that (H1) and (H2) are satisfied for both F = P − and F = P + , and that these hypotheses imply P − ( M ) ≤ F ( M ) ≤ P + ( M ).Every differential equation and differential inequality in this paper is assumed tobe satisfied in the viscosity sense, which is the appropriate notion of weak solutionfor elliptic equations in nondivergence form. For basic definitions as well as a niceintroduction to the theory of viscosity solutions of elliptic equations, we refer to[9] and [7]. The survey [9] is a complete and deep account of the early theory ofviscosity solutions, while the book [7] describes the more recent regularity theory,made possible by the breakthrough article [6].To simplify the reader’s task, we mention some standard results which will beused in this article. In what follows we suppose that Ω is an open subset of R n , theoperator F satisfies (H1), f ∈ C (Ω), and u satisfies the differential inequalities P − ( D u ) ≤ | f | and P + ( D u ) ≥ −| f | in Ω . • Strong maximum principle ([7, Proposition 4.9]). Suppose that u, v ∈ C ( ¯Ω) satisfy F ( D w ) ≤ f ≤ F ( D v ) in Ω and u ≤ v in Ω. If u ( x ) = v ( x )at some point x ∈ Ω, then u ≡ v in Ω. • Harnack inequality ([7, Theorem 4.3]). Suppose in addition u ≥
0. Thenfor each compact subsets K ⊂ K ⊆ Ω, there is a constant C dependingon n, Λ , λ, K , K , Ω , such thatsup K u ≤ C (cid:18) inf K u + k f k L n ( K ) (cid:19) . • Local C ,γ estimates ([7, Theorem 8.3]). For each compact subsets K ⊂ K ⊆ Ω, and each p > n , there exist constants 0 < γ < C dependingon n, p, Λ , λ, K , K , Ω , such that k u k C ,γ ( K ) ≤ C (cid:0) k v k L ∞ ( K ) + k f k L p ( K ) (cid:1) . • Stability under uniform convergence ([7, Proposition 2.9]). Suppose that u k , f k ∈ C (Ω) are such that F ( D u k ) ≤ f k in Ω for each k ≥
1. Alsoassume that u k → u and f k → f locally uniformly in Ω. Then u satisfiesthe inequality F ( D u ) ≤ f in Ω. • Transitivity of inequalities in the viscosity sense ([7, Theorem 5.3]). Sup-pose that
G, H are nonlinear operators satisfying (H1) such that F ( M ) + G ( N ) ≥ H ( M + N ). Suppose also that F ( D u ) ≤ f and v, g ∈ C (Ω)are such that G ( D v ) ≤ g . Then the function w := u + v satisfies H ( D w ) ≤ f + g . • The supremum of a family of subsolutions is a subsolution ([7, Proposition2.7]). Likewise, the infimum of a family of supersolutions is a supersolution.For the rest of this article, we assume that the dimension n of our space is atleast 2. For each α ∈ R we define the radial function ξ α ∈ C ∞ ( R n \ { } ) by (1.5). SCOTT N. ARMSTRONG, BOYAN SIRAKOV, AND CHARLES K. SMART
Notice that we have chosen the signs in the definition of ξ α to ensure continuity inthe following sense(2.2) ξ α − α −→ α ց ξ , ξ α + 1 − α −→ α ր ξ , meant to exploit the fact that if u is a solution of (1.1) then so is au + b , for each a > b ∈ R .For each α ∈ R and all σ >
0, we define the rescaling operator T ασ : C ( R n \{ } ) → C ( R n \ { } ) by(2.3) ( T ασ u )( x ) := (cid:26) σ α u ( σx ) if α = 0 ,u ( σx ) + log( σ ) if α = 0 . Notice that the function ξ α is invariant under the rescaling operator T ασ , that is, T ασ ξ α = ξ α for every σ >
0. For each α ∈ [ − , ∞ ) \ { } , we define the followingspaces of homogeneous functions H α := { v ∈ C ( R n \ { } ) : αv ≥ , T ασ v = v for every σ > } ,H + α := { v ∈ C ( R n \ { } ) : αv > , T ασ v = v for every σ > } , and for α = 0 we set H := H +0 := { v ∈ C ( R n \ { } ) : T σ v = v for every σ > } . We define a special constant α ∗ = α ∗ ( F ) by(2.4) α ∗ ( F ) := sup (cid:8) α ∈ ( − , ∞ ) \ { } : there exists v ∈ H + α such that F ( D v ) ≥ R n \ { } (cid:9) . We call α ∗ ( F ) the scaling exponent of F . In order to estimate α ∗ ( F ), let us calculate D ξ α = ( | α | ( α + 2) | x | − α − x ⊗ x − | α || x | − α − I n if α = 0 , | x | − x ⊗ x − | x | − I n if α = 0 . Observe that for α = 0, the eigenvalues of D ξ α ( x ) are | α | ( α + 1) | x | − α − withmultiplicity one, and −| α || x | − α − with multiplicity n −
1. Similarly, the eigenvaluesof D ξ are | x | − with multiplicity one and −| x | − with multiplicity n −
1. Thusinserting D ξ α ( x ) into the Pucci extremal operators, we discover that(2.5) P − (cid:0) D ξ α (cid:1) = ( | α || x | − α − ( λ ( n − − Λ( α + 1)) if α = 0 , | x | − ( λ ( n − − Λ) if α = 0 , and(2.6) P + (cid:0) D ξ α (cid:1) = ( | α || x | − α − (Λ( n − − λ ( α + 1)) if α = 0 , | x | − (Λ( n − − λ ) if α = 0 . In particular, we see that(2.7) sign (cid:0) P − ( D ξ α ) (cid:1) = sign (cid:18) λ Λ ( n − − − α (cid:19) , (2.8) sign (cid:0) P + ( D ξ α ) (cid:1) = sign (cid:18) Λ λ ( n − − − α (cid:19) . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 9
Since F ≥ P − , it immediately follows from (2.7) and the definition of α ∗ ( F ) that α ∗ ( F ) ≥ λ Λ ( n − − > − . We postpone demonstrating an upper bound for α ∗ ( F ), since for this we need aresult in the next section (see Corollary 3.6).3. Existence of fundamental solutions
In this section we construct fundamental solutions of the equation(3.1) F ( D u ) = 0 in R n \ { } . More precisely, we prove the following result, which represents the existence portionof Theorem 3.
Proposition 3.1.
There exists a solution Φ ∈ H + α ∗ ( F ) of the equation F ( D Φ) = 0 in R n \ { } . The function Φ is unique in the following sense: If α > − and u ∈ H + α is asolution of (3.1) , then α = α ∗ ( F ) and either u ≡ t Φ for some t > , or u ≡ Φ + c for some c ∈ R . To help the reader avoid misunderstandings, we recall that in this paper thespaces H + α contain positive functions if α >
0, and negative functions if α <
Proposition 3.2.
Suppose that α ≥ − and f ∈ H α +2 , u ∈ H α , and v ∈ H + α satisfy the differential inequalities (3.2) F ( D u ) ≤ f ≤ F ( D v ) in R n \ { } . Then (i) if α > , then either u ≤ v or there exists t > such that u ≡ tv ; (ii) if α = 0 , then u − v is constant; (iii) if − < α < and f ≡ , then either u ≥ v or there exists t > such that u ≡ tv .Proof. First we consider the case α >
0. For each s >
1, define the function w s := u − sv . Heuristically, using (H1) we have(3.3) P − ( D w s ) ≤ F ( D u ) − sF ( D v ) ≤ f − sf ≤ R n \ { } , for every s >
1. Using the transitivity of differential inequalities in the viscositysense, we see that w s is a viscosity subsolution of P − ( D w s ) ≤
0. Define(3.4) t := inf { s > w s < R n \ { }} . Our hypotheses imply that 1 ≤ t < ∞ , and w t ≤
0. If t = 1, then we conclude that u ≤ v , and we have nothing left to show.We have left to examine the case 1 < t < ∞ . We must show that w t ≡ w t ( x ) = 0 for some x ∈ R n \ { } . If not, we have − w t > δv on ∂B (and hence on R n \ { } , by thehomogeneity of w t and v ), for some 0 < δ < t −
1. It follows that w t − δ <
0, acontradiction to (3.4). This completes the proof of (i).
Suppose now that α = 0. Define the function w := u − v , which is constant onthe set { tx : t > } for each x ∈ R n \ { } . Moreover, P − ( D w ) ≤ R n \ { } . Set M := max ∂B w = sup R n \{ } w. By the strong maximum principle, w ≡ M . This verifies (ii).Finally, let us prove (iii). If u
0, then by the strong maximum principle u ∈ H + α . Let w s be defined as above, and notice that for s > w s < R n \ { } . Moreover, we have P − ( D w s ) ≤ R n \ { } for all s >
0. Set t := sup { s > w s < R n \ { }} . Then 0 = sup w t = max ∂B w t , and by the strong maximum principle we concludethat w t ≡
0. We have shown that either u ≡ u ≡ tv for some t >
0, from whichthe result follows. (cid:3)
The next lemma establishes that the set of α > − u ∈ H + α of F ( D u ) ≥ R n \ { } is an interval. Lemma 3.3.
Assume that − < α < α ∗ ( F ) . Then there exists a supersolution u ∈ H + α of the inequality (3.5) F ( D u ) ≥ ξ α +2 in R n \ { } . Proof.
