A note on quasilinear equations with fractional diffusion

In this paper, we study the existence of distributional solutions of the following non-local elliptic problem \begin{eqnarray*} \left\lbrace \begin{array}{l} (-\Delta)^{s}u + |\nabla u|^{p} =f \quad\text{ in } \Omega \qquad \qquad \qquad \,\,\, u=0 \,\,\,\,\,\,\,\text{ in } \mathbb{R}^{N}\setminus \Omega, \quad s \in (1/2, 1). \end{array} \right. \end{eqnarray*} We are interested in the relation between the regularity of the source term $f$, and the regularity of the corresponding solution. If $p<2s$, that is the natural growth, we are able to show the existence for all $f\in L^1(\O)$. In the subcritical case, that is, for $p<p_{*}:=N/(N-2s+1)$, we show that solutions are $\mathcal{C}^{1, \alpha}$ for $f \in L^{m}$, with $m$ large enough. In the general case, we achieve the same result under a condition on the size of the source. As an application, we may show that for regular sources, distributional solutions are viscosity solutions, and conversely.


Introduction
Throughout this article, we shall consider the following Dirichlet integro-differential problem (−∆) s u + |∇u| p = f in Ω u = 0 in R N \ Ω, (1.1) mathematics, since Lèvy processes with jumps revealed as more appropriate models of stock pricing. The bounday condition u = 0 in R N \ Ω which is given in the whole complement may be interpreted from the stochastic point of view as the fact that a Lèvy process can exit the domain Ω for the first time jumping to any point in its complement.
Regarding the integro-differential problem that we discuss in the present manuscript, the main results of our research may be summarized as follows • In the sub-critical scenario p < p * := N N −2s+1 , there is a unique non-negative distributional solution u ∈ W 1,q 0 (Ω) of (1.1) for any q < p * . • Moreover, if 1 < p < p * , with similar arguments to those in [2] and [16], we have -If m < N 2s−1 , then |∇u|d 1−s ∈ L q (Ω) for all q < mN N −m(2s−1) . -If m = N 2s−1 , then |∇u|d 1−s ∈ L q (Ω) for all q < ∞. -If m > N 2s−1 , then ∇u ∈ C α (Ω) for some α ∈ (0, 1). In the interval 1 < p < p * the result lies on the estimates for the Green function by Bogdan and Jakubowski in [9].
• For any 1 < p < ∞, u is C 1,α provided the source is sufficiently small. • Any solution u ∈ C 1,α (Ω) with Hölder continuous source is a viscosity solution, and conversely.
Notice that in the local case s = 1, the main existing results can be summarized into two points: If p ≤ 2, then the existence of solution is obtained for all f ∈ L 1 (Ω) using approximation arguments and suitable test function, see [8] and the references therein. However the truncating argument are not applicable for p > 2 including for L ∞ data. In the case of lipschitz data, the author in [28] were able to get the existence and the uniqueness of a regular solution for all p. However this last argument is not applicable for L m data including for p close to two.
In particular, this result implies that the Dirichlet problem (1.1) admits a solution u in W 1,q 0 (Ω) for all q ∈ [1, p * ) and for p < p * . Moreover, for g Hölder continuous and bounded in R, solutions to (1.2) becomes strong for a Hölder continuous source.
The regularity of solutions to (1.1) is strongly related to the corresponding issue for problems As a by-product of the results in [2], [16] and [17], we have the following result which will be largely used throughout our paper. In the case where f ∈ L 1 (Ω) ∩ L m loc (Ω) where m > 1, then as it was proved in [2], the above regularity results hold locally in Ω. More precisely we have Let v the unique solution to problem (1.3). Suppose that m < N 2s−1 , then for any Ω 1 ⊂⊂ Ω and for all p ≤ mN N −m(2s−1) , there exists C := C(Ω, Ω 1 , p) such that (1.5) ||∇v|| L p (Ω1) ≤ C(||f || L 1 (Ω) + ||f || L m (Ω1) ).
