FUNDAMENTAL SOLUTIONS OF A CLASS OF HOMOGENEOUS INTEGRO-DIFFERENTIAL ELLIPTIC EQUATIONS

. In this paper, we study a class of integro-diﬀerential elliptic operators L σ with kernel k ( y ) = a ( y ) / | y | d + σ , where d ≥ 2 ,σ ∈ (0 , 2), and the positive function a ( y ) is homogenous and bounded. By using a purely analytic method, we construct the fundamental solution Φ of L σ if a ( y ) satisﬁes a natural cancellation assumption and | a ( y ) − 1 | is small. Furthermore, we show that the fundamental solution Φ is − α ∗ homogeneous and Lipschitz continuous, where the constant α ∗ ∈ (0 ,d ). A Liouville-type theorem demonstrates that the fundamental solution Φ is the unique nontrivial solution of L σ u = 0 in R d \ { 0 } that is bounded from below.

1. Introduction and main results. The fractional Laplacian arises in many branches of sciences such as phase transitions, particles propagation, stratified materials and others (see [1,15,30]). In particular, it can be understood as the infinitesimal generator of a stable Levy process. This work is devoted to study the following integro-differential equation likes a fractional Laplacian where P.V. stands for the Cauchy principal value. The measurable function a(y) satisfies here 0 < σ < 2 and 0 < λ ≤ Λ. The operator L σ is a non-locally linear operator, corresponding to purely jump processes when diffusion and drift are neglected.
Before proceeding to the precise statements of our results, let us give some additional contexts. We use := to denote a definition.
As is well known, the operator L 0 σ is called the fractional Laplace operator, which has the symbol C(d, σ)|ξ| σ . In that case, u(x) = C/|x| d−σ is a fundamental solution of the equation L 0 σ u = 0 in R d \ {0}, where C depends on d and σ. In recent years, there has been a great amount of interest in nonlocal equations. Cafarelli and Silvestre [11] introduced the extension method which turns nonlocal problems involving the fractional Laplacian into local ones in higher dimensions, then the classical theories for local elliptic partial differential equations can be applied. We refer to [7] for broad applications of this method. Later, Cafarelli and Silvestre [12,13,26] produced a deep study of Hölder estimates of fully nonlinear nonlocal equations in a sequence of papers, and Dong and Kim [18,19] made a study of Schauder and L p estimates of linear nonlocal equations using different methods. We also refer to [17,32] for symmetry of solutions on nonlocal equations.
In this article we want to construct a fundamental solution of the following equation In the literature the term fundamental solution refers to a viscosity solution of (4), which goes to infinity at the origin and is locally bounded in R d \ {0}. It should be noted that, if Φ is a fundamental solution of the fractional Laplacian, then L 0 σ Φ(x) is interpreted as the Dirac mass at the origin.
In addition to the elliptical condition (2), we assume that a(y) is a homogeneous function. Moreover, we make a natural cancellation assumption about the integral kernel.
In particular, we do not assume that a(y) is continuous or symmetric.
There is a number of paper concerned with fundamental solutions of linear and nonlinear equations. For example, some results on linear equations appeared in [5], moreover, Gilbarg and Strrin [22] made a thorough study of fundamental solutions and isolated singularities of linear equations in the view of modern theories. We refer to [29] for more details on fundamental solutions of quasi-linear equations. Fundamental solutions of the extremal Pucci operator were defined by Labutin [24,25] and were used to study the removability of singularities for these operators. Armstrong, Sirakov and Smart [2,3] obtained fundamental solutions for fully nonlinear differential operators which are general, not necessarily radially invariant, and they proved Liouville type theorems for these differential operators. We also refer to [10] more results on singular solutions of the fully nonlinear equations. For nonlocal integro-differential operators, Felmer and Quaas [20,21] generalized the results in [24]. They obtained the existence of fundamental solutions and Liouville-type theorems for a class of Isaacs integral operators with symmetric kernels. In [16], Chen and Zhang obtained the existence of the Gauss kernels for a class of linear parabolic equations involving the nonlocal elliptic operators.
After constructing the fundamental solution of (4), we show a Liouville-type theorem. We refer to [6,14,31] for more details on Liouville-type theorems of fractional Laplacian. The essence of all results on isolated singularity is that, if a function fails to be a solution at an isolated point and it is bounded from below or above in a neighborhood of this point, then it behaves like a fundamental solution of the elliptic operator near this point.