Select β > − α < β < α ∗ and for which there exists a superso-lution v ∈ H + β of the inequality(3.6) F ( D v ) ≥ R n \ { } . First we suppose that 0 < α < β . Define τ := β/α > w ( x ) := ( v ( x )) /τ .Notice that w ∈ H + α . Formally, we have F ( D w ) = 1 τ v /τ − F (cid:0) D v − (1 − /τ ) v − Dv ⊗ Dv (cid:1) ≥ λ ( τ − | Dv | v /τ τ v . As v ∈ H + β , by differentiating v ( x ) = t β v ( tx ) with respect to t we get x · Dv = − βv ,hence | Dv | ≥ β | x | − v . Thus we formally estimate(3.7) F ( D w ) ≥ τ − λ ( τ − β v /τ | x | − = λα ( τ − w | x | − ≥ c (min ∂B w ) ξ α +2 . To verify (3.7) in the viscosity sense, we select a smooth test function ϕ and x ∈ R n \ { } such that x w ( x ) − ϕ ( x ) has a local minimum at x = x . We mustdemonstrate that(3.8) F ( D ϕ ( x )) ≥ λα ( τ − w ( x ) . We may suppose without loss of generality that ϕ ( x ) = w ( x ) and ϕ >
0. Let ψ ( x ) := ( ϕ ( x )) τ . Then v ( x ) = ϕ ( x ) and x v ( x ) − ψ ( x ) has a local minimumat x = x . Recalling (3.6), we have F ( D ψ ( x )) ≥ . A routine calculation revealsthat D ψ ( x ) = τ ( ϕ ( x )) τ − D ϕ ( x ) + τ ( τ −
1) ( ϕ ( x )) τ − Dϕ ( x ) ⊗ Dϕ ( x ) . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 11
Thus 0 ≤ F (cid:16) D ϕ ( x ) + ( τ −
1) ( ϕ ( x )) − Dϕ ( x ) ⊗ Dϕ ( x ) (cid:17) ≤ F (cid:0) D ϕ ( x ) (cid:1) + τ − ϕ ( x ) P + ( Dϕ ( x ) ⊗ Dϕ ( x ))= F (cid:0) D ϕ ( x ) (cid:1) − λ ( τ − w ( x ) | Dϕ ( x ) | . Rearrange to write(3.9) F (cid:0) D ϕ ( x ) (cid:1) ≥ λ ( τ − w ( x ) | Dϕ ( x ) | . We will next derive a lower bound for | Dϕ ( x ) | . Owing to the homogeneity of w ,at any point x = 0 we have ∂∂s ( w ( x + sx )) | s =0 = w ( x ) ∂∂s (1 + s ) − α (cid:12)(cid:12) s =0 = − αw ( x ) . Since w − ϕ has a maximum at x , we see that x · Dϕ ( x ) = ∂∂s ( ϕ ( x + sx )) | s =0 ≤ − αw ( x ) . Hence | Dϕ ( x ) | ≥ αw ( x ) | x | . Inserting into (3.9), we obtain (3.8). Recalling that w ∈ H + α , we see that a largemultiple of w satisfies (3.5).In the case α = 0 < β , we define w ( x ) := β − log v ( x ). Then w ∈ H +0 , andformally we see that F ( D w ) = F (cid:18) D vβv − Dv ⊗ Dvβv (cid:19) ≥ λ | Dv | βv ≥ λβ | x | − in R n \ { } . This differential inequality is easily verified in the viscosity sense, as we arguedabove in the proof of (3.7).Similarly, in the case α < β , we define w ( x ) := − exp( αv ( x )). It is easilyverified that w ∈ H + α , and formally we have F ( D w ) = F ( wD v − αwDv ⊗ Dv ) ≥ − λα | w || Dv | ≥ λ | α || w || x | − in R n \ { } . This inequality can also be routinely verified in the viscosity sense, so that somepositive multiple u of w satisfies (3.5). Likewise, if α < < β we can combine thelast two cases to obtain the desired supersolution.Finally, we consider the case that − < α < β <
0. With τ := β/α <
1, wedefine w ( x ) := − ( − v ( x )) /τ . Formally we compute F ( D w ) = τ − ( − v ) /τ − F (cid:0) D v − (1 /τ − − v ) − Dv ⊗ Dv (cid:1) ≥ λ (1 − τ )( − v ) /τ | Dv | τ ( − v ) ≥ λ ( τ − | x | − βw. Since λ ( τ − | x | − βw ≥ c | x | − α − in R n \ { } , we may again argue as above toconclude that a multiple of w satisfies (3.5). (cid:3) From the previous two results we deduce a maximum principle.
Corollary 3.4.
Assume that − ≤ α < α ∗ ( F ) , α = 0 . Suppose that u ∈ H α satisfies the inequality (3.10) F ( D u ) ≤ in R n \ { } . Then u ≡ . If α ∗ ( F ) > , then there does not exist a function u ∈ H +0 satisfyingthe inequality (3.10) .Proof. According to Lemma 3.3, there exists a function v ∈ H + α which satisfies F ( D v ) ≥ ξ α +2 in R n \ { } . If α = 0, then according to Proposition 3.2, either | u | ≤ c | v | for every c >
0, or u ≡ tv for some t >
0. The first alternative implies that u ≡
0. The second alternativeis not possible since u is a subsolution and v is a strict supersolution of F ( D u ) = 0.If α = 0, then we deduce that u − v is constant, which is impossible. (cid:3) Corollary 3.5.
For any < λ ≤ Λ , (3.11) α ∗ (cid:16) P − λ, Λ (cid:17) = λ Λ ( n − − and α ∗ (cid:16) P + λ, Λ (cid:17) = Λ λ ( n − − . Proof.
Recalling (2.7), from the definition of α ∗ we see that α ∗ ( P − ) ≥ λ Λ ( n − − . However, if α ∗ ( P − ) > λ Λ ( n − −
1, then we see that Corollary 3.4 and (2.7)are incompatible. This verifies the first equality in (3.11). The second equality isproved with a similar argument. (cid:3)
Corollary 3.6.
For any operator F which satisfies (H1) and (H2), λ Λ ( n − − ≤ α ∗ ( F ) ≤ Λ λ ( n − − . Proof.
Since P − ≤ F ≤ P + , the result immediately follows from (3.11) and thedefinition of α ∗ ( F ). (cid:3) We now split the proof of Proposition 3.1 into two parts, and consider separatelythe cases α ∗ ( F ) > α ∗ ( F ) ≤ Lemma 3.7.
Suppose α ≥ , f ∈ H + α +2 , and u ∈ H + α satisfy (3.12) F ( D u ) = f in R n \ { } . Then α < α ∗ ( F ) .Proof. Employing the local C ,γ estimates for uniformly elliptic equations, we de-duce that u ∈ C ( R n \ { } ). Set k := sup ∂B | Du | . By the homogeneity of u , wehave(3.13) | Du ( x ) | ≤ k | x | − α − for every x ∈ R n \ { } . First we consider the case α >
0. Let < τ < w ( x ) := ( u ( x )) /τ . Notice that w ∈ H + β for β := α/τ > α . From (3.13) we easily obtain the estimate(3.14) | Dw ( x ) | ≤ C | x | − β − . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 13
We claim that if τ is selected sufficiently close to 1, then w satisfies the inequality(3.15) F ( D w ) ≥ R n \ { } . Take a smooth test function ϕ and a point x = 0 such that ϕ ( x ) = w ( x ), and x w ( x ) − ϕ ( x ) has a local minimum at x = x . Observe that Dw ( x ) = Dϕ ( x ).Set ψ := ϕ τ . The function x u ( x ) − ψ ( x ) has a local minimum at x = x . Thus F (cid:0) D ψ ( x ) (cid:1) ≥ f ( x ) . Following the calculations in the proof of Lemma 3.3, we obtain the estimate f ( x ) τ ( w ( x )) τ − ≤ F (cid:0) D ϕ ( x ) (cid:1) + Λ(1 − τ ) ϕ ( x ) | Dϕ ( x ) | . Rearranging, we have F (cid:0) D ϕ ( x ) (cid:1) ≥ c | x | − α − ( w ( x )) − τ − C (1 − τ ) w ( x ) | x | − β − ≥ c | x | − β − − C (1 − τ ) | x | − β − . Taking 1 − τ > F (cid:0) D ϕ ( x ) (cid:1) ≥
0, which verifiesthat for such τ the function w satisfies (3.15). It now follows from the definition(2.4) of α ∗ ( F ) that α < β ≤ α ∗ ( F ).Next we consider the case α = 0. Define the function v := exp( βu ), where β > v ∈ H + β ∩ C ( R n \ { } ), and if u ∈ C we check that D u = 1 β D vv − β Dv ⊗ Dvv . Formally, for some c > cβv | x | − ≤ F (cid:18) D v − v Dv ⊗ Dv (cid:19) ≤ F ( D v ) + Λ v | Dv | . This calculation can be made rigorous by arguing with smooth test functions, sothat in the viscosity sense we have F ( D v ) ≥ cβv | x | − − Λ v | Dv | . Using | Dv | = β | Du | v and the estimate (3.13), we obtain F ( D v ) ≥ cβv | x | − − Λ k β v | x | − . Thus F ( D v ) ≥ R n \ { } , provided that we select β := c/ Λ k > (cid:3) The next lemma is the key to the proof of Proposition 3.1 in the case α ∗ ( F ) > Lemma 3.8.