As a consequence we conclude that, if f ∈ L m (Ω) with m > 1, then It is clear that a < a 0 = 1 1−s is optimal. To see the optimality of a 0 in this regularity result, we argue by contradiction. Assume that, for 0 f ∈ L ∞ (Ω), there exists a solution v to (1.3) such that v ∈ W 1,p 0 (Ω) with p > 1 1−s . By using the classical Hardy inequality we obtain that By the results in [32] the solution behaves as v ⋍ d s , therefore, as a consequence, 1 d p(1−s) ∈ L 1 (Ω), that is, p < 1 1−s , a contradiction. Hence, the bound for the exponent of the gradient seems to be natural if we impose that the solution lies in the Sobolev space W 1,p 0 (Ω) for the problem with reaction gradient term.
In the case of absorbtion gradient term, this affirmation seems to be difficult to prove, however, in Theorem 2.6, we will show that the non existence result holds, at least, for large value of p and for all bounded non negative data.
In the case of gradient reaction term and for 2s ≤ p < s 1 − s , the authors in [2] proved the existence of a solution u with |∇u| ∈ L p loc (Ω) using a fixed point argument. In the present paper we will use the same approach to get the existence of a solution for p ≥ 2s. However, in addition to the regularity condition of f , smallness condition on the source term ||f || L m (Ω) is also needed.
The paper is organized as follows. In Section 2, we introduce the functional setting and we precise the notion of solution that we will use throughout this work as the weak sense and the viscosity sense. We give also some useful estimates for weak solution and the general comparison principle. A non existence result is proved using suitable estimate on the Green function for the fractional Laplacian with drift term.
The existence of a solution is proved in Section 3. In the Subsection 3.1 we treat the case of natural growth behavior in the gradient term, namely the case p < 2s. In this case existence of a solution is obtained for all L 1 datum. As a complement of the result proved in [16], we prove that if p > p * , the existence of a solution for general measure data ν is not true and additional hypotheses on ν related to a fractional capacity is needed.
Problem with a linear zero order reaction term is also analyzed. In a such case we are able to show existence for data in L 1 and then a breaking of resonance occur under natural hypotheses on the zero order term and p.
Some additional regularity results are obtained in the subcritical case p < p * .
The general case, p ≥ 2s, is treated in Subsection 3.3. Here and since we will use fixed point theorem, we need to impose some additional condition on the regularity and the size of f . The existence result is obtained in a suitable weighted Sobolev space under additional hypotheses on p. The above existence result holds trivially for the case s = 1 and then can be seen as an extension of the existence result obtained in [28] in the framework of L m datum.
The analysis of the viscosity solution is done is Section 4 where it is also proved that weak solution is a viscosity solution and viceversa if the data f is sufficiently regular and s is close to 1.
Some related open problems are given in the last section.
1.1. Basic notation. In what follows, Ω will denote a bounded, open and C 2 domain in R N with bounded boundary, N ≥ 1. We introduce some functional-space notation. By U SC(Ω), LSC(Ω) and C(Ω), we denote the spaces of upper semi-continuous, lower semi continuous and continuous real-valued functions in Ω, respectively. Moreover, the space C k (Ω), k ≥ 1, is defined as the set of functions which derivatives of orders ≤ k are continuous in Ω. Also, the Hölder space C k,α (Ω) is the set of C k (Ω) whose k−th order partial derivatives are locally Hölder continuous with exponent α in Ω.
For any x ∈ Ω, we set δ(x) := dist(x, Ω c ) the distance of the point x to the set Ω c := R N \ Ω. For σ ∈ R, we define the truncation operator as follows T k (σ) := max(−k, min(k, σ)).

Preliminaries and technical tools.
In order to introduce the notion of distributional solutions, we give some definitions. For s ∈ ( 1 2 , 1) and u ∈ S(R N ), the fractional Laplacian (−∆) s is given by For larger class of functions the fractional laplacian can be defined by density. See [19] or [35] for instance.
Definition 2.1. We say that a function φ ∈ C(R N ) belongs to X s (Ω) if and only if the following holds , a. e. in Ω and for all ǫ ∈ (0, ǫ 0 ).