In this work, we assume that the spacial dimension is at least 2. We establish the existence and the main properties of the fundamental solution of (4) in Theorem 1.1. (2) and (5) hold with 1 − δ ≤ λ ≤ Λ ≤ 1 + δ, then there is a nonconstant viscosity solution Φ of (4), which satisfies For all r > 0, the viscosity solution Φ satisfies the following homogeneous relation where the constant α * ∈ (0, d) depends on L σ . 3. There exists a universal constant 0 < c < 1, such that where the constant c depends on d, σ and δ.

The viscosity solution
is also a viscosity solution with (6) holds, then u = aΦ for some a ∈ R.
Remark 1. The function Φ that appears in Theorem 1.1 is said to be the fundamental solution of (4).
We call the constant α * (L σ ) ∈ (0, d) as the scaling exponent of the operator L σ . Informally, α * (L σ ) characterizes the intrinsic internal scaling of the operator L σ , and we think the scaling exponent as a kind of principle eigenvalue of (4) on the unit sphere. As we see in section 3, α * (L σ ) is defined by α * (L σ ) = sup α > 0 : there exists an(−α)homogeneous supersolution of In order to prove Theorem 1.1, we make use of the strategy developed in [2,3], which discussed the existence and characterization of singularity of the fundamental solution to homogenous fully nonlinear equation F (D 2 u) = 0 in R d \{0}. Of course, some new difficulties arise since the integral operator L σ is nonlocal. In our proof, we use a technique based on the comparison principle and the Perron method, which permit us to solve a Dirichlet problem on space of homogeneous functions. We define the scaling number α * (L σ ) in section 3, and show that L σ satisfies a comparable principle with respect to −α homogeneous functions for any 0 < α < α * .
As an application of Theorem 1.1, we are able to characterize the isolated singularity of a viscosity solution Φ of (4), where Φ is positive in R d and locally bounded except the origin. The following result shows that Φ is the only fundamental solution of (4).
Theorem 1.2. Suppose that all the hypotheses of Theorem 1.1 hold, and u ∈ C(R d \ {0}) ∩ L 1 (R d , µ) is a viscosity solution of the equation (4). If u is bounded from below or above in B 1 \ {0} and u is bounded in R d \ B 1 , then either u ≡ c, or u ≡ aΦ + c for some a, c ∈ R. Remark 2. Theorem 1.2 implies that a bounded viscosity solution of (4) must be a constant.
The paper is organized as follows. In next section, we recall some preliminary definitions and some standard results for integro-differential equations that we use later. The Theorem 1.1 is established in section 4 after we show the scaling number α * (L σ ) ∈ (0, d) in section 3. Finally, we discuss the behavior of a viscosity solution of (4) near the origin in section 5.

2.
Preliminaries. In this section, we prepare some important results for later use. We begin by introducing some notations.
(1) The open ball of radius r with center x is denoted by B r (x).
(3) If f is a L 1 function, its Fourier transformation is written as F(f ).
(4) The letter C with or without subscripts will denote a positive finite constant, whose exact value is not important and may change in different places.
Definition 2.1. A function ψ is said to be C 1,1 at the point x, written as ψ ∈ C 1,1 (x), if there is a vector e ∈ R d and a number M > 0 such that for any |y| small enough.
We say a function is C 1,1 in a set Ω if the previous definition holds at every point in Ω with the uniform constant M .
It is easily seen that the value of L σ u(x) is well defined as long as u ∈ C 1,1 (x) ∩ L 1 (R d , µ)(see [11,14]), where µ is a measure In [8,9,12], Caffarelli and Cabré have defined viscosity solutions by testing nondivergence operators in C 1,1 functions. Thus a continuous function u may be a viscosity solution of the integral equation (4). We state the definition.

Definition 2.2.
Suppose Ω is a bounded domain in R d and f ∈ C(Ω). A function u : R d → R, upper(lower) semi-continuous in Ω, is said to be a viscosity sub-(super-solution) to L σ u = f , and we write as L σ u ≤ (≥)f in Ω if all the following happen, (a): x 0 is any point in Ω, and U is a neighborhood of x 0 in Ω, A viscosity solution is a function that is both a sub-solution and a super-solution, which is the appropriate notion of weak solutions for elliptic equations in nondivergence form. Note that every equation and inequality in this paper is assumed to be satisfied in viscosity sense.