Suppose that < α < α ∗ ( F ) and f ∈ H α +2 . Then there exists aunique solution u ∈ H α of the equation (3.16) F ( D u ) = f in R n \ { } . Moreover, if f , then u ∈ H + α .Proof. According to Lemma 3.3, there exists a supersolution w ∈ H + α of F ( D w ) ≥ f in R n \ { } . Let us define u ( x ) := sup { ˜ u ( x ) : ˜ u ∈ C ( R n \ { } ) is a subsolution of (3.16) , and ˜ u ≤ w } . Obviously the zero function is a subsolution of (3.16), so u is well-defined and u ≥ u ∈ C ( R n \ { } ) is a subsolution of (3.16), then so is T ασ ˜ u for any σ >
0, bythe scaling invariance of the equation. Thus u ∈ H α by construction. Standardarguments from viscosity solution theory (see [9]) imply that u is a solution of(3.16). The uniqueness of u follows at once from Proposition 3.2. If f
0, then bythe strong maximum principle u > u ∈ H + α . (cid:3) We are now ready to prove Proposition 3.1 in the case that α ∗ ( F ) > Proposition 3.9.
Suppose that α ∗ ( F ) > . Then there exists a function Φ ∈ H + α ∗ such that F (cid:0) D Φ (cid:1) = 0 in R n \ { } . Moreover, if β > − and u ∈ H + β satisfy F ( D u ) = 0 in R n \ { } , then β = α ∗ ( F ) and u ≡ t Φ for some t > .Proof. For each 0 < α < α ∗ , let u α ∈ H + α denote the unique solution of F ( D u α ) = ξ α +2 in R n \ { } . We claim that(3.17) sup | x | =1 u α ( x ) → + ∞ as α → α ∗ . Suppose on the contrary that there exists a sequence α j → α ∗ such thatsup j ≥ sup | x | =1 u α j ( x ) ≤ C. By the homogeneity of the functions u α , it follows easily thatsup x ∈ K u α j ≤ C for any compact subset K ⊆ R n \ { } . Therefore we have the estimate k u α j k C γ ( K ) ≤ C. By taking a subsequence, if necessary, we may assume that u α j converges locallyuniformly on R n \ { } to a function u ∈ C ( R n \ { } ). It is immediate that u ∈ H + α ∗ and u is a solution of the equation F ( D u ) = ξ α ∗ +2 in R n \ { } . This contradicts Lemma 3.7 and the definition of α ∗ . Therefore (3.17) holds.Define the functions v α by v α ( x ) := c − α u α ( x ) , where c α := sup | x | =1 u α ( x ) . Then v α ∈ H + α . In fact, using homogeneity and the Harnack inequality we have cξ α ≤ v α ≤ ξ α in R n \ { } for some c >
0. Using the homogeneity of F , we see that v α is a solution of F ( D v α ) = c − α ξ α +2 in R n \ { } . For every compact subset K ⊆ R n \ { } , we have the estimate k v α k C γ ( K ) ≤ C. Thus there exists a function Φ ∈ C ( R n \ { } ) such that, up to a subsequence, v α → Φ locally uniformly on R n \ { } . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 15
It immediately follows that Φ ∈ H + α ∗ and cξ α ∗ ≤ Φ ≤ ξ α ∗ . The uniqueness assertions in the last statement in the proposition are immediatelyobtained from Proposition 3.2 and Corollary 3.4. (cid:3)
The proof of the existence of the fundamental solution in the case α ∗ ( F ) ≤ α < α >
0, we expect supersolutions u ∈ H + α of F ( D u ) = 0 tolie below subsolutions. Thus we do not know how to extend Lemma 3.8 to α < α ∗ ( F ) ≤ Proposition 3.10 (Leray-Schauder alternative) . Let X be a real Banach space, K ⊆ X a convex cone, and A : R × K → K be a compact and continuous mappingsuch that A (0 , u ) = 0 for every u ∈ K . Then there exist unbounded, connected sets C + ⊆ [0 , ∞ ) × K and C − ⊆ ( −∞ , × K such that (0 , ∈ C + ∩ C − and A ( λ, u ) = u for every ( λ, u ) ∈ C + ∪ C − . We use the Leray-Schauder alternative to control the norms of approximatefundamental solutions. We apply it to the Banach space X = C ( ∂B ), and theconvex cone K := { u ∈ C ( ∂B ) : u ≤ } . Observe that for each α < H α is isomorphic to K via the map u ( x ) → ˜ u ( x ) := u ( x/ | x | ).The following lemma will provide the map A to which we are going to applyProposition 3.10. Lemma 3.11.
For every − ≤ α, β < and v ∈ H α , there exists a unique function u ∈ H + α that satisfies the equation (3.18) F ( D u + α | x | − ( u − v ) I n ) = | x | − ( βu − αv ) + α | x | − α − in R n \ { } . Moreover, we have the estimate (3.19) max ∂B | u | ≤ C αβ (1 + max ∂B | v | ) , for some constant C = C ( n, Λ) > .Proof. Notice that the zero function is a smooth, strict supersolution of (3.18) since F (cid:0) − α | x | − v ( x ) I n (cid:1) ≥ > − αv | x | − + α | x | − α − for every x ∈ R n \ { } . Consider the function w ( x ) := − C | x | − α , where we select C > w into (3.18), we discover that F (cid:0) D w + α | x | − ( w − v ) I n (cid:1) = F (cid:0) − Cα ( α + 2) | x | − α − ( x ⊗ x ) − ( αv ) | x | − I n (cid:1) ≤ P + (cid:0) − Cα ( α + 2) | x | − α − ( x ⊗ x ) − ( αv ) | x | − I n (cid:1) ≤ n Λ | x | − ( αv ) . Select C := αβ ( n Λ + 1) (cid:18) ∂B | v | (cid:19) , so that n Λ | x | − ( αv ) ≤ ( − Cβ + α ) | x | − − α − α | x | − v = | x | − ( βu − αv ) + α | x | − α − , allowing us to conclude that w is a subsolution of (3.18).Let us define the function u ( x ) := sup { w ( x ) : w ∈ C ( R n \ { } ) is a subsolution of (3.18) and w ≤ } . It is clear that − C | x | − α ≤ u ( x ) ≤ x ∈ R n \ { } , giving us (3.19) for C := ( n Λ + 1). Moreover, we have u ∈ H α due to the scaling invariance of (3.18)and the definition of u — since if w is a subsolution, then so is T ασ w , for all σ > u is a solution of (3.18).Since the zero function is a smooth strict supersolution, it cannot touch u fromabove, as u is viscosity subsolution. Thus u ∈ H + α .To establish the uniqueness of u , we notice that the equation is “proper” withrespect to the space H + α . By this we mean that the function u + c | x | − α is a strictsupersolution of (3.18) for any c >
0, a fact which is easy to check. Let us supposethat ˜ u is another solution of (3.18) such that c := max ∂B (˜ u − u ) >
0. Then bythe strong maximum principle, we must have ˜ u ≡ u + c | x | − α , which is impossiblesince u + c | x | − α is a strict supersolution. The uniqueness of u follows. (cid:3) Using the Leray-Schauder alternative and the solution operator from the previouslemma, we build approximate fundamental solutions.
Lemma 3.12.
For every k > and − < β < , there exists a number α < satisfying (3.20) min { α ∗ , β } < α < cβ, for some constant < c = c ( n, Λ) < / , and a function u ∈ H + α satisfying theequation (3.18) with u = v , that is, (3.21) F ( D u ) = | x | − ( β − α ) u + α | x | − α − in R n \ { } , and for which (3.22) max ∂B | u | = k. Proof.
Recall that K := { w ∈ C ( ∂B ) : u ≤ } . Given ( α, w ) ∈ [ − , × K , let u ∈ H + α be the unique solution of (3.18) for v ( x ) := | x | − α w ( x/ | x | ), and define anoperator A : [ − , × K → K by setting A ( α, ˜ v ) := ˜ u . Let us extend the domainby setting A ( α, ˜ v ) := 0 for all α ≥
0, and A ( α, ˜ v ) := A ( − , ˜ v ) for α < − A : R × K → K. is easily seen to be continuous and compact.We now apply Proposition 3.10 to deduce the existence of an unbounded andconnected set C ⊆ ( −∞ , × K such that (0 , ∈ C and A ( α, ˜ u ) = ˜ u for every ( α, ˜ u ) ∈ C . We claim that(3.23)
C ⊆ (min { β, α ∗ } , × K. Suppose that − < α < β and ˜ u ∈ K such that ( α, ˜ u ) ∈ C . Then we see that thefunction u ( x ) := | x | − α ˜ u ( x/ | x | ) belongs to H + α and satisfies F ( D u ) ≤ R n \ { } . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 17
By Corollary 3.4, we see that α > α ∗ ( F ). Thus C ∩ [ − , min { β, α ∗ } ] × K = ∅ . Since C is connected, we deduce (3.23).Since C is unbounded and connected, we can find ( α, ˜ u ) ∈ C such that u ( x ) := | x | − α ˜ u ( x/ | x | ) satisfies (3.22). Since − < α <
0, it is clear that u also satisfiesequation (3.21). Finally, we notice that (3.19) and (3.22) give us12 C < kC (1 + k ) ≤ αβ , where C = ( n Λ + 1) is the constant in (3.19). Thus if we select c := 1 / C , thenthe second inequality in (3.20) must hold. (cid:3) We now construct fundamental solutions in the case α ∗ ( F ) <
0, using the ap-proximate fundamental solutions from Lemma 3.12.