Before staring the sense for which solution are defined, let us recall the definition of the fractional Sobolev space and some of its properties.
Assume that s ∈ (0, 1) and p > 1. Let Ω ⊂ IR N , then the fractional Sobolev Space W s,p (Ω) is defined by Notice that W s,p (Ω) is a Banach Space endowed with the norm The space W s,p 0 (Ω) is defined as the completion of C ∞ 0 (Ω) with respect to the previous norm. If Ω is a bounded regular domain, we can endow W s,p 0 (Ω) with the equivalent norm Notice that if ps < N , then we have the next Sobolev inequality, for all v ∈ C ∞ 0 (IR N ), where p * s = pN N − ps and S ≡ S(N, s, p).
In the following definition, we introduce the class of distributional solutions.
Assume that ν is a bounded Radon measure and consider the problem Let us begin by precising the sense in which solutions are defined for general class of data.
Definition 2.2. We say that u is a weak solution to problem (2.1) if u ∈ L 1 (Ω), and for all φ ∈ X s , we haveΩ where X s is given in Definition 2.1.
For σ ∈ R, we set As a consequence of the properties of the Green function, the authors in [17] obtain the following regularity result.
For ν ∈ L ( Ω), setting T : Related to T k (u) and for s > 1 2 , we have the next regularity result obtained in [2]. Theorem 2.2. Assume that f ∈ L 1 (Ω) and define u to be the unique weak solution to problem We recall also the next comparison principle proved in [2] Recall that we are considering problem (1.1), then we have the next definition.
We denote by G s the Green kernel of (−∆) s in Ω and by G s [·] the associated Green operator defined by See [9] for the estimates of the Green function.
The other class of solutions that we shall consider is the class of viscosity solutions. Unlike the distributional scenario, the notion of viscosity solutions requires the punctual evaluation of the equation using appropriate test functions that touch the solution from above or below.
Definition 2.5. An upper semicontinuous function u : R N → R is a viscosity subsolution to (1.1) in Ω, if u ∈ L loc (R N ), and for any open set U ⊂ Ω, any x 0 ∈ U and any φ ∈ C 2 (U ) such that and v ≤ 0 in R N \Ω. On the other hand, a lower semicontinuous function u : and for any open set U ⊂ Ω, any x 0 ∈ U and any and v ≥ 0 in R N \ Ω. Finally, a viscosity solution to (1.1) is a continuous function which is both a subsolution and a supersolution to (1.1).
To end this section, we prove the next non existence result that justify in some way the condition p < 1 1−s that will be used nextly.
Proof. Suppose by contradiction that problem (1.1) has no solution u with u ∈ W 1,p 0 (Ω). It is clear that u solves the problem the Kato class of function defined by formula (30) in [9]. Thus whereĜ s is the Green function associated to the operator (−∆) s + B(x)∇.. From the result of [9], we know thatĜ s ≃ G s , the Green function associated to the fractional laplacian. Hence Therefore, using the Hardy inequality we deduce that Corollary 2.7. Let f be a Lipschitz function such that f 0, then if p > 1 1−s , problem (1.1) has no solution u such that u ∈ C 1 (Ω) with |∇u| ∈ L p (Ω).
Remark 2.8. It is clear that the above result makes a significative difference with the local case and the general existence result proved in [28] for Lipschitz function. We conjecture that the non existence result holds at least for all p > 1 s−1 as in the case of gradient reaction term.
3.1. The problem with natural growth in the gradient: p < 2s. In this section we consider the case of natural growth in the gradient, namely we will assume that p < 2s. Then using truncating argument, we are able to show the existence of a solution to problem (1.1) for a large class of data. We are also we treat the case where a linear reaction term appears in (1.1).
In the case where p < p * , then for more regular data f , we can show that the solution is in effect a classical solution.
We prove that u is a strong solution. Since the term f −|∇u| p is C ǫ in Ω, and then, by appropriate extension, in Ω, we deduce from [17, Lemma 2.1(ii)] that u ∈ X s . Hence the integration by parts formulaˆΩ for almost everywhere x in Ω. By continuity, it holds in the full set Ω.