An important observation of the equation (4) is the following property.
There have considerable works concerning regularity issues of non-local equations, such as Harnack inequalities, Hölder estimates, and non-local versions of L p estimates. We collect some results on integro-differential equations, which will be used in our work. Lemma 2.3. ( [12]). Let u k be a sequence of functions that are uniformly bounded in R d and continuous in Ω such that if (a): u k → u locally uniformly in Ω, and u k → u a.e. in R d , [4,28]). Suppose that the function u is contin- where C is a constant depending only on d, σ, λ, Λ.
Lemma 2.8. (Comparison principle, [14,25]). Suppose Ω is a bounded open set, and u, v are two bounded functions in R d , which satisfy Next, we give maximum principles which are suitable to our analysis.
Proof. On the contrary, we assume that u(x 0 ) > 0 and x 0 = 0. Then there exists a but which is a contradiction.
Proof. We prove it by contradiction. It follows that which is a contradiction.
3. The boundedness of scaling exponent α * . The following spaces of homogeneous functions will play an important role in the sequel.
Proof. We claim that if α > d − 0 , then for any u ∈ A + α , there exists x 0 ∈ ∂B 1 , such that FUNDAMENTAL SOLUTIONS

4.
Existence and Uniqueness of fundamental solutions. In this section we will construct the fundamental solution of (4). Our proof borrows some ideas from the arguments of Armstrong [2], who proved similar results for fully nonlinear elliptic equations F (D 2 u) = 0 in R d \ {0}. We begin by proving a comparison principle on spaces of homogeneous functions, suitable for our purposes.
Then either u ≡ 0 or v ≡ 0 or u ≡ cv and L σ u = 0 = L σ v for some constant c > 0 in R d \ {0}.
Proof. From Lemma 2.10, we have either v ≡ 0 or v > 0 in R d \{0}, that is, v ∈ A + α . If v ≡ 0, the lemma holds. In the latter case, let Thus w t is strictly negative in R d \ {0} for large t, namely, for t > (max ∂B1 u)/ (min ∂B1 v).
We define Formally, we have We know (26) implies that ether u ≡ cv, which proves the lemma, or w c < 0 in R d \ {0}. In the latter case, if u ≡ 0, the lemma holds. Now we assume w c < 0 and , that contradicts with (25).
The next lemma establishes that the set α > 0 for which there exists a supersolution u ∈ A + α is an interval. Lemma 4.2. For any 0 < α < α * , there is a function u ∈ A + α , such that Proof. We may select a constant β, such that α < β < α * . By the definition of α * , there exists a function v ∈ A + β , satisfying Let τ = α/β < 1, and ω = v τ ∈ A + α . We claim that there exists a positive constant 0 > 0, such that for any Without loss of generality, we may assume that v(x 0 ) = 1. To verify (29) in viscosity sense, we select a small neighborhood of U (x 0 ). For any test function One can easily check that ψ(x) = ϕ 1/τ (x) is a test function to v, and ψ(x 0 ) = ϕ(x 0 ) = 1. By (28) we see that where 0 < θ < 1. It follows that where 0 depends only on d, σ, α, β, λ, and v. Thus we estimate since the homogeneity of ω implies that ω ≥ (min ∂B1 ω)|x| −α . So dividing the ω by a constant we obtain (27).
From the previous two results we deduce a maximum principle in A α .
Proof. According to the previous lemma, there exists a function v ∈ A + α , such that Then we conclude u ≡ 0 from Lemma 4.1.
In next lemma we show that the interval α > 0 for which there exists a super- Then α < α * .
The next lemma is the key to the existence of the fundamental solution.
Step I. Let f (t) = 1/t (α+σ) (t > 0). Consider a family of functions f n : In the view of Lemma 2.7, we can set u n ∈ H σ p (R d )(p > d/σ) is the unique solution to the following equation We know that u n is a continuous function from the embedding By Lemma 2.9, we have From Lemma 2.5, there is a universal constant β > 0, such that for any compact subset K ⊂ Ω ⊂ R d \ {0}. By taking a subsequence if necessary, we may assume that u n converges locally uniformly in (43) and (44).