Proposition 3.13. If α ∗ ( F ) < , then there exists a solution Φ ∈ H + α ∗ of theequation F ( D Φ) = 0 in R n \ { } . Moreover, if β > − and u ∈ H + β satisfies F ( D u ) = 0 in R n \ { } , then β = α ∗ ( F ) and u ≡ t Φ for some t > .Proof. Choose a sequence 1 < k j → ∞ , and use Lemma 3.12 to find numbers α j such that α ∗ < α j < cα ∗ < , and u j ∈ H + α j which satisfy the equation F ( D u j ) = | x | − ( α ∗ − α j ) u j + α j | x | − α j − in R n \ { } , as well as max ∂B | u j | = k j . By taking a subsequence, we may assume that α j → α ′ as j → ∞ for some number α ∗ ≤ α ′ ≤ cα ∗ < w j ( x ) := u j ( x ) /k j . Observe that(3.24) max ∂B | w j | = 1 , and that w j is a solution of the equation F ( D w j ) = | x | − ( α ∗ − α j ) w j + αk − j | x | − α j − in R n \ { } . Since the right-hand side of the expression above is locally uniformly bounded,H¨older estimates imply that for any compact K ⊆ R n \ { } , k w j k C γ ( K ) ≤ C, for some constants C, γ >
0. By passing to a further subsequence we may assumethat w j → Φ locally uniformly in R n \ { } , for some function Φ which necessarilybelongs to H α ′ . It follows that Φ is a solution of the equation F ( D Φ) = | x | − ( α ∗ − α ′ )Φ in R n \ { } . Notice that (3.24) implies that max ∂B | Φ | = 1, so Φ
0. Since ( α ∗ − α ′ )Φ ≥ α ∗ implies α ′ ≤ α ∗ , and thus we deduce α ′ = α ∗ . Therefore Φ is asolution of F ( D Φ) = 0 in R n \ { } . By the strong maximum principle Φ ∈ H + α ∗ . The uniqueness assertions follow fromProposition 3.2, Corollary 3.4 and the definition (2.4) of α ∗ . (cid:3) Our construction of Φ in the case α ∗ ( F ) = 0 is a variation of the above argument.It is complicated somewhat by the need to bend the approximate solutions so thattheir limit lies in the set H +0 . Proposition 3.14. If α ∗ ( F ) = 0 , then there exists a solution Φ ∈ H +0 of theequation F ( D Φ) = 0 in R n \ { } . Moreover, if β > − and u ∈ H + β satisfies F ( D u ) = 0 in R n \ { } , then β = 0 and u ≡ Φ + c for some c ∈ R .Proof. Select a sequence ε j → ε j >
0, and use Lemma 3.12 to find numbers α j < − ε j < α j < − cε j , and functions u j ∈ H + α j which satisfy the equation(3.26) F ( D u j ) = | x | − ( − ε j − α j ) u j + α j | x | − α j − in R n , and for which(3.27) max ∂B | u j | = 1 , so 0 < − u j ≤ | x | − α j . We may improve (3.25) by observing that − ε j − α j ≤ α j , as otherwise we wouldhave F ( D u j ) ≤ R n \ { } , in violation of Corollary 3.4. Thus we have(3.28) − ε j ≤ α j ≤ − cε j , In particular, we have α j → j → ∞ , and by taking a subsequence we mayassume that the quantity(3.29) − α − j ( ε j + α j ) → b for some number 1 ≤ b ≤ (1 − c ) /c .The right-hand side of (3.26) is locally bounded by Cε j . Recalling (3.27), wemay use the Harnack inequality to deduce that | u j + 1 | is locally bounded by Cε j .That is,(3.30) max K | u j | ≤ Cε j , for all compact K ⊆ R n \ { } . Moreover, the H¨older estimates imply that(3.31) k u j k C γ ( K ) ≤ Cε j for any compact subset K of R n \ { } .Define w j := α − j log( − u j ). It is straightforward to check that w j ∈ H +0 . Using(3.25), (3.28), (3.30), and (3.31), it is simple to verify that k w j k C γ ( K ) = α − j k log(1 − (1 + u j )) k C γ ( K ) ≤ C. By taking a further subsequence, we may assume that w j → Φ locally uniformly in R n \ { } as j → ∞ , for a function Φ ∈ H +0 .Formally differentiating, we find D u j = α j u j D w j − α j u j Dw j ⊗ Dw j . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 19
Since F is positively homogeneous and α j u j >
0, a standard viscosity solutionargument yields that w j is a solution of the equation F ( D w j − α j u j Dw j ⊗ Dw j ) = − α − j ( ε j + α j ) | x | − + u − j | x | − α j − in R n \ { } . Passing to the limit j → ∞ in the viscosity sense, and recalling (3.29) and (3.30),we discover that Φ is a solution of the equation F ( D Φ) = ( b − | x | − in R n \ { } As b ≥ α ∗ = 0, we may apply Lemma 3.7 to deduce that b = 1. Theuniqueness assertions follow from Proposition 3.2, Corollary 3.4 and (2.4). (cid:3) Proof of Proposition 3.1.
Proposition 3.1 is immediately obtained from Proposi-tions 3.9, 3.13, and 3.14. (cid:3)
Remark 3.15.
In light of Proposition 3.1 and Corollary 3.4, the scaling exponent α ∗ ( F ) may also be expressed as(3.32) α ∗ ( F ) = min (cid:8) α ∈ ( − , ∞ ) : there exists v ∈ H + α such that F ( D v ) ≤ R n \ { } (cid:9) . Examples and discussion
In this section we discuss several examples.
Example 4.1 (Operators with radial fundamental solutions) . Felmer and Quaas[11] observed that a certain class of operators F have radial fundamental solutions.The key hypothesis is that F is invariant with respect to orthogonal changes ofcoordinates, that is,(4.1) F (cid:0) Q t M Q (cid:1) = F ( M ) for every real orthogonal matrix Q and M ∈ S n . In particular, they noticed that if F satisfies (4.1) and some additional hypotheses,then for some α > − F ( D ξ α ) = 0 in R n \ { } . Of course, that (4.1) suffices for Φ = ξ α ∗ is an immediate consequence of Theorem 4.Let us generalize this observation. Notice that (4.1) is stronger than the condition(4.2) F ( ay ⊗ y − I n ) = F ( az ⊗ z − I n ) for all a ≥ | y | = | z | = 1 . For fixed | y | = 1 and a ≥
1, we see that λ ( n − − Λ( a − ≤ P − ( ay ⊗ y − I n ) ≤ F ( ay ⊗ y − I n ) ≤ P + ( ay ⊗ y − I n ) = Λ( n − − λ ( a − . Since F is continuous, there exists a constant 1 ≤ ˜ a ≤ Λ λ ( n −
1) + 1 such that F (˜ ay ⊗ y − I n ) = 0 . If (4.2) holds, then F (˜ az ⊗ z − I n ) = 0 for every | z | = 1 . It follows that F ( D ξ α ) = 0 in R n \ { } , for α = ˜ a −
2. Thus α ∗ ( F ) = ˜ a − F ) = ξ ˜ a − , and so we see that (4.2)implies that the fundamental solution of F is radial. Example 4.2 (Concave and Convex operators) . Let us review some well-known,elementary facts regarding the fundamental solutions of linear elliptic operators.The Laplacian − ∆ has scaling exponent α ∗ ( − ∆) = n − ξ n − . If L is any linear, uniformly elliptic operator with constant coefficients, givenby Lu = − P i,j a ij u x i x j , then there is a change of coordinates with respect to which L is transformed into − ∆. It follows that α ∗ ( L ) = n − L ) are ellipsoids. Moreover, it is clear that the levelsets of Φ( L ) distinguishes L among all linear operators, up to a positive constantmultiple.From these facts, we will argue that if F is a convex operator which is not themaximum of multiples of the same linear operator, then α ∗ ( F ) > n − . For such an operator F , there exists two linear operators L and L such that L = cL for every c >
0, and F ≥ max { L , L } . Let Φ := Φ( L ) and Φ := Φ( L )be the fundamental solutions for L and L , respectively. Since L and L arenot proportional, we see that Φ and Φ are not proportional, by the Liouvilletheorem. Since F ( D Φ i ) ≥ R n \ { } for i = 1 ,
2, Proposition 3.2 implies that α ∗ ( F ) > n − F is concave and not the minimum of multiplesof one linear operator, then α ∗ ( F ) < n −
2. The underlying idea behind this examplewas previously observed in different contexts in [20, 1, 2].It would be interesting to discover more about the relationship between the twoscaling exponents α ∗ ( F ) and α ∗ ( ˜ F ). In particular, we do not know whether there isan operator F satisfying (H1)-(H2) for which both α ∗ ( F ) and α ∗ ( ˜ F ) are negative.We show now that there is no such operator F which also satisfies (4.2). Proposition 4.3.