Remark 3.2.
Observe that the reasoning employed to prove the above result gives the precise way in which the function f transfers its regularity to a solution u. Indeed, if f ∈ C 2ns+ǫ−n locally in Ω, for ǫ ∈ (0, 2s − 1) and n ≥ 0, then u ∈ C 2(n+1)s+ǫ−n locally in Ω.

3.2.
The case p * ≤ p < 2s with general datum. In this subsection we will assume that p * ≤ p < 2s, then the first existence result for problem (1.1) is the following.
Proof. We follow by approximation. Define u n to be the unique solution to the problem where f n = T n (f ). By the comparison principle in Theorem 2.3, it follows that u n+1 ≤ u n ≤ w for all n where w is the unique solution to problem Hence, there exists u such that u n ↓ u strongly in L σ (Ω) for all σ ≤ N N −2s .
We set g n (|∇u n |) = |∇u n | p 1 + 1 n |∇u n | p , and let k > 0, using T k (u n ) as a test function in (3.1) it follows thatˆˆD Hence {T k (u n )} n is bounded in H s 0 (Ω) for all k and then, up to a subsequence, we have T k (u) ⇀ T k (u) weakly in H s 0 (Ω). We claim that {g n } n is bounded in L 1 (Ω). To see that, we fix ε > 0 and we use v n,ε = un ε+un as a test function in (3.1). It is clear that v n,ε ≤ 1, then taking into consideration that (u n (x) − u n (y))(v n,ε (x) − v n,ε (y)) ≥ 0, it follows thatΩ Letting ε → 0, we reach thatΏ g n (|∇u n |)dx ≤ C an the claim follows. Define h n = f n − g n , then ||h n || L 1 (Ω) ≤ C. As a consequence and by the compactness result in Theorem 2.1, we reach that, up to a subsequence, u n → u strongly in W 1,α 0 (Ω) for all α < N N −2s+1 and then ∇u n → ∇u a.e in Ω. Hence g n → g a.e. in Ω where g(x) = |∇u| p . Since p < 2s, then by Theorem (2.2) and using Vitali Lemma we conclude that T k (u n ) → T k (u) strongly in W 1,p 0 (Ω). Hence to get the existence result we have just to show that g n → g strongly in L 1 (Ω).
Notice that, using T 1 (G k (u n )) as a test function in (3.1) it holds that un≥k+1 g n dx ≤ˆu n ≥k f dx → 0 as k → ∞.
Let ε > 0 and consider E ⊂ Ω to be a measurable set, then By (3.3), letting n → ∞, we can chose |E| small enough such that In the same way and since f n → f strongly in L 1 (Ω), we reach that Hence, for |E| small enough, we have lim sup n→∞ˆE g n dx ≤ ε.
Thus by Vitali lemma we obtain that g n → g strongly in L 1 (Ω). Therefore we conclude that u is a solution to problem (1.1).
Ifû is an other solution to (1.1), then by an induction argument we can show thatû ≤ u n for all n and thenû ≤ u.

Remark 3.3.
(1) The existence of a unique solution to the approximating problem (3.1) holds for all p ≥ 1.
(2) As a consequence of the previous result and following closely the same argument we can prove that for all p < 2s, for all a > 0 and for all (f, g) ∈ L 1 (Ω) × L 1 (Ω) with f, g 0, the problem has a positive solution u .
In the case where the datum f is substituted by a Radon measure ν, existence of solution holds for all p < p * as it was proved in [16]. However, if p > p * , then the situation change completely as in the local case, and, additional hypotheses on ν related to a fractional capacity Cap σ,p are needed, with σ < 1.
The fractional capacity Cap σ,p is defined as follow. For a compact set K ⊂ Ω, we define Notice that, using Sobolev inequality, we obtain that if Cap σ,p (A) = 0 for some set A ⊂⊂ Ω, then |A| = 0. We refer to [38] for the main properties of this capacity. To show that the situation changes for the set of general Radon measure, we prove the next non existence result.