Step II. We prove the function u ∈ A α , that is, for any r > 0, Given r > 0, we may select a large k, such that 2 −k ≤ r, r σ < 2 k . Let v n r (x) = r α u n (rx)(n > 3k). According to Proposition 1, we have L σ v n r (x) + 2 −n−k v n r (x) ≤ r α+σ L σ u n (rx) + 2 −n r α+σ u n (rx) = r α+σ f n (r|x|) ≤ f n+k (|x|) = L σ u n+k (x) + 2 −n−k u n+k (x). (46) By Lemma 2.8, we see that By a similar argument, we obtain It is easily seen that (45) holds by taking n → ∞ since r is any positive number.
By (43) we see that u n are uniformly bounded and continuous in R d . For any x 0 ∈ ∂B 1 , we have u n , g n → u, g locally uniformly in since the kernel k(y) is bounded in B 1/2 (x 0 ). For any x ∈ B 1/2 (x 0 ), Thus It is easily seen that (40) holds from (52).
Step IV. We claim u ∈ A + α . It follows that either u > 0 or u ≡ 0 from Lemma 2.10. Then u must be positive.
Proof. On the contrary, we assume that there exists a sequence 0 < α n < α * , such that α n → α * , and where M 0 is a universal constant. By the homogeneity of u αn , it follows that By taking a subsequence if necessary, we may assume that u αn → u α * locally uniformly in R d \ {0}. It is immediate that u α * ∈ A α * . We get u α * ∈ L 1 (R d , µ) since α * < d.
Next we claim that u α * ∈ A + α * , and We obtain (57) through a similar argument in the previous lemma (see the proof of (40)). But this contradicts with Lemma 4.3. Then we see (53) holds.
Proof. In the view of Lemma 4.4, we can choose a sequence 0 < α n < α * , α n → α * , and u αn ∈ A + αn which satisfies (54). Define the function v αn as follows v αn (x) : It is clear that v αn ∈ A + αn and max ∂B1 v αn = 1. We have Hence for every compact subset K ⊂⊂ R d \ {0}, we obtain the estimates from Lemma 2.5. Then there exists a function Φ ∈ A α * , such that, up to a subsequence, v αn → Φ locally uniformly in R d \ {0}.
(61) By Lemma 2.3, we conclude that since M αn tend to +∞. It is easily seen that Φ ∈ A + α * and max ∂B1 Φ = 1. Thus min ∂B1 Φ ≥ c is a consequence of the Harnack inequality(see Lemma 2.4), where the constant c depends only on σ, λ, Λ. Now we prove the uniqueness. Assume that u ∈ A + α (α > 0) is a viscosity solution of (4). Then we get α = α * from Corollary 2. So Lemma 4.1 implies that u = CΦ, where C > 0 is a positive constant.
We have concluded the existence of the fundamental solution of (4) in the previous theorem. The following corollary states that the fundamental solution is C 0,1 except the origin.
On the other hand, setting x 0 ∈ ∂B 1 , for any x ∈ B 1/2 (x 0 ), let where where , , By (65) and where the constant C 1 does not depend on e, h.
Proof of Theorem 1.1. Our results is immediately concluded from Corollary 2, Theorem 4.6 and Corollary 3.

5.
Characterization of singularities. Throughout this section, assuming that u is a viscosity solution of (4), we study its behavior near the origin. Here we always since Φ does not vanish on ∂B r . We divide the proof of Theorem 1.2 into several lemmas. The first two lemmas state ρ(r) ≈ρ(r).
Then there exists a universal constant C depending only on λ, Λ and d, such that for each r > 0, where C is a constant depending only on d, λ, Λ and σ.
Proof. (72) is a simple consequence of the Harnack inequality and the fact that if a function u(x) is a solution of (4), then so is u(x/r). Proof. It is easily seen thatρ(r) ≥ ρ(r) ≥ 0. On the contrary, we may assume that lim sup r→0ρ (r) = 1.