Suppose that Φ and ˜Φ are radial functions. Then (4.3) λ Λ ( n − ≤ max { α ∗ ( F ) , α ∗ ( ˜ F ) } and min { α ∗ ( F ) , α ∗ ( ˜ F ) } ≤ Λ λ ( n − . Proof.
For ease of notation let us write α := α ∗ ( F ) and ˜ α := α ∗ ( ˜ F ). Since Φ = ξ α and ˜Φ = ξ ˜ α , we have that F (( α + 2) x ⊗ x − I n ) = F ( − (˜ α + 2) x ⊗ x + I n ) = 0 for all x ∈ ∂B . Let us suppose that max { α, ˜ α } ≤ k := λ ( n − / Λ ≥
0. Select x, y ∈ ∂B with x · y = 0 and observe that by (H1) we have0 = − k + 2 λ ( n − P − (( k + 2) x ⊗ x + ( k + 2) y ⊗ y − I n ) ≤ P − (( α + 2) x ⊗ x + (˜ α + 2) y ⊗ y − I n ) ≤ F (( α + 2) x ⊗ x − I n ) − F ( − (˜ α + 2) y ⊗ y + I n )= 0 . It follows that k = α = ˜ α , and we obtain the first inequality in (4.3). The secondinequality is obtained by a similar argument, using P + in place of P − in the abovecalculation. (cid:3) UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 21
Remark 4.4.
The inequalities in (4.3) are sharp. To see this, consider the operators F ( M ) := − Λ( µ ( M ) + µ n ( M )) − λ n − X i =2 µ i ( M ) ,F ( M ) := − λ ( µ ( M ) + µ n ( M )) − Λ n − X i =2 µ i ( M ) , where µ ( M ) ≤ µ ( M ) ≤ · · · ≤ µ n ( M ) are the eigenvalues of M ∈ S n . It is easy tocheck that F = ˜ F and F = ˜ F , that F and F satisfy (H1)-(H2) as well as (4.1),and α ∗ ( F ) = λ Λ ( n −
2) and α ∗ ( F ) = Λ λ ( n − . Characterization of singularities and a Liouville theorem
In this section we study the behavior near the origin of a solution u ∈ C ( B \ { } )of the equation(5.1) F (cid:0) D u (cid:1) = 0in B \{ } which is bounded on one side, and the behavior near infinity of a solution u ∈ C ( R n \ B ) of (5.1) in R n \ B which is bounded on one side.Throughout this section, we take α ∗ = α ∗ ( F ) and Φ to be the scaling exponentand fundamental solution, respectively, for the operator F obtained in Proposi-tion 3.1, and ˜ α ∗ = α ∗ ( ˜ F ) and ˜Φ be the scaling exponent and upward-pointing funda-mental solution, respectively, for the dual operator ˜ F given by ˜ F ( M ) := − F ( − M ).We make repeated use of monotonicity properties of the quantities(5.2) m ( r ) := min ∂B r u and M ( r ) := max ∂B r u, (5.3) ρ ( r ) := min ∂B r u Φ and ¯ ρ ( r ) := max ∂B r u Φ , defined if Φ does not vanish on ∂B r . Lemma 5.1.
Suppose u is a solution of (5.1) in B \ { } (resp. in R n \ B ).Then there exists a constant C = C ( n, λ, Λ) such that for each r ∈ (0 , / (resp. r ∈ (2 , ∞ ) ) we have M ( r ) ≤ Cm ( r ) and ¯ ρ ( r ) ≤ C ρ ( r ) . Proof.
This is a simple consequence of the Harnack inequality and the fact that ifa function x → u ( x ) is a solution of F (cid:0) D u (cid:1) = 0, then so is x → u ( x/r ). (cid:3) Classification of isolated singularities.
Our proof of Theorem 5, thoughconsiderably more complicated, borrows some ideas from Labutin [17], who provedTheorem 5 for F = P − λ, Λ , in the case that α ∗ ( P − λ, Λ ) ≥
0. The idea is to show thateither the singularity at the origin of a solution u of(5.4) F ( D u ) = 0 in B \ { } is removable, or else is bounded between two multiples of Φ near the origin. Thenwe use the Harnack inequality and the strong maximum principle to squeeze thegap as we blow up the function u at the origin. A similar idea will establish thecorresponding conclusions in the case that α ∗ ( F ) < We divide the proof of Theorem 5 into five lemmas. The first step is the followingresult, which states that a nonnegative solution u of (5.1) must either be boundednear the origin, or α ∗ ( F ) ≥ ρ ( r ) ≥ c > Lemma 5.2.
Assume that u ∈ C ( B \ { } ) is a nonnegative solution of (5.4) .Suppose that either (i) α ∗ ( F ) ≥ and lim inf r → ρ ( r ) = 0 , or (ii) α ∗ ( F ) < . Then u is bounded in B / \ { } .Proof. We first consider the case (i). By adding a constant to Φ in the case that α ∗ ( F ) = 0, we may assume that Φ > B . Let r k → ρ ( r k ) → k → ∞ . Select a point x k ∈ B r k \ { } such that u ( x k ) ≤ (1 /k )Φ( x k ) . According to the Harnack inequality, there exists a constant
C >
0, depending onlyon n and the ellipticity constants Λ and λ , such that u ( x ) ≤ C (1 /k )Φ( x ) for all | x | = | x k | . According to the maximum principle, u ≤ C (1 /k )Φ + M (1 /
2) in B / \ B r k . Passing to the limit k → ∞ , we obtain(5.5) u ≤ M (1 /
2) in B / \ { } . Thus u is bounded above in the punctured ball B / \ { } .We now consider case that α ∗ ( F ) <
0. By Lemma Lemma 5.1 we know that M ( r ) ≤ Cm ( r ) for every 0 < r < /
2. Define a := − max ∂B / Φ > ψ ( x ) := a − (Φ( x ) + a ) . Then ψ ≤ ∂B / , ψ ≤ B / , and ψ > / B r \ { } . It is clear that F ( D ψ ) = 0 in B / \ { } . According tothe maximum principle, u ≥ m ( r ) ψ in B / \ ¯ B r . Hence u ≥ C − M ( r ) ψ in B r \ ¯ B r . Since ψ > Suppose that u ∈ C ( B \ { } ) is a bounded solution of (5.4) . Then u can be defined at the origin so that u ∈ C ( B ) .Proof. We must show that lim x → u ( x ) exists. Define u := lim inf x → u ( x ) . Choose ε > 0. Define v ( x ) := u ( x ) − u + ε , and fix 0 < r = r ( ε ) < / v > B r \ { } . By making r smaller, if necessary, we can find x ∈ ∂B r such that v ( x ) ≤ ε . Let 0 < s < r . Choose x ∈ B s \ { } such that v ( x ) ≤ ε .By the Harnack inequality, there exists a constant C = C (Λ , λ, n ) > v ≤ Cε on ∂B r ∪ ∂B | x | . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 23 By the maximum principle, v ≤ Cε in B r \ B s . We send s → v ≤ Cε in B r \ { } . Thus sup B r \{ } u ≤ u + Cε. It follows that lim sup x → u ( x ) ≤ u . (cid:3) Lemma 5.4. Assume that u ∈ C ( B ) is a solution of (5.4) such that u (0) = 0 .Suppose that either (i) α ∗ ( F ) ≥ , or (ii) α ∗ ( F ) < and lim inf r → ρ ( r ) ≤ . Then u is a subsolution of the equation F ( D u ) = 0 in the whole ball B .Proof. Consider a smooth test function ϕ for which the function u − ϕ has a strictlocal maximum at the origin. We must show that(5.6) F ( D ϕ (0)) ≤ . We may assume without loss of generality that ϕ (0) = 0 = u (0).We may also assume without loss of generality that Dϕ (0) = 0. To see this,define ˜ u ( x ) := u ( x ) − x · Dϕ (0) , and ˜ ϕ ( x ) := ϕ ( x ) − x · Dϕ (0) , and notice that ˜ u − ˜ ϕ has a strict local maximum at the origin, D ˜ ϕ (0) = 0, ˜ ϕ (0) =0 = ˜ u (0), and ˜ u is a solution of (5.4). Moreover, our hypotheses (i) or (ii) hold for˜ u . To get the second condition in case (ii), notice thatlim inf | x |→ ˜ u ( x )Φ( x ) = lim inf | x |→ u ( x ) − x · Dϕ (0)Φ( x ) ≤ lim inf | x |→ u ( x )Φ( x ) + C lim | x |→ | x || x | − α ∗ The last expression on the right vanishes, since α ∗ > − 1. Finally, we remark thatour conclusion holds for ˜ u if and only if it holds for u . Therefore, we may assumethat u = ˜ u and Dϕ (0) = 0.We claim that(5.7) 0 ≤ max ∂B r u for every r > . In the case that (i) holds, we argue just as we did to obtain (5.5) in the proof ofLemma 5.2, by replacing u by u + C , where C is chosen so that u + C is positive in B , and then showing that u + C ≤ max ∂B r u + C in B r . Next, consider the casethat (ii) holds, and suppose on the contrary that max ∂B r u < 0. By multiplying u by a positive constant, we may assume that u ≤ Φ on ∂B r . Since u (0) = Φ(0), the maximum principle implies that u ≤ Φ in B r \ { } . Thiscontradicts the second hypothesis in (ii). We have established (5.7).Owing to (5.7), there exists a unit vector z ∈ ∂B and a sequence { y j } ⊆ B \{ } such that y j → u ( y j ) ≥ z · y j > | y j | j. For ε > 0, we define ψ ε ( x ) := ϕ ( x ) − εz · x. Select r, δ > u ( x ) − ϕ ( x ) ≤ − δ for every | x | = r. For ε > u (0) = ψ ε (0) = 0 and u ( x ) − ψ ε ( x ) ≤ − δ | x | = r. Notice that ψ ε ( y j ) = ϕ ( y j ) − εz · y j ≤ − ε | y j | + o ( | y j | ) as j → ∞ . Thus for j large enough, we have u ( y j ) − ψ ε ( y j ) ≥ − ψ ε ( y j ) > 0, as well as | y j | < r .Let x ε ∈ B r such that u ( x ε ) − ψ ε ( x ε ) = max B r ( u − ψ ε ) . Since x ε = 0, we deduce that F ( D ϕ ( x ε )) = F ( D ψ ε ( x ε )) ≤ . It is clear that x ε → ε → 0. We now pass to limits to obtain (5.6). (cid:3) Lemma 5.5. Suppose that α ∗ ( F ) ≥ , and u ∈ C ( B \ { } ) is a nonnegativesolution of (5.4) . Then (5.8) lim sup r → ρ ( r ) < ∞ . Moreover, if (5.9) a := lim inf r → ρ ( r ) > , then there is a constant C > such that (5.10) a Φ − C ≤ u ( x ) ≤ a Φ + C in B / \ { } . Proof. In the case that α ∗ ( F ) = 0, we may assume that Φ > B \ { } . Themaximum principle implies that for any 0 < r < / u ≥ ρ ( r ) (cid:18) Φ − max ∂B / Φ (cid:19) in B / \ B r . Since max ∂B / Φ > max ∂B / Φ and max ∂B / u < ∞ , we deduce thatsup 0. By adding a positive constant to u ,we may assume that ρ (1 / ≥ a . We claim that for sufficiently small r > ρ ( r ) = min ¯ B / \ B r u Φ . Select 0 < r < / B r \{ } u Φ ≤ 32 ¯ a. Then for 0 < r < r , we have u ≥ ρ ( r )Φ on ∂B / ∪ ∂B r . By the maximumprinciple, u ≥ ρ ( r )Φ on ¯ B / \ B r . Hence (5.12) holds for every r ∈ (0 , r ). We UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 25 deduce that r ρ ( r ) is increasing on the interval (0 , r ), and thus lim r ↓ ρ ( r ) = a .In particular,(5.13) u ≥ a Φ in B / \ { } . For each 0 < r < r , select x r with | x r | = r such that u ( x r ) = ρ ( r )Φ( x r ). Wenow employ a rescaling argument to show that a = ¯ a . That is, we claim that(5.14) lim x → u ( x )Φ( x ) = a. To prove (5.14) we consider the cases α ∗ ( F ) > α ∗ ( F ) = 0 separately.Suppose first that α ∗ ( F ) > 0. For each 0 < r < r and x ∈ B / (2 r ) , we define v r ( x ) := r α ∗ u ( rx ) . Recalling (5.11) and (5.13), for every compact set K ⊆ R n \{ } we have the estimatesup 0. Thuslim sup x → u ( x )Φ( x ) = lim sup r → max x ∈ ∂B r u ( x ) r − α ∗ Φ( x/r ) = lim sup r → max x ∈ ∂B v r ( x )Φ( x ) = a. This verifies (5.14) in the case α ∗ ( F ) > α ∗ ( F ) = 0. For each 0 < r < r , define the function v r ( x ) := u ( rx )Φ( x r ) , x ∈ B / r \ { } . It is clear that v r satisfies the equation F ( D v ) = 0 in B / r \ { } . To get a lowerbound for v r , we notice that v r ( x ) ≥ a Φ( rx )Φ( x r ) = a Φ( x ) − log r Φ( x r /r ) − log r → a locally uniformly as r ↓ . Since v r (cid:16) x r r (cid:17) = ρ ( r ) , the Harnack inequality provides the bound k v r k L ∞ ( K ) ≤ C K for every 0 < r < r ≤ r and compact subset K ⊆ B / (2 r ) \ { } . As before, using the H¨older estimateswe can find a subsequence r j ↓ 0, a point y ∈ ∂B , and a function v ∈ C ( R n \ { } )for which v r j → v locally uniformly in R n \ { } and r − j x r j → y as j → ∞ . We immediately deduce that F ( D v ) = 0 in R n \ { } , as well as v ( y ) = a and v ≥ a in R n \ { } . The strong maximum principle implies v ≡ a . Therefore,lim sup x → u ( x )Φ( x ) = lim sup r → max x ∈ ∂B u ( rx )Φ( rx )= lim sup r → max x ∈ ∂B v r ( x )Φ( x r )Φ( rx )= lim sup r → max x ∈ ∂B v r ( x ) (Φ( x r /r ) − log r )Φ( x ) − log r = a. This completes the proof of (5.14). In particular, ¯ a = a .We have shown above that by adding a constant to u so that ρ (1 / ≥ a = 2 a ,then we deduce that u ≥ a Φ in B / \ { } . By a symmetric argument, we can showthat by subtracting a constant from u so that ¯ ρ (1 / ≤ a/ 2, then u ≤ ¯ a Φ = a Φ in B / \ { } . Therefore (5.10) holds. (cid:3) Lemma 5.6. Assume that α ∗ ( F ) < and u ∈ C ( B ) is a solution of (5.4) suchthat u (0) = 0 and (5.16) a := lim inf r → ρ ( r ) > . Then < a < ∞ , and (5.17) lim x → u ( x )Φ( x ) = a. Proof. Our hypothesis (5.16) implies that there exists 0 < r < ρ ( r ) > < r ≤ r . For such r , since u ≤ ρ ( r )Φ on ∂B r ∪ { } , the maximum principleimplies that u ≤ ρ ( r )Φ on ¯ B r . In particular, ρ ( r ) = min ¯ B r u/ Φ and u < r ρ ( r ) is decreasing in r ∈ (0 , r ) andlim r ↓ ρ ( r ) = a . By a similar argument, we see that the map r ¯ ρ ( r ) satisfies¯ ρ ( r ) = max ¯ B r u/ Φ for all 0 < r < r , and is therefore increasing in r ∈ (0 , r ). Inparticular, a ≤ lim sup r → ¯ ρ ( r ) < ¯ ρ ( r ) < ∞ .For every 0 < r < r , select x r ∈ ∂B r such that u ( x r ) = ρ ( r )Φ( x r ). Define thefunction v r ( x ) := r α ∗ u ( rx ) , < r < r , x ∈ ¯ B / r . By the homogeneity of Φ, for sufficiently small s > ρ ( s )Φ ≤ v r ≤ ρ ( s )Φ in ¯ B s/r , as well as v r ≤ ρ ( r )Φ on ¯ B , and v r ( x r /r ) = ρ ( r )Φ( x r /r ). Using the H¨olderestimates, we can find a subsequence r j ↓ v r j → v locally uniformly in R n \ { } , and x r j /r j → y for some v ∈ C ( R n \ { } ) and y ∈ ∂B . Passing to limits we deducethat v is a solution of the equation F ( D v ) = 0 in R n \ { } . UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 27 According to (5.18), we have v ≤ a Φ in R n \ { } and v ( y ) = a Φ( y ). By the strongmaximum principle, v ≡ a Φ. It follows that the full sequence { v r } r> converges to a Φ locally uniformly as r ↓ 0. Thuslim sup x → u ( x )Φ( x ) = lim sup r → max x ∈ ∂B r u ( x ) r − α ∗ Φ( x/r ) = lim sup r → max ∂B v r Φ = a. The proof is complete. (cid:3) We now combine the previous five lemmas into a proof of Theorem 5. Proof of Theorem 5. Let us assume that u is bounded below in a neighborhood ofthe origin and first consider the case whenlim inf r → ρ ( r ) ≤ . According to Lemmas 5.2, 5.3, and 5.4, we can define u at the origin so that u ∈ C ( B ), and u is a subsolution of (5.1) in the whole ball B . If α ∗ ( ˜ F ) ≥ 0, orif α ∗ ( ˜ F ) < x → ( − u ( x ) + u (0)) / ˜Φ( x ) ≤ 0, then applying Lemma 5.4to − u we see that u is supersolution of (5.1) in the whole ball, and therefore thesingularity is removable, giving us alternative (i). In the case that α ∗ ( ˜ F ) < a := lim inf x → ( − u ( x ) + u (0)) / ˜Φ( x ) > 0, then Lemma 5.6 implies that alternative(v) holds.On the other hand, if a := lim inf r → ρ ( r ) > , then according to Lemmas 5.5 and 5.6, we have a < ∞ , and α ∗ ( F ) ≥ α ∗ ( F ) < u is bounded below. If u is boundedabove, then we repeat our argument with − u in place of u , and ˜ F in place of F . (cid:3) Classification of singularities at infinity. In this subsection we study thebehavior near infinity of a solution u ∈ C ( R n \ B ) of the equation(5.19) F ( D u ) = 0 in R n \ B . Our approach mirrors the proof of Theorem 5 given in the previous subsection. Lemma 5.7. Assume that u ∈ C ( R n \ B ) is a nonpositive solution of equation (5.19) , and that either (i) α ∗ ( F ) > , or (ii) α ∗ ( F ) ≤ and lim inf r →∞ ρ ( r ) = 0 .Then u is bounded in R n \ B .Proof. According to the Harnack inequality and the homogeneity of F , there existsa constant 0 < C < M ( r ) ≤ Cm ( r ) for all r ≥ 2. Thus it suffices toshow that M ( r ) is bounded below.We first consider case (i). Recall we assume that min ∂B Φ = 1. By the maximumprinciple, for every r > u ( x ) ≤ − M ( r )Φ( x ) + M ( r ) for all x ∈ B r \ B . Evaluating this expression