As in [8], let's now show that if ν ∈ W −σ,p (Ω), then ν << Cap σ,p ′ . Notice that, if in addition, ν is nonnegative, then we can prove that and we deduce easily that ν << Cap σ,p ′ . Here we give the proof without the positivity assumption on ν.
Notice that, for all v ∈ W σ,p ′ 0 (Ω), we know that Since (Ω) for all k.
As a direct consequence of the above Theorem we obtain that for p > p * , to get the existence of a solution to problem (1.1) with measure data ν, then necessarily ν is continues respect to the capacity Cap σ,p for all σ ∈ (2s − 1, 2s).
Let consider now the next problem with g 0. As in local case studied in [3], we can show that under natural condition on q and g, the problem (3.7) has a solution for all λ > 0. Moreover, the gradient term |∇u| q produces a strong regularizing effect on the problem and kill any effect of the linear term λgu.
Before stating the main existence result for problem (3.7), let us begin by the next definition.
Let g be a nonnegative measurable function such that g ∈ L 1 (Ω). We say that g is an admissible weight if Hence we are able to state the next result.
Theorem 3.3. Assume that 1 < p < 2s and suppose that g is an admissible weight in the sense given in (3.8).
Proof. Fix λ > 0 and define {u n } n to be a sequence of positive solution to problem To reach the desired result we have just to show that the sequence {g(x) u n 1 + 1 n n u } n is uniformly bounded in L 1 (Ω). To do that, we use T k (u n ) as a test function in (3.9), hence It is clear thatΩ 1 p−1 dt. By a direct computations we obtain that where C 1 , C 2 > 0 are independent of n.
Therefore, going back to (3.10), we conclude that Since p > 1, then by Young inequality we reach that {gu n } n is uniformly bounded in L 1 (Ω). The rest of the proof follows exactly the same compactness arguments as in the proof of Theorem 3.1.  7) has a u such that u ∈ W 1,p 0 (Ω) and T k (u) ∈ W 1,α 0 (Ω) ∩ H s 0 (Ω) for all α < 2s.

3.3.
The case 2s ≤ p < s 1−s : Existence in a weighted Sobolev space. For 2s ≤ p < s 1 − s and in the same way as above we can show the next existence result.
Proof. The proof follows closely the argument used in [2], however, for the reader convenience we include here some details. Without loss of generality we can assume that N ≥ 2 and that N 2s < m < N 2s−1 . Fix λ * > 0 such that if ||f || L m (Ω) ≤ λ * , then there exists l > 0 satisfies C(l + ||f || L m (Ω) ) = l (Ω) and It is clear that E is a closed convex set of W 1,1 0 (Ω). Using Hardy inequality, we deduce that v ∈ E, then |∇v| 2sm d 2sm(1−s) ∈ L 1 (Ω) and Define now the operator in Ω.
Then using the fact that v ∈ E, we reach that |∇v| 2s d β + f ∈ L 1 (Ω). Therefore the existence of u is a consequence of Theorems 2.1 and 1.1. Moreover, |∇u| ∈ L α (Ω) for all α < N N −s . Hence T is well defined. Now following the argument used in [2] and for l defined as above, we can prove that T is continuous, compact operator on E and that T (E) ⊂ E, T is a continuous and compact operator on E. Therefore by the Schauder Fixed Point Theorem, there exists u ∈ E such that T (u) = u, then u ∈ W 1,2s loc (Ω) solves (1.1).
(1) It is clear that the above argument does not take advantage from the fact that the gradient term appears as an absorption term. (2) The existence can be also proved independently of the sign of f . As in Theorem 3.1, if in addition we suppose that f is more regular, then under suitable hypothesis on s and p, we get the following analogous result of Theorem 3.1.

Corollary 3.7. Assume that the conditions of Theorem 3.5 hold. Assume in addition that
Then if f ∈ C ǫ (Ω), for some ǫ ∈ (0, 2s − 1), then the C 1,α distributional solutions from Theorem 3.5 is a strong solution.