Note that adding a constant to Φ modifies neither the hypotheses nor the conclusion of the lemma. Without loss of generality, we may suppose that We may choose ρ(r k ) → 0 as r k → 0. For every r k , select a x k ∈ ∂B r k such that According to the Harnack inequality, there exists a universal constant C > 0, depending only on d, σ, λ, Λ, such that for any > 0, as long as r k is sufficiently small. By making r k smaller, if necessary, we obtain Let M := max 5r k ≤r≤1ρ (r). Then we have 1 − < M < +∞ from (73) and (74). (76) and (77), we have by taking small enough. But this contradicts with L σ u = 0.
The next result tells us that u/Φ is bounded near the origin.
Proof. On the contrary, we suppose that lim sup Denotingū we obtain that But (81) contradicts with that M is sufficiently large.
In the following three lemmas, we prove that a nonnegative viscosity solution of (4) must be a constant under the condition of that ρ(r) converges to zero at the origin.
Proof. We have lim r→0ρ (r) = lim r→0 ρ(r) = 0 from Lemma 5.2. Choosing a sequence of r k such that ρ(2r k ) → 0, and Letū where Ω k = B 1 \ B 2r k . From the comparison principle, we obtain Passing to the limit k → ∞, we derive that Thus u is bounded.
Then u can be defined at origin so that u ∈ C(R d ). Moreover, u is a viscosity solution of Proof. Suppose u > 0 (if necessary, adding a constant). On the contrary, we assume that lim inf On the other hand, since L σ u(x) = 0 for all x = 0, then for any r > 0, we obtain by passing to the limit r → 0. Then L σ u ∈ L 1 (B 1 ). We deduce that According to Lemma 2.5, there is a constant α > 0, such that for any > 0, where C is a universal constant. But this contradicts with (87). Therefore, we conclude that u is continuous by redefining u(0) = lim |x|→0 u(x).
Proof. Let v(x) = u(rx)(r > 0). By Lemma 2.5, we have where β, C are universal positive constants. Then we have [u] C β (B1) = 0 by making r → 0 + . Hence we deduce that u must be a constant.
The next result says that u/Φ converges to a constant in the case of lim inf Proof. Denoting a = lim inf r→0 ρ(r) andā = lim inf r→0ρ (r), it follows that 0 < a ≤ā < +∞ from Lemma 5.3. We claim that In fact, for any > 0, we can select r 0 ∈ (0, 1/4) and R 0 > 1 such that since Φ(x) → 0 as |x| → +∞. We deduce that u + a > (a − )Φ in B R0 \ B r0 from the comparison principle. Then (93) holds by taking → 0. We now employ a rescaling argument to show that a =ā. That is, we want to show lim Note that the limit a do not change when we add a constant to u. So we may assume that aΦ ≤ u ≤ 2āΦ + 1 in R d \ {0} (96) since u is bounded in R d \ B 1 . Let r 0 < 1. For each 0 < r < r 0 , select x r with |x r | = r such that u(x r ) = ρ(r)Φ(x r ). Let v r (x) = r α * u(rx) 0 < r < r 0 .
By the homogeneity of Φ, we conclude that Using the Hölder estimate (13), we can find a function v ∈ C(R d \ {0}) and a sequence r j → 0 such that v rj → v locally uniformly in R d \ {0} as j → ∞.
By taking a further subsequence, we may also assume that x rj /r j → y as j → ∞ for some y ∈ ∂B 1 . We have v ≥ aΦ in R d \ {0} since Φ is homogeneous. It is clear that v is a viscosity solution of (4) from Lemma 2.3. Since v(y) = lim j→∞ r α * j u(x rj ) = lim j→∞ r α * j ρ(r j )Φ(x rj ) = lim j→∞ ρ(r j )Φ(x rj /r j ) = aΦ(y), we obtain v ≡ aΦ by Lemma 2.10. For any r j+1 < r < r j , Thus the full sequence v r → aΦ locally uniformly in R d \ {0} as r → 0. From this we deduce that lim sup Proof. We obtain that lim inf r→0 ρ(r) = a = lim sup r→0ρ (r) from the last lemma. The comparison principle implies that for every small > 0, where B is a constant. Now we let → 0 to deduce that Since u − aΦ is a bounded viscosity solution of (4), we conclude that b = u − aΦ is a constant by Theorem 5.6.
Proof of Theorem 1.2. We may assume u(or−u) is positive in R d \{0} (if necessary, adding a constant) since u is bounded on one side. Thus Theorem 1.2 is immediately obtained from Theorem 5.6 and 5.8.