at a point | x | = r such that Φ( x ) < / | x | ≥ r , wediscover that m ( r ) ≤ u ( x ) ≤ (1 / M ( r ) for all r > , verifying that M ( r ) is bounded below.We now consider case (ii). By subtracting a positive constant from Φ in the case α = 0, we may assume that Φ < R n \ B . The Harnack inequality implies that ¯ ρ ( r ) ≤ Cρ ( r ) for all r ≥ 2. Thus for any ε > 0, there exists r > ρ ( r ) < ε . By the maximum principle, we have u ( x ) ≥ ε Φ( x ) + m (1) in B r \ B . Let r → ∞ and then ε → u ( x ) ≥ m (1) in R n \ B . (cid:3) Lemma 5.8. Suppose that u ∈ C ( R n \ B ) is a bounded solution of (5.19) . Then lim | x |→∞ u ( x ) exists.Proof. Let u := lim inf | x |→∞ u ( x ). Let ε > 0, and define v ( x ) := u ( x ) − u + ε . Ifwe take r > v > R n \ B r . We can find a point x ∈ R n \ B r such that v ( x ) ≤ ε . For any s > | x | , there is a point x ∈ R n \ B s such that v ( x ) ≤ ε . By the Harnack inequality, there is a constant C = C ( n, λ, Λ) suchthat v ≤ Cε on ∂B | x | ∪ ∂B | x | . By the maximum principle, v ≤ Cε in B s \ B | x | . Letting s → ∞ , we deduce that v ≤ Cε in R n \ B | x | . Therefore, lim sup | x |→∞ v ( x ) ≤ Cε . This implies that lim sup | x |→∞ u ( x ) ≤ u + Cε . (cid:3) Lemma 5.9. Assume that u ∈ C ( R n \ B ) is a bounded solution of (5.19) and lim | x |→∞ u ( x ) = 0 . Suppose that either (i) α ∗ ( F ) > and lim inf r →∞ ρ ( r ) ≤ , or(ii) α ∗ ( F ) ≤ . Then m ( r ) ≤ for all r > .Proof. Suppose on the contrary that (i) holds but m ( r ) > r > 1. Let c > c Φ ≤ m ( r ) on ∂B r . By the maximum principle, for any ε > u ≥ c Φ − ε in R n \ B r . Thus u ≥ c Φ, a contradiction to our assumption that lim inf r →∞ ρ ( r ) ≤ 0. Thiscompletes the proof in case (i). In the case that (ii) holds, we argue as in the lastparagraph in the proof of Lemma 5.7. (cid:3) Lemma 5.10. Suppose that α ∗ ( F ) ≤ and u ∈ C ( R n \ B ) is a nonpositive solutionof (5.19) . Then lim sup r →∞ ρ ( r ) < ∞ . Moreover, if a := lim inf r →∞ ρ ( r ) > , thenthere exists a constant C > such that (5.20) a Φ( x ) ≤ u ( x ) ≤ a Φ + C in R n \ B . Proof. In the case that α ∗ ( F ) = 0, we may assume that Φ < R n \ B . Themaximum principle implies that for any r > u ( x ) ≤ ρ ( r ) (cid:18) Φ − min ∂B Φ (cid:19) in B r \ B . In particular, m ( r ) ≤ ρ ( r ) (cid:0) max ∂B r Φ − min ∂B Φ (cid:1) for all r > 2. Since we havemin ∂B r Φ < min B Φ if r is sufficiently large, we deduce that sup r>r ρ ( r ) < ∞ .The Harnack inequality implies that¯ a := lim sup r →∞ ¯ ρ ( r ) < ∞ . Suppose now that a := lim inf r →∞ ρ ( r ) > 0. By subtracting a positive constantfrom u , we may assume that ρ (2) ≥ a . By the maximum principle, u ≤ ρ ( r )Φ in ¯ B r \ B for all r > ρ ( r ) < a . Sending r → ∞ , we find that u ≤ a Φ in R n \ { } .A scaling argument very similar to the one in the proof of Lemma 5.5 confirms thatlim | x |→∞ u ( x )Φ( x ) = a. In particular, ¯ a = a . By adding a constant to u so that ¯ ρ (2) ≤ a , we find that u ≥ a Φ in R n \ B , by the maximum principle. Thus we have (5.20). (cid:3) Lemma 5.11. Assume that α ∗ ( F ) > and u ∈ C ( R n \ B ) is a solution of (5.19) such that | x |→∞ u ( x ) , and a := lim inf r →∞ ρ ( r ) > . Then a < ∞ and lim r →∞ ¯ ρ ( r ) = a < ∞ .Proof. Notice that ¯ ρ ( r ) is decreasing, since the maximum principle implies that forall ε > u ≤ ¯ ρ ( r )Φ + ε in R n \ B r . Thus a ≤ ¯ a := lim sup r →∞ ¯ ρ ( r ) ≤ ¯ ρ (1). Set v r ( x ) = r α ∗ u ( rx ). By a rescalingargument similar to that in Lemma 5.6, we find that v r → a Φ locally uniformly as r → ∞ . It follows that a = ¯ a . (cid:3) Proof of Theorem 6. The proof is nearly identical to the proof of Theorem 5, withLemmas 5.7, 5.8, 5.9, 5.10, and 5.11 taking the place of Lemmas 5.2, 5.3, 5.4, 5.5,and 5.6, respectively. (cid:3) A Liouville-type result. In this subsection we use Theorems 5 and 6 toprove Theorem 4. Proof of Theorem 4. We proceed by considering each of the alternatives providedby Theorem 5.Case (i): the singularity at the origin is removable. In this case, the function u is a solution of F ( D u ) = 0 in the whole space R n , and u is bounded from aboveor below in R n . It now follows from the Liouville theorem for uniformly ellipticequations that u is constant. (The Liouville theorem is an immediate consequenceof the Harnack inequality, see Remark 4 in Chapter 4 of [7]).Case (ii)(a): α ∗ ( F ) > u ( x ) = a Φ( x ) + O (1) as x → 0. We may assumethat a = 1. We claim that u is bounded below on R n \ { } . Suppose otherwise.Then u is bounded above on R n \ { } and the map r m ( r ) is decreasing and m ( r ) → −∞ as r → ∞ . From the Harnack inequality we deduce that u ( x ) → −∞ as | x | → ∞ . By the maximum principle, it follows that u ≤ 2Φ + k for any k ∈ R .This is obviously a contradiction, as we can let k → −∞ . Thus u is bounded below.By adding a constant to u , we may assume that inf R n \{ } u = 0. Then m ( r ) → r → ∞ , and using the Harnack inequality again we deduce that u ( x ) → | x | → ∞ . The maximum principle immediately yields that (1 − ε )Φ − ε ≤ u ≤ (1 + ε )Φ + ε for every ε > 0. Now we let ε → u ≡ Φ.Case (ii)(b): α ∗ ( F ) = 0 and u ( x ) = Φ( x ) + O (1) as x → 0. We claim that(5.21) ¯ δ := lim sup r →∞ ¯ ρ ( r ) ≥ . Suppose on the contrary that ¯ δ < 1, and select δ > δ < δ < 1. Thenfor every k ∈ R and every s > u ≥ δ Φ + k on ∂B s ∪ ∂B /s . Hence u ≥ δ Φ+ k in B s \ B /s by the maximum principle. Sending s → ∞ and then k → ∞ yields u ≡ + ∞ . This contradiction establishes (5.21). A similar argumentverifies that δ := lim inf r →∞ ρ ( r ) ≤ 1, and the Harnack inequality implies that δ > 0. By inspecting the alternatives in Theorem 6, we see that a Φ − C ≤ u ≤ a Φ + C in R n \ { } , for some a > 0. Since δ ≤ ≤ ¯ δ , we must have a = 1. By adding a constant to u , we may suppose that max ∂B ( u − Φ) = 0. By the maximum principle, for every0 < c < u ≤ Φ + c (Φ − min ∂B Φ) in B \ { } . and u ≤ Φ − c (Φ − max ∂B Φ) in R n \ B . Sending c → u ≤ Φ in R n . Since max ∂B ( u − Φ) = 0, the strong maximumprinciple implies that u ≡ Φ.Case (iii): α ∗ ( ˜ F ) ≥ u ( x ) = − ˜Φ( x ) + O (1) as x → 0. We may repeat ourarguments in case (ii) above, or simply apply them to − u and the dual operator ˜ F ,to deduce that u ≡ − ˜Φ.Case (iv): α ∗ ( F ) < u (0) = 0, and lim x → u ( x ) / Φ( x ) = 1. By the maximumprinciple, for every r > ε > u + ε ≥ ¯ ρ ( r )Φ in B r \ { } . Thus u ≥ ¯ ρ ( r )Φ in B r \ { } for any r > 0. Thus ¯ ρ ( r ) = sup B r \{ } u/ Φ, and so r ¯ ρ ( r ) is increasing. According to our assumption regarding the behavior of u near the origin, we must have ¯ ρ ( r ) ≥ r > 0. A similar argument ensuresthat ρ ( r ) ≤ r > 0, and that r ρ ( r ) is decreasing.In particular, we deduce that u is unbounded from below at infinity, and hencebounded from above in R n . Moreover, it is clear from lim sup r →∞ ¯ ρ ( r ) > ρ ≤ ≤ ¯ ρ we haveΦ − C ≤ u ≤ Φ + C in R n \ B . The monotonicity of ρ and ¯ ρ now immediately imply that ρ ≡ ¯ ρ ≡ 1, and thus u ≡ Φ.Case (v): α ∗ ( ˜ F ) < u (0) = 0 and lim x → − u ( x ) / ˜Φ( x ) = 1. We may apply theresult we have proven in case (iv) to − u and ˜ F to deduce that u ≡ − ˜Φ. (cid:3) Proof of Theorem 3. The theorem is immediately obtained by appealing to Propo-sition 3.1 and Theorem 4. (cid:3) Applications to stochastic differential