Notice that the condition (3.13) is used in order to show that |∇u| p−1 ∈ L σ loc (Ω) for some σ > N 2s−1 which is the key point in order to get the desired regularity. In the case where f 0, we can prove also that u 0, more precisely, we have Corollary 3.8. Assume that the above conditions hold. Let f ∈ C ǫ (Ω) ∩ L ∞ (Ω), for some ǫ ∈ (0, 2s−1). If f (x) ≥ 0 for all x ∈ Ω, then the solution from Theorem 3.5 is non-negative. Moreover, if f 1 ≤ f 2 and u 1 and u 2 are the corresponding strong solutions to f 1 and f 2 from Corollary (3.7), respectively, then u 1 ≤ u 2 .
Proof. Suppose that there is a point x 0 ∈ Ω so that u(x 0 ) < 0. Since u is continuous in R N (see Proposition 1.1 in [32]), we have u attains its negative minimum at an interior point x 1 of Ω. Hence We next prove the last statement in the Corollary. Let f 1 ≤ f 2 . Let u 1 and u 2 be the corresponding strong solutions from Corollary (3.7), and assume that Hence ∇(u 1 − u 2 )(x 0 ) = 0 and (−∆) s (u 2 − u 1 )(x 0 ) < 0, so we have the contradiction

Equivalence between distributional and viscosity solutions
In this section, we investigate the relation between distributional solutions and viscosity solutions. Let us recall that according to Theorem 3.1 and Corollary 3.7, to obtain strong solutions to (1.1) it is sufficient that f ∈ C ǫ (Ω) and that for λ * defined in Theorem 3.5. In this section we show that strong solutions to (1.1) are viscosity solutions. The converse is also true provided a comparison principle for viscosity solutions proved in the next subsection.

A comparison principle for viscosity solutions.
We prove a comparison result for viscosity solutions of problem (1.1). This result requires a continuous source term f .
In order to state the result, we shall need some technical lemmas that could have interest by themselves. For related results see [27].
We start with a usual property for the fractional Laplacian of smooth functions.
In the definition of viscosity solutions do not evaluate the given equation in the solution u. However, the following lemma state an extra information when u is touched from below or above by C 2 -test functions.
Lemma 4.2. Let u be a viscosity supersolution to (1.1) and suppose that there exists φ ∈ C 2 (U ), U ⋐ Ω, touching u from below at x 0 ∈ U . Then (−∆) s u(x 0 ) is finite and moreover: A similar result holds for subsolutions.
Proof. We assume that x 0 = 0 and u(0) = 0. For r > 0 so that B r := B(0, r) ⊂ U , define: Hence for all 0 < ρ < r where we have used that φ touches u from below. As ρ → 0, the last integral converges since φ ∈ C 2 (B r ). Hence Also, from the fact that u is a supersolution, we have u ≥ 0 in R N \ Ω. Thuŝ Since u ∈ LSC(Ω), there is a constant m so that u(y) ≥ m, for all y ∈ Ω \ B r .
We now give the main result of this section. . Assume that f ∈ C(Ω). Let v ∈ U SC(Ω) be a subsolution and u ∈ LSC(Ω) be a supersolution, respectively, of (1.1). Then v ≤ u in Ω.
Proof. We argue by contradiction. Assume that there is x 0 ∈ Ω so that: As usual, we double the variables and consider for ǫ > 0 the function By the upper semi continuity of v and −u, there exist x ǫ and y ǫ in Ω so that By compactness, x ǫ → x and y ǫ → y, up to subsequence that we do not re-label. From and the upper boundedness of v and −u in Ω, we derive As a consequence, by letting ǫ → 0 in (4.4) and using the semicontinuity of u and v, we obtain Also, observe that x ∈ Ω, because otherwise there is a contraction with v ≤ u in R N \ Ω.
Define the C 2 test functions Then φ ǫ touches v from above at x ǫ and ψ ǫ touches u from below at y ǫ . By Lemma 4.2, we have and (−∆) s u(y ǫ ) + |∇ψ ǫ (y ǫ )| p ≥ f (y ǫ ) Therefore: Since f ∈ C(Ω) and ∇ y ψ ǫ (y ǫ ) = −∇ x φ ǫ (x ǫ ), we have that the right hand side in (4.6) tends to 0 as ǫ → 0. Thus, we obtain (4.7) Let A 1,ǫ := {z ∈ R N : x ǫ + z, y ǫ + z ∈ Ω}. Hence for z ∈ A 1,ǫ , we have from the inequality Define A 2,ǫ := R N \ A 1,ǫ . We will justify that we are allowed to use Fatou's Theorem in by showing that the integrand is bounded from below by an L 1 function. Firstly, let r > 0 so that B 3r (x) ⊂ Ω and take ǫ 0 small enough such that x ǫ , y ǫ ∈ B r (x) for all ǫ < ǫ 0 . Take z ∈ A 2,ǫ . We show now that |z| ≥ 2r. Indeed, to reach a contradiction, assume that |z| < 2r. Since z / ∈ A 1,ǫ , it follows that x ǫ + z or y ǫ + z does not belong to Ω. Without loss of generality, assume x ǫ + z / ∈ Ω. Hence |x ǫ + z − x| < 3r, and so x ǫ + z ∈ B 3r (x) ⊂ Ω which is a contradiction. Next, notice that Hence, using that z / ∈ B 2r when z ∈ A 2,ǫ , we havê On the other handˆA Observe that the last integral is finite since v ∈ L 1 loc (R N ) by definition. In this way, recalling (4.10), the term v(x ǫ ) − v(x ǫ + z) |z| N +2s is bounded from below by an L 1 -integrable function. A similar result follows for Hence, we may use Fatou Lemma in (4.9) and derive lim inf (4.11) Here A x := {z ∈ R N : x + z ∈ Ω} and we have used the a. e.-pointwise convergence of χ A2,ǫ to χ Ax , [10, Lemma 4.3] together with a diagonal argument to conclude for a subsequence for a. e z ∈ R N . Moreover, the inequality u ≥ v in R N \ Ω implies that the last integral in (4.11) is non-negative. Then v(x ǫ ) − u(y ǫ ) − v(x ǫ + z) + u(y ǫ + z) |z| N +2s dz ≥ 0.
Therefore by Fatou Lemma, (4.12) and (4.7), we deducê almost everywhere in A 1,ǫ . In particular for z ∈ A x . We then have by the lower semicontinuity of −v and u in Ω and (4.5), that Since z ∈ A x is arbitrary, we conclude σ ≤ v(x) − u(x) for a.e. in Ω, which implies for x ∈ ∂Ω (v(y) − u(y)) ≥ σ.
A contradiction with the hypothesis.

4.2.
Equivalence between strong and viscosity solutions. In this subsection we prove that strong and viscosity solutions coincide. Proof. The proof is straightforward, we give it by completeness. Let u ∈ C 1,α (Ω) be such that (−∆) s u(x) + |∇u(x)| p = f (x), for all x ∈ Ω.
By the assumption on φ, we have that (−∆) s φ(x 0 ) ≤ (−∆) s u(x 0 ) and so the u is a viscosity sub-solution. In a similar way, u is a super-solution and the conclusion follows.
Proof. To prove the converse, assume that u is a viscosity solution to problem (1.1). In view of Theorem 3.5 and Corollary 3.7, there exists a distributional solution v (which is also strong in view of the assumptions on f ). Since any strong solution is of viscosity, we consequently infer from the Comparison Theorem 4.3 that u = v. This ends the proof of the theorem. 5. Some open problems.
(1) For the existence of solution using approximating argument, the limitation p < 2s seems to be technical, we hope that the existence of a solution holds for all p ≤ 2s and for all f ∈ L 1 (Ω). For p > 2s, this is an interesting open question, even for the Laplacian, with L m data. Notice that this is not the framework of the paper [28]. (2) For p > 2s, it seems to be interesting to eliminate the smallness condition ||f || L m (Ω) and to treat more general set of p without the condition (3.13). (3) In order to understand a bigger class of linear integro-differential operators, is seems necessary to obtain alternative techniques independent of the representation formula.