games In this section we give an interpretation of the scaling exponent α ∗ ( F ) in termsof two-player stochastic differential games. In particular, we generalize the well-known fact that Brownian motion is recurrent in dimension n = 2 and transientin dimensions n ≥ 3. For a review of the connection between viscosity solutions ofsecond-order elliptic and parabolic equations and stochastic differential games, werefer to Fleming and Souganidis [12] and Kovats [15].Let us briefly describe the probabilistic setting (see [15] for more details). Weare given a probability space (Ω , F , P ), a filtration of σ -algebras {F t } t ≥ whichis complete with respect to ( F , P ), and a d -dimensional Weiner process { W t } t ≥ adapted to F t . Also given are compact metric spaces A and B , which are the control sets for Players I and II, respectively, and a function σ : A × B → M n × d .We are interested in a random process { X t } t ≥ governed by the stochastic dif-ferential equation(6.1) ( dX t = σ ( a t , b t ) dW t ,X = x ∈ R n . Here a t and b t are A and B -valued F t -progressively measurable stochastic processes,called the admissible control processes for Players I and II, respectively. The setof admissible control processes for Player I is denoted by M , and for Player II isdenoted by N . Here we do not distinguish between controls { a t } , { ˜ a t } ∈ M forwhich P [ a t = ˜ a t for almost every t ≥ 0] = 1 . An admissible strategy for Player I is a mapping γ : N → M , and similarly an admissible strategy for Player II is a mapping θ : M → N . We denote the set ofadmissible strategies for Players I and II by Γ and Θ, respectively.For each r > 0, we let the random variable τ r = τ x,a,b,r denote the first timethe process X t hits the sphere ∂B r . Similarly, we define τ = τ x,a,b, to be the firsttime the process X t touches the origin, and also denote τ ∞ = ∞ .We first consider a game played in the annulus B R \ B r , for 0 < r < | x | < R ≤ ∞ ,and for which the payoff functional is the map J = J x,r,R : M × N → R given by J x,r,R [ a t , b t ] := P [ τ x,a,b,r < τ x,a,b,R ] . Player I wishes to maximize the payoff and Player II wishes to minimize it. Thus,Player I wishes the process to exit the annulus B R \ ¯ B r on the inner boundary ∂B r while Player II tries to force the process to exit on the outer boundary ∂B R .We define the upper value of the game by v + r,R ( x ) := sup γ ∈ Γ inf b ∈N J x,r,R [ γ ( b ) , b ] , and the lower value of the game by v − r,R ( x ) := inf θ ∈ Θ sup a ∈M J x,r,R [ a, Θ( a )] . For each M ∈ S n , we define the upper Isaacs operator by F + ( M ) := − min a ∈ A max b ∈ B (cid:26) 12 trace( σ ( a, b ) σ ( a, b ) T M ) (cid:27) , and the lower Isaacs operator by F − ( M ) := − max b ∈ B min a ∈ A (cid:26) 12 trace( σ ( a, b ) σ ( a, b ) T M ) (cid:27) . It is clear that F + ( M ) ≤ F − ( M ) for all M ∈ S n . According to [15, Theorem 4.3],we have the following characterization of the upper and lower value functions. Proposition 6.1. Let < r < R < ∞ . The upper value function v + r,R is the uniqueviscosity solution of the boundary-value problem (6.2) F + ( D v ) = 0 in B R \ ¯ B r ,v = 1 on ∂B r ,v = 0 on ∂B R . Similarly, the lower value function v − r,R is the unique viscosity solution of theboundary-value problem (6.3) F − ( D v ) = 0 in B R \ ¯ B r ,v = 1 on ∂B r ,v = 0 on ∂B R . As F + ≤ F − , it is clear from the maximum principle that v − r,R ≤ v + r,R in B R \ B r .We now show that we may use the fundamental solutions of F + and F − to estimatethe value functions. We only treat v + r,R , since v − r,R can be estimated similarly.We let Φ and α = α ∗ ( F + ) > − F + . Denote m ( r ) := min | x | = r Φ( x ) and M ( r ) := max | x | = r Φ( x ) foreach r > 0. Fix two radii 0 < r < R , and notice that, by comparison,(6.4) Φ( x ) − M ( R ) M ( r ) − M ( R ) ≤ v + r,R ( x ) ≤ Φ( x ) − m ( R ) m ( r ) − m ( R ) , r < | x | < R. Let us suppose first that α = 0. From these inequalities we obtain m (1) | x | − α − M (1) R − α M (1)( r − α − R − α ) ≤ v + r,R ( x ) ≤ M (1) | x | − α − m (1) R − α m (1)( r − α − R − α ) , r < | x | < R. Suppose now that α > 0, and fix x ∈ R n \ { } . Then we obtainlim r → log v + r,R ( x )log r = α ∗ ( F ) , and this limit is uniform in R , for large R . That is, for any R > α ∗ ( F + ) = lim r → sup γ ∈ Γ inf b ∈N log P (cid:2) τ x,γ ( b ) ,b,r < τ x,γ ( b ) ,b,R (cid:3) log r . We can let R → ∞ to obtain α ∗ ( F + ) = lim r → sup γ ∈ Γ inf b ∈N log P (cid:2) τ x,γ ( b ) ,b,r < ∞ (cid:3) log r . That is, for small r > γ ∈ Γ inf b ∈N P (cid:2) τ x,γ ( b ) ,b,r < ∞ (cid:3) ∼ r α ∗ . We conclude that if α > 0, then whichever strategy γ ∈ Γ Player I selects, thesecond player can find a control process b ∈ N so that the resulting diffusionprocess is transient . That is, with probability 1, the process { X t } converges toinfinity in the sense that it eventually leaves every bounded set and never returns.To see this, recall that the random process { X t } eventually must leave any ballsince its variance is positive. Furthermore, we see from (6.5) that the process { X t } returns to any given ball infinity often with probability zero.Let us suppose instead that α = α ∗ ( F + ) < 0. Then (6.4) can be rewritten as m (1) r − α − M (1) | x | − α m (1)( r − α − R − α ) ≤ − v + r,R ( x ) ≤ M (1) r − α − m (1) | x | − α M (1)( r − α − R − α ) , r < | x | < R, and we obtain lim R →∞ log (1 − v r,R ( x ))log R = α ∗ ( F ) , UNDAMENTAL SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 33 and this limit is uniform in r > 0. Sending r → R > γ ∈ Γ inf b ∈N P (cid:2) τ x,γ ( b ) ,b, < τ x,γ ( b ) ,b,R (cid:3) ∼ − R α ∗ . We can let R → ∞ to obtainsup γ ∈ Γ inf b ∈N P (cid:2) τ x,γ ( b ) ,b, < ∞ (cid:3) = 1 . That is, Player I can find a strategy which ensures the process { X t } returns to theorigin almost surely. Therefore the process { X t } is recurrent in a very strong sense.The case α = α ∗ ( F + ) = 0 is a compromise between the two cases discussedabove. The estimate (6.4) implies m (1) − M (1) + log R log R − log r ≤ v + r,R ( x ) ≤ M (1) − m (1) + log R log R − log r . From these inequalities, we see that v + r,R ( x ) → R → ∞ , and v + r,R ( x ) → r → 0. Thus, Player I has a strategy γ which ensures that the diffusion { X t } willalmost surely return to every neighborhood of the origin infinitely many times, butPlayer II may select a control process which ensures that the process { X t } nevertouches the origin (almost surely). Since α ∗ ( − ∆) = 0 in dimension n = 2, this ishow Brownian motion behaves in the plane. Acknowledgements. This research was conducted in part while the first authorwas a visitor at Le Centre d’analyse et de math´ematique sociales (CAMS) in Paris. 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Singularities of solutions of second order quasilinear equations ,volume 353 of Pitman Research Notes in Mathematics Series . Longman, Har-low, 1996. Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803. E-mail address : [email protected] UFR SEGMI, Universit´e Paris 10, 92001 Nanterre Cedex, France, and CAMS, EHESS,54 bd Raspail, 75270 Paris Cedex 06, France E-mail address : [email protected] Department of Mathematics, University of California, Berkeley, CA 94720. E-mail address ::